A Much Bigger Piece Of Pi

Punk_Rock_Johnny points to an AP story on Pi-obsessed Professor Yasumasa Kanada. A snippet from the story: "Kanada and a team of researchers set a new world record by calculating the value of pi to 1.24 trillion places, project team member Makoto Kudo said yesterday. The previous record, set by Kanada in 1999, was 206.158 billion places." Trillion! "

Comments

One simple question

[Tafs]

Why?

Re:One simple question

[Nyh]

Because pi is there. And they still have found only the tiniest fraction of the total of decimals of pi...

Nyh!

Re:One simple question

[Tafs]

Well, I could accept it if was possible to find all the decimals, and be done with it. But that can't be done.

Re:One simple question

[Dexter's Laboratory]

No matter how many decimals they calculate, that "tiny fraction" will always be just as tiny as always...

Re:One simple question

[badansible]

Damn, if I could only compute e^(i*pi) with a trillion digits precision...

Re:One simple question

[MoonBuggy]

I'm not entirely up on complex math, but they want to know if it has a reccurring pattern.

Just like 1/3 makes 0.3333etc. which reccurs after 1 digit, 1/7 makes 0.142857142857 which reccurs after 6 digits. Pi could reccur after, say, 1.5 trillion digits. I don't know why that would mean anything, but I'm sure it would be a big discovery ;-)

Re:One simple question

[Servants]

Nope. Any number with a recurring pattern is automatically rational, and pi is not.

This isn't too hard to see. For example, if pi repeated after 1.5 trillion digits, we could write its value (where [1.5tril] represents all those repeating digits:

pi = 3.14159[1.5tril]..14159[1.5tril]..14159[..]

Then multiply this number by 10^(1.5 trillion).

10^(1.5trillion) * pi = 314159[1.5tril].14159[..]

such that the repeating part starting with .14159 still follows the decimal point.

Then subtract the top equation from the bottom one, so the repeating part gets subtracted away.

(10^(1.5 trillion) - 1) * pi = 314159[1.5tril]

Then just divide both sides by (10^(1.5 trillion) - 1) and you've written pi as a ratio of two integers.

Re:One simple question

[SpaceRook]

Why?

Well, if you read the article, you would know why. Mapping out a very large number like that is useful for testing the accuracy of supercomputers. Also, the research process spins off lots of discoveries. Someone who mapped out pi to 1.24 trillion decimal places probably learned a couple neat tricks along the way.

Re:One simple question

[SavingPrivateNawak]

Mmh... funny...

It's true that there are so many uses of "-1" (You know "-1 Offtopic", "-1 Troll", etc...), that getting a ten trillion digits for it would be a great breakthrough!!

Re:One simple question

[CableModemSniper]

I think thats the whole point. They haven't found the repeating digits yet and they are hoping to.

Re:One simple question

[jrexilius]

ow! ow! stop.. you hurt my brain.. haven't had the morning coffee yet. put a disclaimer on these posts moderators ;-)

Re:One simple question

[Anonymous Coward]

They don't need to do so, as one can prove that Pi is irrational in advance.

Re:One simple question

[sco08y]

put a disclaimer on these posts moderators ;-)

I don't need a 20 page proof to tell me the moderators are irrational.

Re:One simple question

[Deadstick]

Zero percent, to be precise. And thus it will stay.

rj

Re:One simple question

[zome]

Because he had crazy question in his head and dare to find the answer.

Much like Newton did when he had crazy question in this head (why apple falls on my head) .

Or when Einstein asked hinself why the speed of light doesn't change no matter how fast the observer goes.

When they found the answers, I don't think they saw immediate uses of their finding, but in the future, who knows

Re:One simple question

[spaceyhackerlady]

I'm not entirely up on complex math, but they want to know if it has a reccurring pattern.

No.

Pi was proven to be irrational long ago. Pi was proven to be transcendental long ago too.

So not only can it not be expressed as any ratio of integers a/b, it cannot be the solution of any poynomial equation with a finite number of integer coefficients.

They aren't looking for repetition, because it has long since been proven that there is no repetition to be found.

...laura

Re:One simple question

[MrSpock]

But to do that, you'd need trillions of digits of precision for both e and pi!

Re:One simple question

[Servants]

What I meant was, if pi had repeating digits, then you'd be able to show that it was rational. But it isn't rational, therefore it can't have repeating digits without creating a contradiction.

Although nobody's looking for exact repeating patterns, I think some people are on the lookout for digits and patterns that occur with greater probability than chance would predict, like if 12% of the digits of pi seemed to be "6" instead of the expected 10%... dunno if they've had any luck though.

Re:One simple question

[martyn s]

They want to find a message from God. Read Contact, Carl Sagan.

Re:One simple question

[Directrix1]

How about no percent. In order to calculate a percentage you have to have a value to compare with. Now if you take the limit as total digits approach infinity then it will be zero percent. I see no point in calculating pi to this many places. I don't even find it news worthy. Uses for a trillion places of precision are far and few between, what the hell is wrong with this guy, and why is everybody so fucking obsessed with pi. e is so much more useful.

Re:One simple question

[Directrix1]

I concur. Doesn't this seem about as relevant as calculating a trillion digits of one over infinity (the most boring number on earth).

Re:One simple question

[Directrix1]

You know Newton didn't question why the apple fell on his head. He did what every other scientist does: he made observations and he derived formulas to predict within reasonable accuracy the behavior he is observing. Same with Einstein. Physics doesn't explain the universe people, it only predicts behavior within certain systems of it.

Re:One simple question

[parallactic]

Pi has already been proven to be an irrational and transcendental number. A proof of its irrationality (which implies non-repetitiveness):

http://dspace.dial.pipex.com/town/way/po28/maths /d ocs/pi.html

All this proves is that Kanada has a bigger penis. Seriously. Clearly we can't use this value in computations, and the value of pi to, say, 40 places is enough to calculate the circumference of the observable universe to the precision of a proton.

Re:One simple question

[MillionthMonkey]

They will not find simple repetition. That would mean that pi = p/q for two ints p and q and we know for a fact that pi is not only irrational (there is no p and q) but that it's transcendental (not the root of any polynomial with integer coefficients).

However, that doesn't mean they won't find a pattern. It isn't known whether pi is a normal number. A normal number is an irrational number where each digit 0-9 occurs 1/10 of the time, each pair of digits 00-99 occurs 1/100 of the time, etc. Pi is believed to be a normal number because it looks like one, but nobody has proven it.

A normal number may or may not occur in a predictable sequence. For example, this is the Champernowne constant:

0.12345678910111213141516171819202122232425.....

This is irrational and transendental and still there's an obvious pattern.

This is the Copeland-Erdos constant, which is like the Champernowne constant except you only use primes:

0.235711131719232931374143475359616771737983...

The Thue constant is an example of an irrational, transcendental number that is not normal. The nth digit is 1 if n is not divisible by 3 and is the complement of the (n/3)-th bit if n is divisible by 3. This is what it looks like in base 2:

0.110110111110110111110110110110110111110110...
(and in base 10: 0.85909979685470310490357250...)

Try writing that as a fraction.

It's possible that on the way to the trillionth digit of pi they might find that something weird happens, like there's no digits except 0 and 9 after a certain point, but I doubt it.

Re:One simple question

[Pig Hogger]

What it does, actually, is when you compute pi in base 12, at around two trillion decimal places, only ones and zeroes appear, and by putting those ones and zeroes in a 2-D array, you end up with the picture of a circle.

Re:One simple question

[Old Wolf]

Not necessarily. Suppose that some sort of pattern is found after 10 trillion places, such that one could claim that all places are then known (in the same way that you can claim all the places of 1/3 are known). This search for patterns is surely the driving force of pi computation.

Re:One simple question

[Old Wolf]

Well, yes it can

Consider:
0.110100100100001000010000010000001.. .
This has repeating digits (in some sense of the expression, anyway) but is irrational (perhaps someone could remind me of the proof). Finding this sort of pattern in pi would be groundbreaking for number theory, and is not ruled out by results such as 'pi is irrational'.

Re:One simple question

[coloth]

This search for patterns is surely the driving force of pi computation.

Maybe a little bit, but if a pattern started after 10 trillion digits, like "1234123412341234...", the search would then begin for the *end* of the pattern. That end *must* exist, by definition of "irrational number", then the pattern search would start again.

'course humans are pattern-seeking machines.

I personally liked the end of Contact, where a bitmap of a circle is found somewhere deep in pi.

Re:One simple question

[suricatta]

I'll be nice and let you in on the joke :)

Euler's formula is e^(i*x) = cos(x) + i*sin(x)

If you let x = pi, then you get

e^(i*pi) = cos(pi) + i*sin(pi)
e^(i*pi) = -1 + i*0
e^(i*pi) = -1 ...so you don't REALLY need a trillion places for that calculation, unless you're slightly desperate.

Re:One simple question

[elveu]

thus proving that god exists, yet since god refused to prove his existence that in turn proves god does not exist

Re:One simple question

[Jace of Fuse!]

personally liked the end of Contact, where a bitmap of a circle is found somewhere deep in pi.

I loved that little tid-bit. Correct me if I'm wrong, but that was one of the many things left out of the movie, no?

I don't seem to recall it having been mentioned.

Re:One simple question

[Dexter's Laboratory]

Yes it wasnt it the movie, sadly. The closest they got, was the star like circle of sand grains in her hand. And in the novel she didn't go alone, but I'm guessing the movie focused even more on Ellie and her father.

Re:One simple question

[Dexter's Laboratory]

But the thing is, I have written a few programs to calculate pi, perhaps I'm thinking how cool it would be if *I* could find a pattern... ofcourse, that's not likely, considering real mathematicians hasn't, yet. I know it's unnecessary to calculate as many digits of pi as possible, but somehow I still think they should. I don't know why. It's a bit irrational :)

Re:One simple question

[Directrix1]

Unless we plan on doing large scale measurements of the volume of the universe in some infintesimally small unit, I don't see this being very useful even if we did find a pattern. I wonder how one goes about proving that a number is irrational.

Re:One simple question

[coloth]

I don't seem to recall it having been mentioned.

Sad, but true.

They probably figured that it fell too far on the techie side of the "cosmic message," where they were looking for the human interest side.

For me, though, that circle just blew my mind even more.

The funny thing is, that circle is there! (along with mpeg4 renditions of every movie ever made, plus a disturbing one of Genghis Khan and Janet Reno making out)

We'll just never calculate enough digits of pi to reach any of it.

Re:One simple question

[matrix29]

Why?

Uh, why not?

The thing is either human beings have to come to grips with the concept of infinitely long numbers or become uncaring about them.

It mirrors the issue of finding a wonderously marvelous thing and then trying to find more functional useage for it. Imagine discovering a magic dildo that changes all wombats to umbrellas. Surely it would be of strictly limited productive useage, however, the act of exploring HOW IT ACHIVES ITS FUNCTION would yield far more value than anything that could be gained by using the device itself.

As with most learning it is the journey that is more rewarding than the destination though the completion of the journey has a reward as well.

And again... Why Not?
It is a far better way to waste one's time than completing a crossword puzzle though less useful than making a method for converting salt water to fresh using the minimal amount of energy with the minimal amount of time.

Re:One simple question

[packeteer]

Pi cannot be solved with in a sense that would make anyone with only a high school education happy. When you figure out Pi its not actually the real number your going after. You are trying to measure an nth sided figure. If you found the measurements of a 10^10th sided figure it would be close but its still not a REAL circle. You must measure an infinity sided figure to get the REAL number.

Re:One simple question

[DavMac]

But you could never *prove* that the pattern you had found would continue forever... unlike 1/3 where it's quite easy to prove that all digits will be "3". (Similar to another poster, who mentioned calculating the decimal digits of 1/infinity - they would, of course, all be nought). And if you really want a reason for calculating digits of pi, I recommend reading "Contact" (a novel) by Carl Sagan. I won't give a spoiler, but essentially: What if we found some pattern in the digits which had some meaning in the context of pi itself? or a message of some sort? ("Contact" does also raise the point of trying not decimal digits, but other bases).

Re:One simple question

[SpryGuy]

... or you just want to be REALLY, REALLY accurate :-)

"Dammit, Jim! I've calculated it out to a trillion decimal places, and I'm POSITIVE the answer is EXACTLY -1!"

"Keep going, Bones! We have to be Absolutely SURE!"

Re:One simple question

[ShavenYak]

Imagine discovering a magic dildo that changes all wombats to umbrellas. Surely it would be of strictly limited productive useage, however, the act of exploring HOW IT ACHIVES ITS FUNCTION would yield far more value than anything that could be gained by using the device itself.

Get real. Everyone knows wombats are far too prudish to use dildos. On the other hand, I can't count the number of times I've been stranded in a rainstorm without an umbrella, but had easy access to a wombat. I think more research is called for.

Re:One simple question

[Directrix1]

Why did you reply to my comment. This does not seem to be directed at it.

Re:One simple question

[bplipschitz]

-- ... or you just want to be REALLY, REALLY accurate :-)

"Dammit, Jim! I've calculated it out to a trillion decimal places, and I'm POSITIVE the answer is EXACTLY -1!"

"Keep going, Bones! We have to be Absolutely SURE!"
--

"I'm a *Doctor*, damnit, NOT a MATHMATICIAN!"

Well ... what is it?

[LoudMusic]

How about we see this bad boy!? I'd sure like to paste it into my "info.txt" file for future referance. It could come in handy sometime.

Re:Well ... what is it?

[WesG]

So if I did my math correctly:
1240000000000 characters * 8 bits/character = 9920000000000 bytes

9920000000000 bytes/ 1024000 = 9687500 MB

9687500 MB = 9.6875 TB

Thats a pretty darn big info.txt file!

I think I'll just use the 32 byte version for my SIG. :-)

3.14

Re:Well ... what is it?

[mfos.org]

Actually, since this is not text data, but numbers, you don't need to waste a whole byte to store a number, if my calculations are correct (probably aren't, hey its early) you only need 514 billion bytes

Re:Well ... what is it?

[mfos.org]

Here's the magic

You have a 1.24 trillion digit base ten number

10^1.24e12

Now we find out how many digits long it'll be in base 2, x

10^1.24e12 = 2^x
x = ln(10^1.24e12)/ln(2)

x = 1.24e12 * ln(10)/ln(2) = 4119190837660.6

Now divide by 8 to get bytes, and viola!

515e9

Re:Well ... what is it?

[isorox]

Why waste all that space? You can write any number from 0 to 255 in 1 byte, any number from 0 to 65355 in 2 bytes, and 0 to 4.2billion in 4 bytes.

Anyone care to do the math about how much space you need (uncompressed)? (I cant cause I'm dumb)

P.S. Does Pi compress very well?

Re:Well ... what is it?

[WesG]

Well this is a info.txt file afterall :-)

Last time I checked text files use 8 bits per character.

3.14

Re:Well ... what is it?

[EvanED]

Others in a different branch of this thread calculate it to be 515 GB.

Re:Well ... what is it?

[Hater's Leaving, The]

What the heck is your calculation supposed to calculate?

1.24T characters = 1.24TB
at 1 character per byte.

Simple as that.

The data _could_ be represented as 4.12Tb = 515GB if it were converted to binary.
However, you _really_ don't want to do radix conversion on numbers that large if you have the chance of avoiding it.

If you wanted to store it in packed decimal instead:
2 digits/byte (PBCD) : 620GB
(4 digits/short likewise)
9 digits/word : 551GB
19 digits/64bits : 522GB

So you can get within 1% of the minimum size (515GB) simply by packing the digits into 64bit words in chunks of 19 digits.

(256bit chunks could hold 77 digits, and compress the size down to 515.3GB, which is .1% wastage)

THL.

Re:Well ... what is it?

[Breakfast Pants]

the data for pi would be very hard to compress (unless you simply recalculated it on the fly, but thats not really compression)

Re:Well ... what is it?

[kasperd]

However, you _really_ don't want to do radix conversion on numbers that large if you have the chance of avoiding it.

But if there are too many digits to convert the base what is really the purpose of computing the number in the first place? It is not like there is anything magical about the base 10. It is surely not the easiest base to compute the number in, and it is surely neither the the most usefull base to have the number in for future use.

Re:Well ... what is it?

[pyrros]

Well, your numbers are quite off:

1240000000000 characters * 8 bits/character = 9920000000000 bytes
No, that would be 9920000000000 bits. So, all your nubmers below that are off by a factor of 8.

9920000000000 bytes/ 1024000 = 9687500 MB
wtf? either use 1048576 to be exact or 1000000 for convenience.

Finally, using a whole byte for a single decimal digit is very very inefficient. A byte can hold 256 values (0-255) while a decimal digit can hold only 10 (0-9).

You could save a lot of space by using the full range of values that a bit can hold. (We can treat the nubmer as an integer since we know beforehand where the decimal point exists)

So the correct amount would be:

1240000000000 characters * log(10) (base 2)/character = 514898854708 bytes (rounding UP to the nearest integer)

So, you'd need just a little less than 515GB

Re:Well ... what is it?

[spac]

Actually, to store a number this large it will have to be stored in IEEE 754 format.

In IEEE 754 a string of either 32 bits is divided into a sign bit, 8 or more bits for the exponent, and 23 or more bits for the mantissa.

You'll probably need a little more than 4119190837661 bits to tell you the truth.

Then again, who cares.

Only on slashdot....

My Blog: http://gozman.org

Re:Well ... what is it?

[Ryan Amos]

He was calculating the size of a text file containing pi, and for that, he was correct. Actually, by its nature, pi should not compress very well, if at all. The numbers in pi have no pattern (I forget the proof, but there is one) so most compression algorithms wouldn't be able to do much with it.

Re:Well ... what is it?

[machine of god]

how many library of congress's is that?

In soviet russia, pi stores you.

Re:Well ... what is it?

[Hater's Leaving, The]

To be honest I don't know what base he used for the calculations. However, storage space is cheaper than CPU time, and so now he's got it in base 10 it makes sense to just store that base 10 version. If he has it in binary too, that again, he should dump that too.

And then it's easier to always access the version that's more appropriate.

People like looking for '1234567890' and stuff like that in Pi, so it does make sense to have a base 10 version. Assuming anything to do with it makes "sense"!

THL.

Re:Well ... what is it?

[SWPadnos]

With a few extra bits, you can make it easy to extract any digit of the number.

Split the 1.24TB number into triplets, and store each triplet in 10 bits:

(leaving off the initial 3):
141 592 653 ...
encode into 30 bits:
0010001101 1001010000 1010001101 ...

Pack these together (and on and on), and you use only ~517 GB, but you can index to any digit by a simple division, shift, and decimal conversion.

(assuming you decided to put together a RAID array for the storage in the first place :)

Re:Well ... what is it?

[timeOday]

an ASCII representation of PI would compress quite nicely, because each character it represented with 8 bits yet has only 10 possible values.

Re:Well ... what is it?

[kasperd]

People like looking for '1234567890' and stuff like that in Pi

Stunning, does the sequence 1234567890 actually appear in the digits of Pi? And does it even appear as frequent as any other 10 digit combination? ;-)

Hey, they ought to do that with the binary representation. I bet it won't take long time to find the sequence 10 somewhere in the binary representation of Pi. That would save them a lot of work.

Re:Well ... what is it?

[scotch]

You only need 1 digit in base 1.24_trillion*!!! I can store it in one byte (for sufficiently large bytes).

(*) give or take. probably mostly take.

Re:Well ... what is it?

[Scarblac]

You only need 1 digit in base 1.24_trillion!!!

No. That's like saying you only need one digit in base 5 to store 43232, which has five digits...

1.24 trillion digits is really amazingly large. But you could make do with 1 digit in base 10^1.24 trillion - how to store this digit while distinguishing it from all the other possible digits is left as an exercise for the reader :-).

Re:Well ... what is it?

[*xpenguin*]

Results

The string 1234567890 did not occur in the first 100000000 digits of pi after position 0.


Pi-Search page

Re:Well ... what is it?

[Jerf]

Pi, like everything else, compresses down to one bit, given the correct decompression algorithm. (It is generally nonsense to talk about how well something compresses without specifying something about the algorithm you mean to use.)

Usually, "X compresses down to one bit for a correct algorithm" is a snarky answer, but in this case, it actually makes sense. Generally one has to define those algorithms as a table, where "X" is what the decompression function returns for "1", which definately feels like cheating. In this case, though, one can provide a finite algorithm to compute as many digits of pi as you please, so it makes sense.

In fact, we compress pi down to one or two bytes, with a mathematically defined decompression sequence you can use if you want, all the time. In fact, I've done it three times in this post already, where two different two byte sequences stood in for the infinite series that is that number. Can you find them?

Re:Well ... what is it?

[schmink182]

Statistically speaking, there was about a 10% chance of finding that string in the first 100,000,000 digits of pi, assuming pi is effectively random, since there are just under 1,000,000,000 10-digit strings in it. But go on another few trillion places, it's in there somewhere.

Re:Well ... what is it?

[bergeron76]

I wonder if running a few compression algorithms against said file could help in determining the meaning of pi or a way to calculate it to the nth position.

Or would that unlock a new dimenions or something?

Re:Well ... what is it?

[efflux]

so most compression algorithms wouldn't be able to do much with it.

Except of course for Packed BCD. Not so much a compression algorithm as representation, but this would effectively cut the storage size in half.

Re:Well ... what is it?

[CableModemSniper]

well... $ calc pi\(\) > pi.txt $ ls -sh pi.txt 4.0K pi.txt $ bzip2 pi.txt $ ls -sh pi.txt.bz2 4.0K pi.txt.bz2 Obviously, thats an issue with the filesystem more than anything else....so... $ wc -c pi.txt 24 pi.txt $ wc -c pi.txt.bz2 57 pi.txt.bz2 apprently pi has the mathematical property of getting larger when compressed :)u

Re:Well ... what is it?

[de Selby]

You can find the nth digit of Pi in hexadecimal, but not decimal.

See: http://www.maa.org/mathland/mathtrek_3_2_98.html

Re:Well ... what is it?

[CableModemSniper]

Doh! Shoulda previewed! well... $ calc pi\(\) > pi.txt $ ls -sh pi.txt 4.0K pi.txt $ bzip2 pi.txt $ ls -sh pi.txt.bz2 4.0K pi.txt.bz2 Obviously, thats an issue with the filesystem more than anything else....so... $ wc -c pi.txt 24 pi.txt $ wc -c pi.txt.bz2 57 pi.txt.bz2 apprently pi has the mathematical property of getting larger when compressed :)

Re:Well ... what is it?

[SpinyNorman]

The best ASCII representation to send would be "pi".

Re:Well ... what is it?

[Junior J. Junior III]

Why would you want to compress the answer? The most efficient compression would be just to provide the algorithm and re-compute the number. Of course, that would have a large cost in terms of processor time, but in terms of storage it'd be by far the most efficient method of storing such a large amount of information.

Re:Well ... what is it?

[sco08y]

I'd love to see a version of cc that can work with terabyte large bigints in "10 lines or so"

Re:Well ... what is it?

[scotch]

Opps, yeah, you're right base 10^1.24 trillion. How to store the digit? We'll skip the uninteresting digits and just implement the one for pie. That funny pi symbol might be a good candidate. 1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, c, d, e, f, .......pi. Let's just ignore the decimal.

Re:Well ... what is it?

[martyn s]

actually, no. The most compact/efficient representation is to do it in ternary, since 3 is the closest to e, which is the most efficient base.
Here's the article. I read this in a slashdot article, but since slashdot's search sucks, I can't find it, but here's the article that was linked to.

Re:Well ... what is it?

[Directrix1]

Statistics is bullshit, especially when dealing with an irrational domain like pi. I would say its more along the lines of 50% its either in there or it isn't. And also wouldn't the percentage chance of finding it (using commonly heralded stupid statistical methods) be 0.00000001%? There is no such thing as random chance. Random is just a word we use to describe situations that we can't describe. That doesn't mean that there is equal distribution of the output amongst all possible values in the output range given any value in the input domain.

Re:Well ... what is it?

[Sparr0]

there is no proof because there IS a pattern. a while back someone produced a function for determining the Nth digit of pi (in base 8 i think it was). being able to produce a given digit at will indicates a pattern of some sort, even if it is a nonrepeating one.

Re:Well ... what is it?

[Directrix1]

IEEE is just a standards body. Base 2 exists outside of IEEE standards bud, I hate to brake it to you.

Re:Well ... what is it?

[Directrix1]

Data requirements are the same for any base you choose in this case, as it is always stored in binary. And, no, base 3 is not always the most damn efficient base. I read that article too. e is just the most efficient base if you use the efficiency formula (radix)(number of digits) for any given number, and that formula assumes there is an equal cost for changes in radix as there are for changes in digits. This is kind of like comparing two independant vector spaces though, really. It is completely situationally dependant. It essentially assumes that the same mechanism is used to store the number as to place it, when thats just not the case.

Re:Well ... what is it?

[addaon]

Compressing pi is not particularly hard. As posted elsewhere, one of the more convenient algorithms for pi is:

pi=sum as j goes from 0 to infinity of 1/16^j (4/(8j+1)-2/(8j+4)-1/(8j+5)-1/(8j+6))

So the data calculated compresses, quite readily, to a representation of the above algorithm (machine code, c code, lambda calc, your choice), plus the number of digits, the number of digits being necessary to restore the file to its original form. Of course, the number of bits needed to store the number of digits is logarithmic, so we get O(logn+c) bits to store n digits of pi.

Most compression algorithms wouldn't be able to do much with it. But most /good/ compression algorithms (with about five exceptions you should be able to think of off the top of your head) take advantage of the special form of the data they're compressing.

--

Re:Well ... what is it?

[pediddle]

You're obviously an idiot. Is it a 50% chance, or is it 0.00000001%? Where did you come up with those numbers?

Statistics is perfectly non-bullshit in this case, because pi is random, and there is an infinite amount random data to search. Okay, you're right that pi is not just a "random number". But the digits themselves are statistically random, and somewhere, the entire works of Shakespere is embedded inside.

How in the hell do you arive at "50% chance" for finding a 10 digit number? Okay, let's do it this way: what are the chances of finding the number "1"? Pretty damn high (hint: it's the first digit). Okay, how about the number "12"? "123"? What makes "1234567890" so special? Now, consider what happens because of the fact that the digits of pi extend infinitely: the chances of anything happening approach 100%, because the chances of something not happening are size(string)/size(pi), which approaches 0 as the size of pi approaches infinity.

Now if you can prove that I'm wrong by proving that the digits of pi follow some non-random pattern, then go get yourself a doctorate.

Re:Well ... what is it?

[Rubyflame]

Storing a number as a number (instead of ASCII) is not "compression." It's just the only thing it should reasonably be stored as.

Re:Well ... what is it?

[isorox]

Whatever happend to 7bit ascii?

Re:Well ... what is it?

[fredrikj]

You can compress ALL of Pi in a couple of hundreds of bytes. Just make a program that uses a simple algorithm to calculate Pi ad infinitum.

Re:Well ... what is it?

[Directrix1]

You're obviously an idiot. Is it a 50% chance, or is it 0.00000001%? Where did you come up with those numbers?

sorry I was wrong in correcting the value. The corrected value (of the corrected value) is 0.000000001% = dervived by assuming that there is a 1 in the 100000000000 possible values of ten digit combinations. But see thats assuming completely random distribution (which does not exist, these digits are generated by iterating through a mathematical formula). There is a 50% chance in it happening because in a system where not all of the outputs are known the distribution of the system is very likely unkown, not random. It is either in there or it isn't in there. People attach way too much meaning to random: its random because I don't know the outputs of a function as it progresses, its random because I don't know how something behaves, its random therefore there is equal distribution among the output of the function. Random is just a word to excuse ignorance in the investigation of the behavior of the output of a function.

Statistics is perfectly non-bullshit in this case, because pi is random, and there is an infinite amount random data to search. Okay, you're right that pi is not just a "random number". But the digits themselves are statistically random, and somewhere, the entire works of Shakespere is embedded inside.

OK, do a couple things for me. Define random, show me a proof that the digits of pi satisfy this "randomness" condition, and quit trolling about shit you have absolutely no idea about.

How in the hell do you arive at "50% chance" for finding a 10 digit number? Okay, let's do it this way: what are the chances of finding the number "1"? Pretty damn high (hint: it's the first digit). Okay, how about the number "12"? "123"? What makes "1234567890" so special? Now, consider what happens because of the fact that the digits of pi extend infinitely: the chances of anything happening approach 100%, because the chances of something not happening are size(string)/size(pi), which approaches 0 as the size of pi approaches infinity.

Arg, I hate arguing with idiots. First of all, your talking about the chances of finding a sequence of digits with the idiotic assumption that any function with an as to of yet uncalculated set of outputs will tend to exhibit the characteristics of random distribution (which does not exist). Also, the possibility of existence dooes not increase with the size(string)/size(pi). That number is just inversely proportional to the number of possible combinations you can get out of it.

Now if you can prove that I'm wrong by proving that the digits of pi follow some non-random pattern, then go get yourself a doctorate.

I'm simply declaring something that pi is not, RANDOM. I believe you should be the one retreiving proofs to back up your original statements. And there are many formulas on the net for genereating digits of pi to your hearts delight. I guess we should start calling those random number generators then huh?

Re:Well ... what is it?

[Alsee]

x = ln(10^1.24e12)/ln(2)
515e9

For those of you who looked at the above and said "HUH?"...

What he was trying to said is that you'd need a 480 gig hard drive to store it.

480 gig hard drive:
Myinfinity.com 888-474-6988 212-404-6188--P.O.'s IN STOCK, Snap 4100 Server 480GB NAS Network Attached Storage 480Gig 480.0GBENET 10/100 SERVER MODEL 4100 480.0 Gig Part # 5325301448 manufacture package with $250 Mail-In Rebate ends 10/31
Price: $ 3649
Ship: FREE
Updated:11/22,?5:40PM

In other words you'd need $3649 to store the number.

-

Re:Well ... what is it?

[Alsee]

Blah, I just woke up, brain not functioning yet.

You need a 480 gig harddrive, but I googled up and linked to a $3649 SERVER with a 480 hard drive. Massive massive brainfart. I'll just go hide in the corner for a while.

-

Re:Well ... what is it?

[Old Wolf]

That brings to mind the old conundrum,

"The smallest number that cannot be described in fewer than thirteen words"

(the catch, of course, being that that description has 12 words, thus describing the number that supposedly couldn't be described like that)

Re:Well ... what is it?

[Uncle Jimmy]

Err... your description of IEEE754 is correct, but it doesn't help when trying to store pi, because pi isn't a very large number, it just has an infinite number of digits after the decimal point. IEEE754 can store around 7 decimal places if i remember correctly, nowhere near 1.24 trillion.

Of course if you were trying to represent 1.24 trillion, IEEE754 would be the way to go...

Re:Well ... what is it?

[spac]

It can store approx 7 places in its 32 bit incarnation, but much more in its 64 bit incarnation.

Still, 64 bits isn't enough for storing a number this large. However, since pi is irrational it is a floating point number so it is conceivable that IEEE 754 can be adapted to a larger memory space in order to store all those digits that the kind professor approximated pi to.

I'm not saying my way is the only way, there are probably better ways, but since pi is a floating point number, any approximation is likely to be stored in some incarnation of IEEE 754.

I guess I should have given my post a little more thought, thanks Uncle Jimmy!

My Blog: http://gozman.org

Re:Well ... what is it?

[spac]

Yup, Base 2 does exist outside of the IEEE standards, but how do you store a floating point binary number? You need a standard way to do this or different hardware manufacturers will come up with different implementations and incompatibilities will ensue.

And if you don't think that the IEEE has a standard for dealing with these numbers: http://grouper.ieee.org/groups/754/

Re:Well ... what is it?

[fferreres]

From a neofite I could tell you you don't need to be an expert to know that a compression algorithm is any ruleset that, given a source data, can bring back the original uncompressed version of the compressed data.

Thus, just descrining how to calculate pi up to any number of predefined precision is as valid as any other algorithm for compressing whatever other data you might find. It will only work for PI, and fail miserably for other data, but that's a different story all together.

The fact is PI can anly be understood in a compressed manner. You can NEVER operate on the uncompressed version of PI, unless you live in a infinite time-space world which, unluckly, is not our universe.

So, yes, PI can be used because we know it's compressed version. Even more, we only know how to calculate PI and what is PI, but we'll probably never know the uncompressed version of PI.

Re:Well ... what is it?

[rweir]

Uh, If I go to the trouble to calculate a trillion digits of anything, I'm not going to be storing them in a 'lossy' inaccurate format like IEEE FP.

Re:Well ... what is it?

[ShavenYak]

Well, if you're going to use ridiculous bases, why not just use base pi? Then the value of pi is 10. Nice and simple.

Re:Well ... what is it?

[ShavenYak]

The string 8675309 was found at position 9202591 counting from the first digit after the decimal point.

Ah hah! That must be Jenny's new number.

Re:Well ... what is it?

[sco08y]

I don't think STL supports bigints, so you'd have to define all the functions to manipulate the array, those would take more than 10 lines.

OK, now this is overkill

[Rhinobird]

I think I read somewhere that to draw a circle to circumscrible the known universe with a an error of +/- the width of a proton, you only need to know Pi to about 20 places. What practical purpose is there to know pi to 1.whatever trillion places. Unless, of course, you're Count Duckula, in which case it's a party trick.

Re:OK, now this is overkill

[DoctorNathaniel]

Hmm..

size of the proton: ~ 1 fm = 10^-15 m
age of the universe: ~15 Gyr
speed of expansion ~ c = 3 x 10^8 m/s

gives:
proton/cosmic radius ~ 10^-42

So you need about 40 places for this. Of course, you might want to calculated it to the Plank scale, so maybe tack on a few more.. say 100 for safety. Yes, a trillion digits does seem a bit like overkill.

Re:OK, now this is overkill

[sielwolf]

Well I guess the problem is statistical round-off error (i.e. as the number of values multiplied together increases, the error also is going up, so you have to round down as it is no longer accurate to that number of places). What I was taught:

When multiplying a*b = c, the place of c is the lesser place value of a OR b MINUS one.

So no matter what, to be accurate/not cheat, you are slowly losing granularity as you churn out the calculations.

If there is a God, I hope that he DOES have Pi out to some ludicrous number of digits. I don't want to see the round off error of those calculations ;)

Re:OK, now this is overkill

[grahamlee]

size of the proton: ~ 1 fm = 10^-15 m
age of the universe: ~15 Gyr
speed of expansion ~ c = 3 x 10^8 m/s

[my emphasis above]
I'm not sure that the age of the universe isn't known well enough for it to be worth using as many as 40 sig. figs. in calculating the circumference - given that various claims run from ~10-~18Gyr then claiming pi=4 will provide a smaller contribution to your error than the contribution from the deviation on the age of the Universe.


BTW hello from an undergrad prep'ing for next term's finals (waves). :-)

Re:OK, now this is overkill

[wass]

I believe the poster was assuming that one needed to calculate one of the largest possible circles allowed in the universe (order of magnitude for us, obviously), and would be limited only by the accuracy of pi.

In other words, for most all practical use, where one would be limited by their precision of pi, there is no need to go beyond probably 8-10 digits max.

I also read a similar thing somewhere putting the accuracy of pi into perspective. The above example shows that 40 sig figs is pretty damn huge, how about 100 sig figs?

Suppose the nearest star (Proxima Centauri) forms the diameter of a LARGE sphere (about 4 LY, or roughly 4e+16 m). Now suppose this sphere is filled with paramecia (I ain't no biologist, so I'm assuming each paramecia takes up a cubic micron). Like jam-packed filled. Now take each of the paramecia within the sphere, and space them each the same 4 LY distance to make an enormously-gigantic line (Line is roughly 2*10^84 m). If this line is the diameter of a circle, and one wished to calculate the circumference, limited only by their precision of pi, known to 100 digits, they could do so within a few fermis (FYI - 1 fermi = femtometer = 1e-15 meter = length scale of atomic nucleus). That one kind of blew my mind when i first read it about it too. (just did calculation to check it out). damn. 100 sig figs. crazy.

Re:OK, now this is overkill

[Ryan Amos]

Hrm.. Well, as one of my Computer Science teachers once told me (in a discrete math class).. Mathemeticians do things because it interests them. The fact that it often has no practical application is why they are often cold, bitter and broke. :)

Re:OK, now this is overkill

[Doug Neal]

Because we can!

Re:OK, now this is overkill

[Mattsson]

> "Yes, a trillion digits does seem a bit like overkill."

No! More is *always* better.
We know this from software version numbering.
That's why Redhat 8 obviously is better than Linux 2.4 and Windows 2000 beats everything else! =-)

Re:OK, now this is overkill

[mbogosian]

Hrm.. Well, as one of my Computer Science teachers once told me (in a discrete math class).. Mathemeticians do things because it interests them. The fact that it often has no practical application is why they are often cold, bitter and broke. :)

Maybe the article just got it wrong about what kind of pie he's obsessed with.... ;)

Re:OK, now this is overkill

[Pig Hogger]

I think I read somewhere that to draw a circle to circumscrible the known universe with a an error of +/- the width of a proton,

What you want is to calculate it to the width of a NEUTRINO or a PHOTON.

Re:OK, now this is overkill

[grahamlee]

This would also explain why more people like "Space: 1999" than "Galactica 1980".

Actually, I'm going to stick my neck out and claim to be back on topic. The reason why calculating pi was traditionally done on any new fast computer was to check the logic of the computer - the algorithms used to calculate pi are well-known and simple, so the values the computer gives for any term in the expansion (usually a Taylor expansion of atan(1.0)) can be readily cross-checked - by hand if necessary. Then if there is a problem with the ALU the error will be noticed and troubleshooting may commence. Once it has been done, it can be repeated to check for hardware problems (although other checks are much quicker).

Of course these days, the computers are designed on computers and the logic is tested before the box is built, so calculating pi is no longer a consistency check. I'm afraid they just do it for fun.

If Pi were made into a classic video game...

[Jace of Fuse!]

We would have either found the end by now or discovered a pattern.

heh.

YAY!

[b0ycheese]

That's gonna make for one accurate circle!

How?

[PoiBoy]

Just how is Pi calculated? Are there any free apps available for Linux that I could look at to see how it is done?

Re:How?

[Gudlyf]

You could always just do it with Good ol' Calculus.

Re:How?

[Speare]

Here's a program written in BrainF*ck to calculate pi: http://www.people.fas.harvard.edu/~jafowler/pi/pi. b

Here's the analysis of the program, and a link to what the Turing-inspired BrainF*ck programming language is about.

Re:How?

[Anonymous Coward]

Just type
echo "scale=500;4*a(1);"|bc -l
in a shell, where 500 is the number of decimals you wish.

Re:How?

[jericho4.0]

Aaaaaaaaaaaaaahhhhhhhhhhhggggggggg!!! What a stupid, useless, contribution to a useless chat about pi.

Thank you. I really like the bit where he says 'If you want to calculate more digits, you can add more plus signs at the obvious place in the file.' Yeah, right. My head hurts.

Re:How?

[Jeremy Erwin]

wow: that's terse.
Unfortunately, this algorithm (4*arctan(1), for those of you who don't use bc), doesn't converge as swiftly as the Kanada method.

Go on! I dare you:

echo "scale=124000000000;4*a(1);"|bc -l

Re:How?

[Artemis]

I really like the bit where he says 'If you want to calculate more digits, you can add more plus signs at the obvious place in the file.' Yeah, right. My head hurts.

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
How many digits do you want? ++++++++++
>[-]++++++++++>>>>>>>>>>>>>>>>>>>>>

Considering this is the beginning of the file it seems pretty obvious unless you're blind.

Re:How?

[Alsee]

Considering this is the beginning of the file it seems pretty obvious unless you're blind.

Some of us DO read the web in braille you know.

-

Re:How?

[Artemis]

Ooh, you really got me! Maybe that's why I said "unless you're blind"?

Re:How?

[Alsee]

Sigh. It was supposed to be humor.
I hate killing a joke by disecting it, but here goes anyway...

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> ;
How many digits do you want? ++++++++++
>[-]++++++++++>>>>>>>>>>>>>>>>>>>>> ;
Considering this is the beginning of the file it seems pretty obvious unless you're blind.

Ok, so what about blind people? Well, they read the web with a braille browser. Therefore their broswer displays the following:

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> ;
How many digits do you want? ++++++++++
>[-]++++++++++>>>>>>>>>>>>>>>>>>>>> ;

Except it's in braille.

Just in case the joke's not completely dead yet, here's the post-mortem: it would be obvious even to a blind person who checks the file.

-

no purpose in math?

[slavemowgli]

From the article:

Figuring out pi to much more than about 1,000 decimal places serves little purpose in math or engineering.

I cautiously agree with regard to engineering, but math?

Re:no purpose in math?

[nusuth]

1000 decimal places of pi for engineering is an overkill of more than two magnitudes. 4 to 9 decimal places are quite enough, depending on what exactly you want to calculate. Even one (that is 3, not 3.1) may do sometimes.

As for math, I don't think there is anything at all learnable from actual digits of pi. We know they neither end nor repeat. Actual values are just trivia. It could as well have been 3.76421403038164659... and nobody would care.

Re:no purpose in math?

[Hater's Leaving, The]

Numerically, it has no mathematical use at all. It's fun as an exercise in algorithm designing, optimising the parallelism of these supercomputers, and I guess that could make it a momentous engineering feat. However, it wouldn't have needed to be the digits of Pi that they calculate in order to accomplish that feat.

THL.

Re:no purpose in math?

[slavemowgli]

Mathematically, pi is the ratio of the length of the circumference of a circle to the length of its diameter. No number of digits will ever beat the abstract definition.

Whether the digits of pi are NORMAL is another matter, something a little more interesting in current math research.

Of course, the definition of pi is something different from it's actual numerical representation. I don't think that what you said above counts as an exact mathematical definition; I personally define pi to be two times the smallest positive zero of cos, where cos is defined as \sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{(2n)!}, but that's just a minor point. :)

Anyway, what I really disagreed with was the article's implication that the calculation of significicantly more digits of pi than 1000 or so was something that couldn't possibly be of use for anything aside testing supercomputers.

Re:no purpose in math?

[slavemowgli]

Numerically, it has no mathematical use at all.

Sorry - I just don't buy that. :)

Re:no purpose in math?

[taverngeek]

There is the open question of whether there might be a pattern in Pi. Obviously, it cannot have a sequence that repeats after n digits as then Pi would be rational and it is proven to be transcedental. But it has been proven that one can construct a number that is transcendental. A function like the sum of 1/(10^(2^n)) where n=1 to infinity has been proven to be transcendental. Actually, it is far easier to prove such a number is transcendental than a number like Pi or e because of how the number is constructed.

I may have the exact functional slightly wrong, but the point is that a transcendental number most definitely can have a nonrandom pattern.

Re:no purpose in math?

[slavemowgli]

This is definitely true (and the sum you gave indeed gives a transcendental number, as can be seen quite easily). And while even knowing even trillions of pi's digits won't prove that there exists any kind of pattern in pi's decimal representation, they might be quite helpful as evidence for such a pattern, i.e, spark the initial idea about what the pattern may look like. One question I'm still wondering about myself, for example, is whether there exists a formula that allows one to compute the n-th decimal digit of pi without having to compute the ones before that one. (such a formula exists for pi's hexadecimal representation, for example). Knowing a large number of pi's digits may help in finding such a formula for the decimal case.

Re:no purpose in math?

[nusuth]

Care to enlighten us with a possible use then?

The 1.24th trillion digit of pi is ..

[gargle]

The number Six!

Re:The 1.24th trillion digit of pi is ..

[edbarrett]

Bert: My favorite number is 6.
Ernie: Bert, nobody's favorite number is 6!

Re:The 1.24th trillion digit of pi is ..

[Soul-Burn666]

Actually, in 1.24 trillion digits, it's a pretty good assumption that every sequence of numbers up to 5-6 digits appears in it...

Re:The 1.24th trillion digit of pi is ..

[seanadams.com]

In fact, there are already efficient algorithms for calculating the nth digit of pi, but they're relatively slow if you use them to calaculate all the digits in between. IIRC there are formulas now for base 2 and base 16, but I don't think these have any practical applications... maybe good for a PRNG though?

Re:The 1.24th trillion digit of pi is ..

[gidds]

ObQuote: "I am not a number! I'm a free man!!!"

Re:The 1.24th trillion digit of pi is ..

[Pig Hogger]

I can tell, with 10% probability, what is the value ANY digit of PI.

math question about pi

[selectspec]

Does the problem that pi can't be expressed in decimal notation extend to other base systems? For example, if you tried to write pi out in binary or hex would you encounter the same problem? Is there a special base system (other than base pi) which can describe pi in a finite number of digits?

Re:math question about pi

[Moeses]

You can write Pi as 1 (base Pi).

Re:math question about pi

[DJPenguin]

No - pi is irrational... as far as I know this would be the case for base-n where n is of course an integer.

Re:math question about pi

[agdv]

No - pi is irrational

Okay, I've heard this many times, and I don't doubt it's true. But are there any simple elegant proofs of this (like the one for proving that the square root of 2 is irrational), or are the proofs very involved, or are there no proofs at all except "well, nobody has found the end yet"?

Re:math question about pi

[Dunark]

Pi is worse than irrational - it's trascendental. Merely irrational numbers can be expressed as simple expressions with finite numbers of terms, but transcendentals require an infinite number of terms.

Re:math question about pi

[SpaceLifeForm]

No. Conversion to any other base does not make Pi become non-irrational. The base does not make any magic transformation occur that changes Pi from an irrational number to a rational number. It's just not possible.

Re:math question about pi

[isorox]

He said other then base Pi.

You can write it as 0.5 (base 2Pi)

Re:math question about pi

[Fruit]

You'd have to write it as 10 then :)

Re:math question about pi

[slavemowgli]

Well, at least pi is a computable number - other interesting ones, like Chaitin's Omega, aren't, at least not ahead of time. In fact, if I recall correctly, it is even (provably) possible to construct Turing machines for which no single digit of Omega can be computed at all, but I'm not really sure about this anymore.

Re:math question about pi

[wass]

Nope. Do you write the number 2 as '1' in binary (base 2)?
sorry, but in base pi, pi would be written as 10.

(fyi, i made the same mistake back in the day also)

Re:math question about pi

[atomicdragon]

There is a pattern to pi in Hex. There is a well known equation to calculate the digits of pie in hex (im sure google knows). Of course converting it to decimal requires you to add smaller and smaller fractions of 16^n, so it doesnt get you anywhere.

Re:math question about pi

[hanssprudel]

That pi is irrational was first proved by Johan Heinrich Lambert in 1761, and there are a whole bunch of other proofs for it since then. The final nail was placed by Lindemann who proved that pi is not even the root of a polynomial with rational coefficients in 1882 (basically by combining Hermites proof that this holds for e with Euler's well known equation regarding the relation of the two constants).

Compared to many others, the proofs are not that difficult, but but anybody without a couple of years of university algebra would probably find them daunting. They are certainly not /. level.

Re:math question about pi

[khslinky]

Proving that pi is irrational is not as simple as proving that sqrt(2) is irrational, but neither is it ridiculously difficult. One proof is given as exercise 7.33 in Apostol's mathematical analysis (2nd edition).

Re:math question about pi

[Hater's Leaving, The]

The proof is about 100 years old.
One famous prover was the infamous "Bourbaki" (who wasn't a single person, it was a pseudonym used by a secret society of mathematicians)
I don't believe this has any "oh wow" to it, but here's a pretty typical statement of the proof:
http://www.math.clemson.edu/~rsimms/neat/m ath/pipr oof.html

THL.

Re:math question about pi

[Nathanbp]

Actually it would just be pi.0 in base 2Pi.

Re:math question about pi

[jpetts]

...but transcendentals require an infinite number of terms.

Not if you use other transcendentals. IIRC, e^(-i*pi)=1.

Re:math question about pi

[CoolVibe]

pi can not be computed, it can merely be approximated.

Re:math question about pi

[Lionel Hutts]

You can make much stronger statements. Even Omega is part of the miniscule class of reals which are finitely describable: there's a string of English words which uniquely identifies it. The same is true of all computable numbers, including the trivially small classes like the algebraic numbers.

There are only aleph-null many finite strings of words, but 2^aleph-null reals, so the finitely describable are an insignificant minority.

Re:math question about pi

[slavemowgli]

Indeed - good point. :) One question, though: does there exist a number which is finitely describable but whose digits cannot, for example, be computed one after one by a Turing machine with infinite running time?

In other words, what's the formal definition of "finitely describable"?

Re:math question about pi

[user flynn]

Base 10: 180 degrees

binary : 10110100 degrees

hex : B4 degrees

base 19 and 4/90ths : 98.6 degrees

Re:math question about pi

[Florian Weimer]

...but transcendentals require an infinite number of terms.

Not if you use other transcendentals. IIRC, e^(-i*pi)=1.

"e^(-i*pi)" is thus an abbreviation for a power series involving pi, so there's actually "an infinite number of terms". (Of course, this concept doesn't make much sense. Obviously we can only talk about specific numbers which can be described with a finite amount of symbols.)

Re:math question about pi

[Lionel Hutts]

Yes: Omega is such a number. It is not computable, but it is describable. (Unless you mean something stronger by "Turing machine with infinite running time." Within a finite time, an ordinary TM cannot produce any digits of Omega (or any beyond a small fixed number, for a weaker definition).)

Reasoning about noncomputable numbers gets tricky, but there are formal ways to go about it. Consider even a Turing machine augmented with an oracle for Omega. It can compute (or accept) a much larger class than an ordinary Turing machine, such as numbers algebraic over Omega.

Obviously, you would want a more formal system than English for defining finite describability formally, but any sufficiently rich language will give you a definition that comes out about the same.

Re:math question about pi

[slavemowgli]

OK, thanks for the information - I gotta admit that while I find this whole field very interesting, I'm just a layman, so occasionally I have to ask. :) Concerning finite describability, what properties would a suitable formal language for defining it have to have?

Re:math question about pi

[matrix29]

There is a pattern to pi in Hex. There is a well known equation to calculate the digits of pie in hex (im sure google [google.com] knows). Of course converting it to decimal requires you to add smaller and smaller fractions of 16^n, so it doesnt get you anywhere.

Actually there is a trick that speeds things up somewhat in base converting from decimals.

Take the number 0.57 (Base 10) and convert that to (Base 7). Here is the routine.
0.57 * 7 = 3.99
Your first digit is 3 and your converted number begins 0.3
Now subtract 3 from 3.99 to get .99
Now multiply by 7
0.99 * 7 = 6.93
Now your number is 0.36
(Get the pattern? It is the opposite of RADIX conversion)
Wash - Rinse - Repeat to get 0.366336633663... (or a repeat of .3663)

Okay. Now we begin converting HEX to Base 10. You have to convert your multiplier into the Base you're converting to. So you multiply the Hexdecimal Pi by "A" and your result will be in HEX base. Wash - Rinse - Repeat.

For example.
Take HEX (0.ACE) * A = HEX (6.C16)
(and you multiply the number and convert back to HEX. For example 6 * A = 60 which converts into HEX 3C, carry the 3, multiply A * 1 = 10 + 3 =
13, convert 13 to HEX and get "D", multiply A * C = 120, convert to HEX to get 78. That is how I arrived at the multiplication results below).
(All numbers following are in HEX)
A * 0.C16 = 7.8DC
A * 0.8DC = 5.898
A * 0.898 = 5.5F
A * 0.5F = 3.B6
A * 0.B6 = 7.1C
A * 0.1C = 1.18
A * 0.18 = 0.F
A * 0.F = 9.6
A * 0.6 = 3.C
A * 0.C = 7.8
A * 0.8 = 5
Result = HEX (0.ACE) = DEC (0.675537109375)

It actually is astoundingly simple to base convert decimals this way. To do the reverse, just divide the decimals.

Re:math question about pi

[matrix29]

Result = HEX (0.ACE) = DEC (0.675537109375)

By the way, to verify that I'm not pulling this out of my ass, just multiply DEC (0.675537109375) by 16.
DEC (0.675537109375)* 16 = 10.80859375
Subtract the 10 and convert 10 to HEX = .AC
Now take 0.80859375 = 12.9375
Convert 12 to HEX = 0.AC and subtract from the number.
Multiply 0.9375 * 16 = 15
That gives us a decimal number in HEX of 0.ACE

SWEET ???
I play with non-integer decimal bases and have to find shortcuts to speedy math.

I just figured I'd follow up on that for the people who thought my numbers just whizzed over their heads.

Re:math question about pi

[William Tanksley]

There are only aleph-null many finite strings of words, but 2^aleph-null reals, so the finitely describable are an insignificant minority.

Interesting!

Is there a smallest positive number which cannot be finitely described?

Whoops, no, there isn't; because if there was, I just finitely described it.

So there's something wrong with your hypothesis.

-Billy

Re:math question about pi

[jonadab]

As others have pointed out, there are an infinite number of such
systems; however, none of them are natural numbers. IMO a more
interesting question is, can you represent 1 (decimal) exactly
in base pi (or any of the other bases in which pi can be expressed
exactly)? Sure, 1 can be expressed _algebraically_ in base pi as
pi^0, but my intuition says it is probably possible to prove by
induction that you cannot express it exactly as a string of digits
raised to a power of pi (i.e., conventional notation for floating
point numbers -- I wanted to use the term "decimal point", but
obviously it wouldn't be that; we could call it a pial point I
suppose...) in any of the same bases in which pi can be so
expressed -- i.e., my intuition says these two sets of bases are
disjoint sets and can be considered as equivalence classes, in
terms of which numbers can be expressed in them. (That is, I
suspect that *none* of the numbers representable in the pi class
of base notations can be represented in any of the natural class
of base notations, and vice versa.) I also suspect that there
are provably an infinite number of such classes and that an
interesting study could be had from determining the cardinality
of the set of all of them. (My first guess is aleph-sub-one, but
that's a shot in the dark, as I haven't studied the question.)

Anybody know anyone who would know whether any work has been
done on this concept?

Re:math question about pi

[commodoresloat]

Pi is worse than irrational - it's trascendental.

Are we discussing mathematics here, or psychology, or philosophy?

Re:math question about pi

[ASMprogrammer]

Who says there is a smallest positive one?
There's no smallest positive rational number, either, yet positive rational numbers exist... in math terms, the set is not well-ordered under the usual inequality.

Re:math question about pi

[Lionel Hutts]

No, you didn't.

What's the smallest positive number, period? There isn't one.

All you've shown is that the non-describable numbers are not closed.

Re:math question about pi

[kazad]

Nice point. It's interesting that the number N is written as 10 in its own base, I had never thought about it that way.

Number ten in base ten is "10"

Number two in base two is "10"

Number sixteen in base sixteen is "10"

Obvious in hindsight, but interesting nonetheless. The "ones" digit of any number system is, by definition, never big enough to hold that number.

Re:math question about pi

[preternatural]

There are an infinite number of bases that you can write pi in a finite number of digits. You found one, base pi. There's also base sqrt(pi), base pi^2, etc. If you write pi in base pi^q where q is rational, its expansion will contain a finite number of digits.

Re:math question about pi

[Alsee]

Are we discussing mathematics here, or psychology, or philosophy?

The answer is a definite maybe.

-

Re:math question about pi

[Old Wolf]

0.5 means 5 * 1/(base), ie. 5/2pi , which isn't pi.

Perhaps you should have stuck to "2" (base pi/2) :)

Re:math question about pi

[Old Wolf]

'computable' means that , given enough time, you can compute any given digit of it, which is certainly true for pi.

Re:math question about pi

[Old Wolf]

You forget about 1/infinity

[Mod -1, Troll]

Re:math question about pi

[ilyag]

OK, I guess 10 base pi is pi. 100 base pi is pi^2. 120 is pi^2 + 2pi.

However, can someone explain to me what 1 base pi is? 11 base Pi?

BTW, how many digits do you have in the base Pi system?

Re:math question about pi

[Ninja Programmer]

Well as long as we are being anal retentive -- you will have difficulty writing it as 10 as well. Since this is an irrational base, you are going to need some sort of digit seperator. So something like (1,0)_pi would be more appropriate (so we could write silly things like (1,e)_pi = e + pi)

Re:math question about pi

[Ninja Programmer]

• No - pi is irrational... as far as I know this would be the case for base-n where n is of course an integer.

n in fact could be any rational, or complex rational, and PI would still not have a finite expansion in that base.

Re:math question about pi

[Lionel Hutts]

Like, duh. We're talking about reals here, my cowardly friend. I mean the ordinary topology of R, as was totally obvious from the message I was replying to.

Now, did you really not understand that, or did you just want a chance to quote a textbook?

Re:math question about pi

[Ninja Programmer]

• Pi is worse than irrational - it's trascendental. Merely irrational numbers can be expressed as simple expressions with finite numbers of terms, but transcendentals require an infinite number of terms.

Being "transcendental" just means that its a real number which is not the solution of any polynomial. For example, sin(1), and exp(2), are *probably* trancendental as well even though I wrote them as expressions with finite terms.

This is an important distinction, because there are 5th degree polynomials whose solution I cannot write down in a finite number of symbols, even though that solution is not transcendental.

Re:math question about pi

[Smurf]

However, can someone explain to me what 1 base pi is? 11 base Pi?

I think 1 base Pi would simply be: 1*Pi^0 = 1

And 11 base Pi: 1*Pi^1 + 1*Pi^0 = Pi + 1

And 0.1 base Pi: 1*Pi^-1 = 1/Pi

Re:math question about pi

[eatdave13]

What do *nix people have against vowels?

Not as much as OS/400 people...

Re:math question about pi

[fferreres]

Actually, since you can always find a number which is smaller that any other number, there must exist only exist one number that is smalled than every other number (talking about reals).

If such number did not exist, then it would not be the smallest number, so another number must take his place. The smallest number thus must exist and can only be defined as an idea, but not any particular number.

But in any case, we can't compute the number because when we talk about reals, we are defining an infinite set, so of course you will never be able to compute the numbers of the set. So it all boils down to our axioms when we define the rules which we use. And supposedly, we use these systems for mirroring something that actually happens in our universe. And infinity is something we are not made of.

I could think of infinite sets of axioms that would entail infinite teorems infinitely uncomputable "whatevers".

I am 90% sure what I say makes no sense to you or is blatanly wrong, but it may have to do with another problem which is axioms theory (tough it's a relatively old field).

Re:math question about pi

[fferreres]

"given enough time"

Given enough time, you'll only be able to aproximate pi. That is, it'll never be any closer to the actual pi than just saying "pi". It's not very usefull though. Given infinite would you be able to compute pi? That's just a definition for saying what you already said "i can always get closer to the real pi, but never reach IN time".

Re:math question about pi

[Old Wolf]

You didn't read what I wrote. Let me repeat:

'computable' means that, given enough time, you can compute any one digit of it

(NOT necessarily compute ALL digits, as you seem to want to do).

Or, to put it another way, 'computable' means that there exists a function like this:

int find_nth_digit_of_pi(int n);

that returns after a finite time, if you call it once. Also note that this is a different property of numbers than the property 'can be written down in its entirety'.

Re:math question about pi

[Grizzlysmit]

Pi is worse than irrational - it's trascendental.

It's a very naughty number :-)#

Re:math question about pi

[fferreres]

Yes, you are semantically correct, as everyone else that follows the "computable" definition. I know it's computable in the usual sense and for all practical matters, though it will still take the function an infinite amount of time to just read in the (infinitly long) number i am thinking to supply the function. That was my point, to make clear separation of what computable means vs. being able to compute the ith pi digit up to _any_ precision (obviously cannot be done, as one could take an infinite time to supply the number and if there was such a thing as "infinite time", then the function would also need "infinite time" to compute th result.

Re:math question about pi

[Idarubicin]

Math (in particular number theory) really is philosophy--but a variety that follows stricter rules than most.

Psychology is the technique you use to try to figure out where number theorists come from.

More Please

[dirkdidit]

A Much Bigger Piece Of Pi

Doesn't matter, I still want seconds. With ice cream!

Why?

[SlamMan]

How could this ever be useful? I mean that as an honest question, what could anyone, ever, use this for?

Re:Why?

[lirkbald]

Mathematics is often about doing "useless" things. Mathematicians often do things not because they are useful, but because they are "interesting".

Number theory used to be a "useless" field.

Then someone discovered you could do cryptography with it.

That said, I *still* can't think of a good reason to calculate that many digits of pi :-p

Re:Why?

[jkramar]

Well, it has been conjectured and not disproved that pi is normal; in other words, its digits are entirely, comprehensively, and thoroughly random. Thus, a list of its digits makes an excellent randnum table.

Re:Why?

[k_187]

Pi-level compression my friend.

They keep taking pi out so damned far to see if it contains all possible patterns of numbers (which is a lot). Since its irrationial its possible and if we convert these 1.4 trillion places into binary then we can start and stop at any point along the way and have stuff. then instead of sending a binary package out to somebody, you can give them where to stop and start along the path, of course that number may be larger than what you want to send, but it'd still be cool.

Re:Why?

[Nynaeve]

You could probably do the same thing with rand() since it always outputs the same sequence of digits if you don't use srand()

Re:Why?

[matrix29]

Pi-level compression my friend.

They keep taking pi out so damned far to see if it contains all possible patterns of numbers (which is a lot). Since its irrationial its possible and if we convert these 1.4 trillion places into binary then we can start and stop at any point along the way and have stuff. then instead of sending a binary package out to somebody, you can give them where to stop and start along the path, of course that number may be larger than what you want to send, but it'd still be cool.

Now before any one starts screaming "COUNTING ARGUEMENT" and "PIGEONHOLE PRINCIPALS" I would like to mention to the crowd that although Pi is indeed finite, any number of irrationals or transcendentals used as dictionary sets can do this very thing. Pi alone cannot be used for efficent dictionary transmission compression (using two identical dictionarys on two computers and citing the page number and word number in place of sending the word itself and gaining speed by the reduced data requirements in transmission), BUT using the cube root of 2 and the Golden Ratio and perhaps any number of precalculated numbers will allow huge shortcuts in data transmission (it is not perfect, but still a workable method for more rapid data transmission).

Re:Why?

[jonadab]

> You could probably do the same thing with rand() since it
> always outputs the same sequence of digits if you don't use
> srand()

On the same computer system it does...

Re: Why?

[Omniscient Ferret]

Why not?

How to calculate PI yourself

[renosteve]

One way to calculate for yourself the value of pi is to drop a lot of toothpicks onto a large piece of paper that has lines drawn on it!

Here's how it works. You'll need several boxes of toothpicks. Get a large piece of chart paper, and draw parallel lines on it, from one side to the other. The lines should be separated by a distance just slightly larger than the length of a toothpick.

From a height of about one metre, drop a measured number of toothpicks onto the chart paper, so that they all fall randomly somewhere on the paper. Count how many toothpicks are touching a line (or would be, if they weren't resting on another toothpick).

Repeat this process as many times as you can. Lots of people can do it at once. All that's important is that, each time you drop some toothpicks, you write down how many you dropped, and how many of those ended up touching a line. When you're done, find a total for each quantity.

You now have all the numbers you need to calculate Pi:

c ... toothpick length (in mm) <BR>
a ... line separation (in mm) <BR>
N ... total number of toothpicks dropped <BR>
M ... total number of intersections <BR>
(c must be less than a) <BR>

Now here's the formula you need to calculate Pi:

PI = 2cN / aM

Fill them in the formula, and work out your own value of Pi!

Re:How to calculate PI yourself

[DJPenguin]

Informative? come on now...!

Re:How to calculate PI yourself

[Jacek Poplawski]

One way to calculate for yourself the value of pi is to drop a lot of toothpicks onto a large piece of paper that has lines drawn on it!

You are toothpicks seller, aren't you?

Re:How to calculate PI yourself

[bokmann]

Of course, you could also write a program that just printed out "3.141592654" and then just started generating random numbers...

Run the program, and impress your friends. Tell them you wrote a simulation of the toothpick technique described on slashdot... just don't release the source.

Re:How to calculate PI yourself

[Anonymous Coward]

Are you advocating closed-source software now?

Re:How to calculate PI yourself

[jkramar]

That's very wrong; the ninth digit after the decimal point is not 4 but 3. Not only that, but the toothpick technique is not fast at all.

Another well-known but slow algorithm is as follows:
pi=4/1-4/3+4/5-4/7+4/9-4/11+4/13-4/15+4/ 17-4/19+.. .

Re:How to calculate PI yourself

[masterkool]

I think you are refering to the Monte Carlo method of Pi calculation. Its specifically called the "Buffons needle" method. There is also a cool screensaver that utilizes this method to calculate Pi! Sweet, even in an idle state, my computer can still do something allmost useful!

Re:How to calculate PI yourself

[Kenrod]

So...what happens none of the toothpicks dropped intersects a line? You would have division by zero. Not beyond the realm of possibility...

Re:How to calculate PI yourself

[quantaman]

I've tried that in simulation and while it does work it works VERRYYYYYYY slowly (I seem to remember after 1000 simulation I was upto pi+-0.02) If you want an actual algorithm try a fourier transformation with
(pi^2)/8= (1 + 1/9 + 1/25 + 1/44 + ... + 1/n^2 for all odd n. I garantee you this will converge much faster than the monte carlo method. It also actually has a chance of getting more digits than your calculator (though I've never tried this algorithm on a computer) and you can actually be pretty sure that a number is correct once the numbers a couple decimal places down have stopped changing quickly. Enjoy!

Re:How to calculate PI yourself

[seanadams.com]

Cute, but its accuracy depends on your measurement of the length of the toothpick and the separation between the lines. If you're going to measure something, you may as well just measure a friggin circle. :)

Re:How to calculate PI yourself

[forgotmypassword]

That is using the power series for the arctangent right. A much faster and better method is to use the rational approximation of arctangent, which yields the continued fraction for pi. Almost as easy, but much better.

Re:How to calculate PI yourself

[wik]

>pi=4/1-4/3+4/5-4/7+4/9-4/11+4/13-4/15+4/17-4/19+. . .
When I was 12, I wrote a little gwbasic program on my IBM PC using this algorithm. Granted, my algorithmic skills were really bad then (and I didn't know much about floating point errors). After running for three weeks straight, it settled on 3.14288...

I was horribly disappointed.

Signature of God?

[Speare]

In the book version of Contact by Carl Sagan, but skipped in the Jodie Foster movie, was the notion that the aliens had discovered proof that the universe was created by a higher intelligence. A God or society of Gods far higher and more advanced than the aliens. The whole point of dragging Human-kind to that remote beach to talk with daddy was to tell Human-kind that it was time for them to look for God's signature on this universe.

As any artist, the creator signed the creation. Where? Deep into the insignificant but irrefutably valid digits of several of the fundamental mathematical constants such as pi and e.

The main character finds one of the signatures at the end of the book: if calculating digits of pi in base 11, after a few million or billion places, a 500x500 digit span is almost entirely zeros. If the span was rendered as a square of pixels, the non-zero digits drew a perfect circle inscribed in the square. A circle in a square. The key concept defining pi, in the digits of pi itself. The whole way the universe works is affected by that constant, so any such 'design' in it has, if you pardon the pun, a transcendental import.

Why base 11? It's left to the reader to decide, but I expect Sagan wrote it because it is considered one of the possible designs of the universe, one of the string theories is based on an 11-dimensional all-inclusive physics model. As the alien explains to the main character, it wouldn't be base 10, because what's the likelihood that the creator also happened to have ten fingers?

Re:Signature of God?

[whovian]

I still like Euler's formula (not the one for polyhedra) relating e, pi, i, 0 , and 1:

e^{\pi i} + 1 = 0

Re:Signature of God?

[bokmann]

When I was in college, I wrote a distibuted computing project that drew detailed plot of the Mandelbrot Beetle. I fantasized that if I could zoom in on JUST the right spot, I'd see "God was here" as if in graffiti. Wouldn't that just be kick ass?

Re:Signature of God?

[JanneM]

Well, it seems pi is normal, which means any finite sequence appears somewhere along the expansion of the number. So trivially, that image of a circle is in there somewhere, as is an image of a triangle, the source to Linux 4.0, an image of Bush playing with G.I. Joe dolls on his desk and so on.

Re:Signature of God?

[Speare]

Of course. Though in terms of probabilities, the chances of a few million digits with equal distribution, followed by a square image of a quarter million digits with very limited distribution, followed by billions more digits with equal distribution... we're talking about a big "whoa."

I also find it funny in that it's the skeptics and agnostics who are then brought to the argument, "for without faith, He is nothing." Sure, even the circle-in-square is possibly coincidence, and sure, you could go out to the digits at 11^(-800) and find a similar stretch with a triangle, just based on the statistics. A religious God needs plausible deniability, or so says traditional theology.

Re:Signature of God?

[alienw]

If you have an infinite source of random numbers or you are calculating an irrational number you could potentially have anything in that signal given enough numbers - a circle, a copy of this post, or anything else. You could certainly have 500 zeroes in a row.

Re:Signature of God?

[nusuth]

So trivially, that image of a circle is in there somewhere, as is an image of a triangle, the source to Linux 4.0, an image of Bush playing with G.I. Joe dolls on his desk and so on.

Existance of those does not imply we could be able to find them or identify if happen to come across them. Until we can calculate and scan ~1E250000 digits of pi in a few years, such a finding would strongly support "signiture of god" hypothesis.

Re:Signature of God?

[JanneM]

I don't think you really understood the strength of Pi being normal (if it is). It means all those images, as well as anything else codable in a finite length by necessity exist somewhere along the decimal expansion. The probability for it is one. There is no "whoa" if you find any of those things, as we already know for certain they will be there somewhere.

Re:Signature of God?

[Zone-MR]

Yes and no... Pi and e are not just random numbers found under a rock ;) The current formulas used for them were devised by human beings, and not written in the skies by god. Not to mention that an irrational number will have every sequence you want, repeated an infinite number of times, somewhere along the line.

Re:Signature of God?

[cybercuzco]

But this is already true. Given an INFINITE string of whole numbers, any noninfinite string of whole numbers is included.Digitally encoded, all the books of humanity, all the MP3s, all the pr0n on the internet are found somewhere in pi, Because the finite string of numbers that defines that file is located somewhere in the infinite string of numbers that is pi. Not only that but the file is in every conceivable format that has ever been discovered or ever will be. And every alien file in the universe is in there too. Its the same principle as having an infinite number of monkeeys at an infinite number of typewriters. Not only will the monkeys write all the greatest books of mankind, theyll write every book ever. Heres an interesting infinity problem to ponder: Given a horse race with an infinite number of horses, what are the odds of you winning a bet on a horse? Keep in mind that 1 horse does actually win. What would the payoff be if you won, given the odds are simply calculated at 1/# of horses)

Re:Signature of God?

[Mazzaroth]

Actually, considering that pi is infinite, there is a non-null probability that such sequence occurs. It is even pretty sure... infinite concepts being what it is, it contains all probability, even the smallest.

Now, considering that statistically, this 500x500 of 0 and 1 digit span probably do occur, and we would have a statistical explanation for it, swould it be considered as God's signature?

Re:Signature of God?

[Tim C]

True, but I do think there's a "whoa" factor to finding them, especially if you have to start changing the base and arranging numbers into squares, rectangles, etc.

Just because it's definitely in there somewhere, doesn't mean you'll definitely find it. Pi may be infinite, but you and I are not.

Re:Signature of God?

[JanneM]

No. The rearranging of bases, sizes, images and so on is just diluting it even more.

Say we get a trillion digit sequence. Chances are that if you look long and hard enough, widening your parameters for what's acceptable enough, you will find something. Say you accept not just a perfect (according to some pixellization algorightm) circle exactly filling a 500x500 square in base eleven, but a pretty good approximation of any geometrical figure in any base up to some base and with an image size of anything from 32x32 up to those 500x500 - you suddenly have not just one chance per position in the sequence, but millions. That "Bible Code" scam worked exactly the same way - cast your net wide enough and you can't fail to find something.

Have fun.

Re:Signature of God?

[khslinky]

cybercuzco said: But this is already true. Given an INFINITE string of whole numbers, any noninfinite string of whole numbers is included.

It really bothers me that so many people on Slashdot seem to believe this. The statistical distribution of the digits of pi is still (to the best of my knowledge) an open problem, but the notion that any non-repeating inifinite sequence of digits must contain every possible finite sequence of digits is plainly false. For an example of an irrational number whose decimal expansion does not have this property, consider the number (in base 10) 0.10110111011110111110....

Re:Signature of God?

[nmg]

That reminds me of the old problem I saw on Slashdot a while ago... If you wrote a program to randomly generate an 800x800 image, you'd eventually see a picture of yourself, or a picture of yourself shooting JFK, or a picture of your future children shooting JFK... etc.

Re:Signature of God?

[Anonymous Coward]

C'mon guys, don't you have any idea in math?

Consider of what a probability would be to find such a pattern (250000 specific characters!!!) after just a few billion places in Pi. If anything can be close to zero, this is.

Another problem would be why it is 500 by 500? If these guys love base 11 so much, I'd expect it to be 121x121 or something.

Re:Signature of God?

[Mattsson]

So...
What we need right now is a distributed client to search for the ISO of a Linux 5.6 based distribution so that we can replace windows on the desktop. =-)

Re:Signature of God?

[joeytsai]

I once heard the quote that a million monkeys on a million typewriters would eventually produce Shakespeare. Well, thanks to the World Wide Web, I know this isn't true.

Re:Signature of God?

[cybercuzco]

but pi does include the numbers 0 thru 9, and is not just composed of random digits of 2 or 3 numbers. Even if it were, you could convert it to binary or some other base and extract the information. Take your example, convert the sequence to binary and it does contain any posible combination of binary numbers. Yes it doesnt contain 9 directly, but it does contain the binary representation of 9 (1001)

Re:Signature of God?

[Anonymous Coward]

Proof that God is a male! He used his fingers and his, er... divine rod!

Re:Signature of God?

[quintessent]

Pretty cool. Did Carl Sagan make it up, or is it truly in PI?

Re:Signature of God?

[nhavar]

Wait I thought the signature of God wasn't in pi 3.14... but in phi 1.618...

Re:Signature of God?

[khslinky]

Even including all the digits doesn't get you what you want. Take the irrational number 0.123456789112233445566778899111222333444.... This contains all nine digits (with equal frequency even), but not the sequence 13. As far as changing bases, I'm not really sure what effect that has on the form of a sequence of digits. For all I know, it might be true that you can always find a base that will produce whatever distribution you want, but I've never seen anything saying so (and I don't see any particular reason to think that this is true).

Re:Signature of God?

[Teach]

Chances are that if you look long and hard enough, widening your parameters for what's acceptable enough, you will find something.

Granted. Though a lot of people go from there into assuming that certain things are much more probable than they actually are. For example, though I haven't looked through the digits of pi itself, I feel pretty confident that no 500x500 string of mostly zeros occurs. In fact, the chances of it doing so are so astronomically slim that it would be easier to believe that an intelligent designer had put it there than that it occurred by chance.

The Mathematics of Monkeys and Shakespeare is one of my favorite articles to point intelligent readers to that believe that whole infinite number of monkeys typing would eventually produce Hamlet idea.

Re:Signature of God?

[Lionel Hutts]

It may not be in the first trillion digits, but it's not just in there, it's in there infinitely many times (assuming pi is normal). There are no probabilities about it.

Does anyone seriously doubt that pi is normal?

Re:Signature of God?

[Teach]

You still don't get it. If in fact pi is normal (and the evidence leans this way now), it is in fact *certain* that a 500x500 string of mostly zeros occurs in it somewhere.

Actually I do get it, I just didn't express myself accurately. What I meant was that that 500x500 string of mostly zeroes is staggeringly unlikely to occur in the first trillion digits. In fact, using one of the pi digit searches out there, I was able to determine that even a string of just eight zeroes in a row never occurs in the first 100 million digits, and neither does my ten-digit phone number. And increasing the number of digits to 1.25 trillion only increases the odds by a factor of twelve or so, I think.

In fact, even the string "8479326669" (ASCII digits of "TO BE") never occurs in the first 100 million digits. Yes, all of Hamlet in ASCII would show up if we had enough digits, but it'd probably take more digits than there are particles in the universe and probably take longer than the expected age of the universe to find it using a linear search, even accounting for Moore's law.

My point is, it's unfathomable just how unlikely things can be. Do read The Mathematics of Monkeys and Shakespeare if you haven't. It's quite good, even (gasp!) despite its pro-God stance.

Re:Signature of God?

[sco08y]

...was the notion that the aliens had discovered proof that the universe was created by a higher intelligence.

That'd be evidence, not proof, and it'd be fundamentally no different than a bible.

Re:Signature of God?

[yomegaman]

You still aren't understanding the difference between a priori and a posteriori probabilities. If I take the digits of pi and fool around converting them into different bases, plotting them, whatever, and suddenly I see something that looks like 'Elvis lives', I can't use the a priori probability of that particular combination to prove that it's "impossible" for this to have happened merely by chance. Now that you know it's there, the a posteriori probability is 100% and the a priori probability is meaningless.

Re:Signature of God?

[rmdyer]

Yes, but, it is an interesting exercise nonetheless.

Take a 1x1 grid (pixel) with 1 bit color depth. Choose whether you want the pixel on or off. Once you have made your choice, allow a random source generator to randomly flash that pixel on an off at a given rate. What is the likelyhood in a given time period that the pixel will equal your choice? Pretty darn good for a 1x1.

Now change to a 2x2 grid (pixels). Still a 1 bit color depth. Of the four pixels choose which ones you want on and which ones you want off. This is the same as saying...choose a pattern of pixels in the grid that you want enabled. Once you have made your choice, allow a random source generator to randomly flash those pixels on an off at a given rate. What is the likelyhood in a given time period that the pixel will equal your choice? Hmm, this gets to be a little longer, but still reasonable. Within a short period the random generator chooses a pattern that matches yours.

Now keep expanding on this idea for larger and larger grids of pixels. If you choose for example an 8x8 grid, with the letter "A" as your pattern, the likelyhood that the image will be chosen randomly is very poor. It is almost as if it would be easier to choose a random pattern to match up to a random source than to choose something with regularity like the letter "A".

This is very similar to brute force crypto. It is inconceivable that an 800x800 image with 32 bit depth would ever be randomly selected. But, it nonetheless remains possible. As the rate of choosing randomly generated possibilites goes to infinity, the time required to find such a pattern in a given time period goes to zero.

What does this have to do with us? I suppose not much. The universe is not working itself out using random numbers. The universe is working itself out logically. Whenever a possiblity happens that may have an opposite, the universe splits into two. The universe is working out every logically possible state of existance. Only those things which are totally illogical will not happen. For example, if I have a desk with a coffee cup on it at a given location it is certainly possible that the cup could also be at another location on the desk that might contain the cup. The cup can spontaneously vanish and then reappear at the new location. But this is not very likely within a given time period. It is more likely that the cup will remain ever so close to where its current location is. This is the pixel problem referred to above. Ie, it is randomly easier for the universe to choose an alternate logical choice that is quantumly closer to its current state. This is unless we can provide excess energy to boost the rate at which these choices are made.

This is the connection of E=mc2 to quantum mechanics. Adding energy increases the rate at which quantum processes can randomly choose alternate logical possibilities. Given enough energy, all logically possible states are possible within very very short time frames. At the big bang, every conceivably possible logical state was being worked out quantum mechanically.

Of course there is more to this story that includes black holes, light, Beckenstien bound, and unit circles, but that is a subject for another time.

Have fun! :)

Re:Signature of God?

[Teach]

I can't use the a priori probability of that particular combination to prove that it's "impossible" for this to have happened merely by chance.

Yes, but if something has occurred, I think it's reasonable to look at the relative a priori probabilities of all potential causes to see which one is most likely to be the actual cause. Sometimes the percentages greatly favor one cause over another, and sometimes all causes seem equally (un)likely, and you get to use some other method to disambiguate (e.g. personal preference).

Re:Signature of God?

[commodoresloat]

Why base 11? It's left to the reader to decide, but I expect Sagan wrote it because...

Oh come on. We all know that it goes to 11 because it's louder than 10.

Re:Signature of God?

[cameldrv]

There's a perfectly good way of quantifying this. The key issue here is that we perceive a pattern in the data. The question is how we define a pattern. A very good way is measuring the Kolmogorov entropy of the sequence.

So, find an upper bound on the Kolmogorov entropy of the sequence. In the case of the digit sequence producing a 500x500 image of a circle and a square, I would guess the entropy is less than a couple of hundred digits. This is based on the fact that the program to produce this would be a few lines long in almost any programming language. Now, programming languages are heavily based on our bias towards math, so a turing machine or another more value-free language would probably be a bit bigger. Let's say it's 100,000 base-11 digits to be conservative.

If we make the conservative assumptions that all machines halt and produce unique output, thus providing a bijection between machines and sequences, then the probability of a sequence of length n bits having a machine representation = m bits is 2^m/2^n. 11^100000/11^250000=1/11^150000. Thus in a random sequence, we would expect 1/11^150000 to have such a distinct pattern.

Hence, if we see this pattern significantly before the 11^150000th digit, we should be very surprised. 11^150000 is a huge number, so probably if we find it at all, we should be surprised.

Re:Signature of God?

[xmda]

Well, it seems pi is normal, which means any finite sequence appears somewhere along the expansion of the number. So trivially, that image of a circle is in there somewhere, as is an image of a triangle, the source to Linux 4.0, an image of Bush playing with G.I. Joe dolls on his desk and so on.

Exactly. A friend of mine thought he had come up with an ingenious compression scheme using this. Instead of storing the actual numbers in a file, store the position where that number can be found inside pi!

This sounded a little bit too good to be true though, I supected that to find any number in pi you would sometimes have to scan quite far in the range of endless decimals. And I was right, often the position where the string could be found was a higher number than the number itself and in the end you're not compressing anything at all... :)

Interesting idea though...

Re:Signature of God?

[Ninja Programmer]

• Well, it seems pi is normal, which means any finite sequence appears somewhere along the expansion of the number. So trivially, that image of a circle is in there somewhere, as is an image of a triangle, the source to Linux 4.0, an image of Bush playing with G.I. Joe dolls on his desk and so on.

Actually, it has not been *proven* that PI is normal. It just happens that "most" transcendental numbers are normal, and thus PI, probably is as well.

Re:Signature of God?

[yomegaman]

Probably we are both right and both wrong. In a situation like this, you really can't use a frequentist definition of probability anymore, where you just divide the number of hits by the number of trials. You need to use something like a Bayesian probability, and there we could argue about the proper prior until the cows come home without getting anywhere, since there's no unambiguous recipe for getting the right answer (assuming there even is one). Your comment about 'personal preference' is right on, that's pretty much what it comes down to. I just don't like to see these "the probability of obtaining our universe is 10^(-57), that means there must be an intelligent designer" stories, since that's a frequentist a priori probability and therefore a meaningless number. It doesn't mean there is or is not a God, it just means the author doesn't understand statistics. :-)

Re:Signature of God?

[Hrothgar The Great]

That was probably the funniest comment ever, right there.

Re:Signature of God?

[superyooser]

As any artist, the creator signed the creation.

An artist doesn't need to sign his name if there is no other artist.

I hate to use an oddball example, but this is the only one I can think of right now. Edward ScissorHands didn't need to sign his creations, because nobody else made creations like he did. There was no doubt as to who created those amazingly unique shrubberies. The creations were the signatures! He was ONE OF A KIND. (I didn't actually see the movie, but even if my assumptions about it are wrong, you get the point.)

The universe is the signature of God! No other entity has the powers to produce anything remotely comparable.

Expecting the universe creator's signature to be embedded in the universe is like expecting a logo creator's signature to be on the very logo he created. It's like saying that Microsoft needs to have its signature on its logo; nevermind that the logo consists of nothing but the word "Microsoft". Is there any doubt about which company created the logo which is its owner's signature? Does Bill Gates need to have his name embedded in the logo just to make sure nobody is confused as to its ownership?

The Earth is God's signature. We are God's signature. Look, anything pertaining to God is bigger than we can imagine. The eyes of a gnat will never see the Statue of Liberty (comprehend it beyond being a green solid substance) even if it runs smack into her nose. We encounter God with every breath and blessing of our lives, but we don't see Him because we've become myopic fools from squinting at a trillion digits of Pi trying to find His alleged mathematical signature. This is like a silly British comedy: "Monty Slashdot and the Search for the Holy Pi Signature."

Let me give you a little hint. If you want to see God's name in a written form, you'll have much better luck from studying the meaning of Hebrew letters rather than Greek letters.

Re:Signature of God?

[doubtless]

To quote the conclusion of the 'monkey's paper:

In light of this, I find it impossible to believe that "chance" had anything to do with the process that created life

The most important difference in the pure chance of monkeys typing out Hamlet and the creation of life is the lack of heredity in the former. If everytime a monkey hit the next sequence of character(s) that eventually will form Hamlet, that sequence is recorded and carry into the next 'generation'. Repeat the whole process in a finate number of times... viola, Hamlet!

The most important aspect of the process that created life is heridity. You only need get the 'chance' to stumble upon something that can replicate with the property of heridity. In our planet, it is in the form of DNA.

Re:Signature of God?

[Speare]

I understand your comment, but why do you think our god is the only god? Many authors and theologans have considered that our god has peers, who may have created other universes. Our god is without peer in our universe. Heck, some religions promise we can become the gods of our own universes in our own afterlives if we're "good" enough in this lifetime.

As for how the signature fits into the book, Contact, the signature was there to prove to mankind (and alienkind) that it was indeed designed and not a freak of chance. Knowing there was a creator, beyond the reach of skepticism, changes the creatures' outlook on their universe in many ways. Only the sentient and technically adept species would be able to find such a signature. The aliens just helped Earthlings find it, as circumspect as possible, because the initial radio broadcasts from Earth made the aliens worry about whether we'd grow out of our self-destructive ways soon enough.

I'm not trying to convince anyone to challenge their own theology, I'm just relaying the interesting concept that was in a book, which was on point with the discussion of trillions of digits of pi. If you feel there's only one god, then for you, that is true. It harms me none. If I acknowledge that there are skeptics, agnostics, athiests, and pantheists in this world, it really should harm you none. The "creation" is large and we are two separate points of observation: your point of view is valid, but it doesn't invalidate my different point of view.

Re:Signature of God?

[merlin_jim]

Well, it seems pi is normal, which means any finite sequence appears somewhere along the expansion of the number. So trivially, that image of a circle is in there somewhere, as is an image of a triangle, the source to Linux 4.0, an image of Bush playing with G.I. Joe dolls on his desk and so on.

As a matter of fact, there have even been cryptosystems proposed on that very fact; randomize either your input stream or your cipher stream by finding at what position in pi it first appears. This isn't an encryption method, just a way of making it more difficult to detect pseudo-random number sequences.

The basic problem being, how do you create randomness from mathematical certainty. True mathematical randomness is a very difficult thing to do. Any purely software based method of computing random numbers is susceptible to a sophisticated enough method of prediction, and is therefore not random at all.

All of these cryptosystems are merely proposed and not implemented, because they don't really add randomness while adding a lot of computation. They don't add randomness because they are STILL purely computational methods of deriving random numbers, and even if the numbers are mathematically completely random and indistinguishable from static, there's still a sophisticated algorithm that can predict the next number.

BTW, if your PKI keypair was created without either a hardware random number generator, or you wacking on your spacebar a couple hundred times, you should throw it away and make a new one with a program that makes good keypairs.

Re:Signature of God?

[superyooser]

If you feel there's only one god, then for you, that is true.

I don't know what it means to "feel" truth. That doesn't make any sense. I study and observe truth which points to further truth which must be taken on faith. It is reasoned belief.

Reality is not based on belief; it is for belief to be based on reality. I cannot will something into existence by believing in it. If I believe myself to be a physician, will you let me do surgery on you? Won't it be true if I believe in it? According to your reasoning, yes. I'd also like to believe in world peace. Voila! Now it's true! Now I'm believing that Santa Claus is real. This is cool! And grandly delusional.

It harms me none.

You and I are standing in a busy street. My point of view is that a Mack truck is ten feet away and driving toward us at 60 MPH. My belief is that if you and I and everybody else in the lane does not immediately run to an area of safety, we will all be run over by the truck and die. Somebody glued to their GameBoy will have a different point of view, but the reality of the situation will affect him as equally as it will affect me. The Mack truck will run over you even if you don't believe it's real or harmful. Other examples: Failure to believe that fire is hot will harm you. Believing that you are using Unix when in fact you are using DOS is sure to drive you nuts. Belief won't change your OS. You have to install an OS to change the reality. This is really common sense.

Reality is universal and transcendental. You have to adapt your beliefs to reality. Reality will not adapt itself to your beliefs. It couldn't even if it "wanted" to, because people believe different things. It is what it is, and you will be adversely affected if you believe it to be something it's not.

Re:Signature of God?

[The Famous Brett Wat]

You say:

The most important aspect of the process that created life is heridity. You only need get the 'chance' to stumble upon something that can replicate with the property of heridity. In our planet, it is in the form of DNA.

Bear in mind that your initial life form (which has this all-important property of heredity) must be simple enough to form by chance. I have heard of self-catalysing substances which are sufficiently simple to form by chance, but there is a world of difference between "self-catalysing" and "reproducing with heredity". There is no known DNA-based organism which is anywhere near simple enough to form by chance, so how does your explanation address the issue?

Re:Signature of God?

[ShavenYak]

That'd be evidence, not proof, and it'd be fundamentally no different than a bible.

Well, it'd be a bit different from a bible. A bible is written by humans, pi is not. Thus, a bible can contain lies, half-truths, invalid reasoning, and all the other no-nos you learned about in debate class. Pi can't try to mislead or trick you, it is what it is.

Re:Signature of God?

[superyooser]

Because of the dominance of the Roman Empire, Greek was the lingua franca at the time the Gospels were written. It is believed that Galileans were bilingual, speaking either Hebrew or Aramaic (a precursor to Hebrew) plus Greek. No one disputes that the Gospels were originally written in one of these three languages. Although all of the earliest surviving manuscripts are in Greek, many people believe that they could've originally been written in Hebrew or Aramaic. We know for certain that both Jesus and Mary could speak Aramaic, because the Aramaic words were not translated; the literal words were preserved "as is".

In any event, we are absolutely certain that the original language was not Latin. A complete Latin Bible didn't exist until Jerome's Vulgate Bible in 400 A.D.

While I can't say for sure that the Gospels were originally written in Hebrew, there are hundreds of references and quotes from the Old Testament in the Gospels, which we know was originally in Hebrew. Jesus himself quotes liberally from the Old Testament. Jesus lived according to the Hebrew Torah, and his life as the Messiah, from birth to death to resurrection, fulfills hundreds of prophecies in the Hebrew books of prophecy.

I've personally found Sanskrit more meaningful than Hebrew anyway.

You mean the Vedas? Hindu scripture? I suppose it would be satisfying to believe yourself to be divine. I think that being One with the universe would make for a lonely existence. If everything is one entity, there can be no fellowship. The Hebrew scripture speaks of God as One with whom we can have communion (not union) - a personal relationship. He is the Father; when we put our trust in His Son, Jesus the Christ, God in human form, as our Lord and Savior, we can be adopted as sons and daughters into His family and have fellowship with Him forever.

Re:Signature of God?

[doubtless]

Well, the earliest 'duplicators' are most probably not the DNA we are seeing now. Also, we have got time scale measured in billions of years.

I'm not drawing any conclusions, but this is the best explanation I agree with. Anyway lets not get into the creationist/evolutionist debate.

I love this Quote

[Rhinobird]

I love this quote:

Among the most puzzling mysteries: Mathematicians are pretty sure, but still cannot prove conclusively, that the numbers following 3.141592 occur randomly.

"I don't think we're any closer to answering this question than the Greeks were 2,500 years ago," Borwein said.

Um, you have 1.24 trillion digits of pi. I think you can begin a statisticall analisys now.

Re:I love this Quote

[Dexter's Laboratory]

But what if the resolution of the pattern is 1.24 trillion places?

And if they occur randomly, how the heck can we know that the formulas we're using to calculate pi are correct?

Re:I love this Quote

[WhiteDragon]

Of course, 'randomly' is a bit of a misnomer. If they were random, they would be different every time. What the article should have said is that the digits are distributed normally, or if they thought that sounded too technical, that every digit has an equal probability.

Re:I love this Quote

[Zone-MR]

No. They should say they are distributed pseudo-randomly.

The distribution is NOT normal. The normal distribution states that the mean digit will have the highest probability of occuring, and the further away from the mean you go the lower the probability. This contradicts your next statment.

"that every digit has an equal probability."

This would define a rectangular distribution, and whereas correct, is not the same as saying they occur randomly. "123456789123456789..." has the same probability of each digit occuring, but is hardly random.

Re:I love this Quote

[WhiteDragon]

ok, my bad. I didn't know about the term rectangular distribution, but it makes sense.

How To Calculate Pi

[DrDevil]

You can calculate Pi by doing:

(1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + ..) x 4

Obviously the more iterations you do, the closer you will be to the 'true' value of Pi.

Re:How To Calculate Pi

[wass]

This iteration is correct, but it converges very slowly, so it's not too practical.

Re:How To Calculate Pi

[cafelatte]

Here is a faster converging series:
(16/5 - 4/239) - 1/3(16/5^3 - 4/239^3) + ... = 3.14159

Re:How To Calculate Pi

[apalachicola]

i tried this series to obtain Pi. one thing i observed was that if ur tolerance is 1e-m then it takes exactly 2em iterations to reach the desired value.

Re:How To Calculate Pi

[deppe]

You can also use GNU bc for this:

$ bc -l
bc 1.06
Copyright 1991-1994, 1997, 1998, 2000 Free Software Foundation, Inc.
This is free software with ABSOLUTELY NO WARRANTY.
For details type `warranty'.
scale=50
a(1)*4
3.14159265358979323 8462643383279502884197169399375 08

Re:How To Calculate Pi

[daemonboy]

You've essentially given the power series representation for 4*ATAN(1) which will converge to pi, albeit very slowly I believe O(-4). There are some great formulas due to Ramanujan that have quartic O(4) and quintic O(5) convergence.

In English the series you've given will give an additional correct digit of pi for every 4 (on average) more terms you calculate. A quintic algorithm will give you five *times* more correct digits for each additional term. See here

I believe most of the calculations done now on supercomputers use quartic algorithms for memory efficiency though I don't know what they used for this latest one.

Re:How To Calculate Pi

[wljones]

All values of pi used in calculations are approximations. Taking pi to equal 3.14 is pretty crude, but a lot of schools use it. Be a smartass. Use 355/113 as pi. It is still an approximation, but a good one, and will work on the cheapest four-function calculator.

Irrational (pi != 22/7)

[Omkar]

Some people here seem to bee a little uninformed. pi has been proved irrational and trancedental (duh).

Re:Irrational (pi != 22/7)

[polv0]

While it is an advanced undergraduate proof to show that PI is irrational, this does not mean that the digits of PI are without "order."

First, recall that irrational simply means that PI cannot be expressed in the form PI = P/Q for P and Q!=0 integers. Alternatively, this means that the digits of PI will never form a repeating sequence, such as 237/137 which is

1.729927007299270072992700... (keep adding 72992700)

But this does not mean that PI has no order! For example, a special value of the famous Riemann Zeta Function gives PI by the formula

PI*PI/6 = 1 + 1/4 + 1/9 + 1/16 + 1/25 + ...

So if you want to calculate the digits of PI, just keep adding 1/n^2 on the right hand side. Isn't this an ordering of the digits? Does this have enough meaning to defunct movies such as PI - who rotate around a mythical quest to find some religious numeracy within the digits?

I think so. However, it comes down to this question, how do we define order? I think that order is best defined as the lack of disorder, the lack of randomness. Just as trust is best defined as the lack of distrust. In this definition, we have found order in PI, we have an analytic formula (above) to produce each next digit! Thats as far from random as you get.

Re:Irrational (pi != 22/7)

[coult]

Your formula does indeed compute Pi, but it does not, as you imply, "produce each next digit [from the previous one]". If that were true, then each new term in the series would add a new correct digit to the approximation, and would leave the previous digits unchanged, which it clearly does not. For example, to get 14 digits correct with that formula, you need approximately 10 million terms in the series.

Re:Irrational (pi != 22/7)

[polv0]

I stated only that it produces each next digit, you entered the "from the previous one." As n grows, it leaves unchanged a finite string of consecutive digits following 0. The index of the last digit of this string is a monotone increasing function of n. Therefore, you can use this formula to determine any digit with absolute certainty.

Now, the fact that it takes 10 million terms to get the 14th digit is irrelevant for this conversation. We are talking about the ability to determine the digits of PI, not the practical computation of these digits on contemporary computers. Whether it takes 2 terms or 2 trillion is of no consequence. The formula provides constructive order for the digits of PI.

OMG! That's 4+2 !!!!!

[Anonymous Coward]

42 really is the answer to life, the universe, and everything!!!

I'll be impressed when he memorizes it...

[JohnDenver]

This story reminds me of conversation we had in High School at the computer club about guys memorizing pi up to the 10 thousandth decimal. At which point, one of the less cool geeks, who happened to pronounce DOS, dose, chimmed in enthousiastically, "I once hear of a guy who memorized 30,000 numbers!"

You can bet your ass the room filled up with Louis Skolnick type laughter, along with ribbing along the lines of, "Once I hit 30,000 I stop counting..."

That was BEFORE we had beowulf cluster jokes!

Quiz

[Jacek Poplawski]

What is the best way to waste CPU power?

a) calculating another piece of PI
b) running miss Setia Thome software
c) installing Windows

Share your opinion.

Uh oh...

[NiKnight3]

3.1415926535... ummm... I'll be back in a few thousand years. I have some memorizing to do.

Pi info

[Omkar]

Dr. Math's Pi FAQ. Very informative.

Re:Pi info

[fredrikj]

Also check out Pi at Eric Weisstein's World of Mathematics.

Oops!

[PhipleTroenix]

We've discovered an error in the 175,342,986,462th digit. We're going to have to start over...nah, nobody will ever notice.

Re:You know ... you would think ...

[mdwh2]

Pi is represented usually by a fraction or relatively simple equation, it's just the division that makes the number go on for ever. I don't understand why we must break pi down into a decimal when it can already be represented by a simple fraction.

This is a bit misleading - since Pi is irrational, representing it as a fraction (eg, 22/7) is only an approximation. Representing these divisions usually produce an infinite expansion in decimal (if that's what you mean by "it's just the division that makes the number go on for ever"), but that number is recurring, and thus easy to work out any arbitrary digit since it repeats. This article is about working out the true value of Pi, whose decimal expansion is infinite and non-recurring, and this has nothing to do with divisions.

Taking the equation two divided by three I have found the 100000 trillionth digit ... it's "3"

Yes.. working out digits of rational numbers is slightly easy than irrational ones. Irrational numbers, by definition, can't be represented as the ratio of two integers.

Precise

[dark-br]

Finaly i can be REALLY precise on my circumferences calculations!

the right answer

[svachi]

Is there any way to check if their value of Pi is really correct, short of calculate those trillion places myself? I have heard that long time ago there was someone annouced the value of Pi to around 1000 places, only to be found wrong afterward.

Re:the right answer

[mindstrm]

Yes, there is now an algorithm for calculating the Nth digit of pi (that's just the Nth digit, not all the rest) so you can randomly verify any digit you want.

Also, this IS math.. you don't need to prove the results if you can prove the method used to calcualte them.

If 1+1 = 2, and 1+1+1 = 3, and 1+1+1+1 = 4, I don't have to calculate 1+1+1+1+1 to prove it equals 5.

Re:You know ... you would think ...

[SuperDuG]

22 / 7 = PI

hitting the end

[buttahead]

wouldn't it be great if they found that there was a last place, and that PI is really only 3 trillion places long. how do we know that PI will continue out forever?

Re:hitting the end

[Zone-MR]

The irrationality of pi has been proved (at least until it is disproved by someone later):

http://www.pjmitchell.free-online.co.uk/pi/index .h tm

(Its probably best if you just look at those equations and nod ;)

Form Fitting Condoms

[cyberlotnet]

Now they can make condoms that fit better..

Trojan 2003 condoms in YOUR Size
( We use the trillion place pi method to ensure out condoms are the most precise sized condoms out there, Other companys claim better fit but only use a the million place pi method to compute there condom size )

ummm timothy ...

[SuperDuG]

"A trillion is more than a billion numbnuts ... "

(c) Austin Powers and MPAA and protected by the DMCA

A more straightforward approach

[Waffle Iron]

For simplicity, I prefer to work with the 1897 Indiana pi system. Here's a quick program that quickly prints out pi to any number of digits. It works fine on any PC; performance is bound only by your I/O bandwidth and disk capacity:

#!/usr/bin/python
import sys
sys.stdout.write('3.2')
s = '0' * 4096
while 1: sys.stdout.write(s)

Re:A more straightforward approach

[iggymanz]

The Indiana legislature at the time came within 1 vote of voting pi as "de jure 3.2"? Dumb-ass politicians couldn't even round the correct way, and we allow them to make budgets?

Not sure Pi is irrational?

[SashaM]

Hmm, quoting the article:

Among the most puzzling mysteries: Mathematicians are pretty sure, but still cannot prove conclusively, that the numbers following 3.141592 occur randomly.

Last time I checked, we were pretty sure Pi was irrational, no?

Re:Not sure Pi is irrational?

[Vexler]

I don't remember the exact details, but I think the proof by counter-example of a number's irrationality starts out assuming that a given number is in fact representable by an arbitrary a/b (a divided by b). (This, of course, is the very definition of rationality.) The proof then goes to demonstrate, through a series of algebraic manipulations, how it would then lead to a contradiction if the number in question were in fact a rational number.

Can someone back me up on this, and perhaps even provide the details?

Re:Not sure Pi is irrational?

[SashaM]

Well, I'm sure that's not the only way to prove it, but here's the most popular proof that sqrt(2) is irrational:

Let's assume that sqrt(2) is rational. Then it can be represented by p/q where p and q are integer numbers with no common factors. So sqrt(2) = p/q. Then 2=(p^2)/(q^2) or 2*(q^2)=p^2. Since q^2 is an integer, p^2 must be even. This can only be if p itself is even. So we can write p=2*n where n is an integer. Then p^2=4*n^2. Then 2*(q^2)=4*(n^2) or (q^2)=2*(n^2) which in turn means that q^2 is even and so is q. So both p and q are even. But we assumed that p and q had no common factors, hence the contradiction, so sqrt(2) is not a rational number.

Re:Not sure Pi is irrational?

[buttahead]

smcv (529383) Said :

http://www.math.clemson.edu/~rsimms/neat/math/pipr oof.html

and some other stuff. there is your back-up.

Re:Not sure Pi is irrational?

[mathematician]

A number can be irrational but still have non-random digits. For example

0.101001000100001000001.....

is irrational.

Re:Not sure Pi is irrational?

[SashaM]

Hmm, what is the definition of "random" then? By the definition you seem to be using here, it's quite obvious that the digits of Pi are *not* random, since anyone can calculate them. Furthermore, they can all be compressed by a 100% ratio, from an infinite amount of digits to two bytes representing the characters 'P' and 'i'.

Re:Not sure Pi is irrational?

[mathematician]

The question that interests mathematicians (i.e. the definition of random that I have used) is this. In the base 10 representation of pi, consider the number of digits less than the Nth digit that are 0. Does the ratio of this number to N converge to 1/10 as N converges to infinity? Consider the same question with any other base and/or digit? Numbers with this property are often called "normal numbers."

Nobody has a clue as to the answer to this question.

I believe that there was even a previous slashdot article on this very question.

No, pi is irrational

[smcv]

Pi is represented usually by a fraction or relatively simple equation, it's just the division that makes the number go on for ever.

Nope. If pi was rational (a fraction), it wouldn't go on for ever without repeating. (reference)

In fact pi is irrational, i.e. there are no integers p, q such that pi = p / q. (proof)

You can approximate pi as a fraction, which is what projects like this do. (pi is approximately equal to 31/10, or 314/100, or 31416/1000, or ... but these are just approximations; 22/7 is a good enough approximation a lot of the time, but that's just an approximation too)

Re:You know ... you would think ...

[Pius II.]

While I agree that this kind of precision is rather useless, it is not true that pi can be described by a fraction. If that were so, pi would be a rational number, rather than a real one. Pi is also transcendent (proven 1882 by Lindemann), i.e. it can't be written as the solution of a polynom with integral coefficients.
BTW, under http://pi314.at/math/irrational.html is proof that pi isn't a rational number.

Woopdie doo!

[sopwath]

Just imagine what that 400 hours of supercomputer time could do. I mean, a trillion digits in interesting I guess, but I'd much rather see that processing time go to something useful like protien folding so we can pehaps find a cure for cancer or AIDS or Alzheimers or Parkinsens or ...

Re:Woopdie doo!

[scotch]

So what are you doing to cure cancer, besides bitching about what other people are not doing? Donating money? Computer cycles? Time? Research? Sheesh, there's one in every article.

Re:Woopdie doo!

[sopwath]

I donate money, when I can, and I've got permission to use the lab for Distributed Folding. When I finish my generals, I will be focusing on a biology major. Is that enough whining for today?

why?

[Anonymous Coward]

In all seriousness, there is really no need for Pi to that degree of accuracy.

Re:You know ... you would think ...

[Anonymous Coward]

Taking the equation two divided by three I have found the 100000 trillionth digit ... it's "3"

Actually, if you divide two by three the 100000 trillionth digit would be "6" ... but what do I know ;)

Hidden humor

[mattr]

I found it hilarious that the story "Professor breaks own record -- for thrill of pi" ended with a link named "Subscribe to the P-I".

And well it should! For it is from the Seattle Post-Intelligencer, whose logo is a globe with the initials "P-I". Someone should get those guys to put it on their top page.

Perhaps they held back since it also was posted exactly 61 years after the invasion of Perl Harbor. Oh well.

FWIW, I've been hoping desperately that they'd find some neat geometrical patterns in Pi. My guess is that the reason the mathematicians cannot prove that all those digits are random is that they aren't.. they are just using an extremely good hash algorithm to encrypt the darn thing.

Re:Hidden humor

[Hater's Leaving, The]

Re the US/Japan issue - I was watching the "Ladder to Heaven" Southpark episode last night.
This reminded me of it more than just a little bit.

To explain it would spoilt it - give it a watch.

THL.

Pi are ROUND! Cornbread are square!

[drdanny_orig]

... that's whut Uncle Daddy taught me in school.

We know because it's been proved.

[smcv]

http://www.math.clemson.edu/~rsimms/neat/math/pipr oof.html

That's the great thing about maths, you can prove things like pi being infinitely long without actually calculating any digits.

Is this really necessary?

[Pedrito]

I'm curious. These guys spent 5 years writing the software and then used some 400 hours of computer time on this supercomputer to calculate it. Is there really any advantage besides getting into Guiness to justify this expense? I'm not bashing it, I just don't know. Seems kind of wasteful to me, personally.

Re:Is this really necessary?

[Mattsson]

Everyone needs a hobby...

Search for strings in pi

[Dexter's Laboratory]

http://directory.google.com/Top/Science/Math/Recre ations/Specific_Numbers/Pi/Digits/

I can't believe I find this funny :)

Search for strigns of numbers among the 100 million forst digits

Verification of precision?

[Vexler]

I also posted a question for you all a few minutes ago regarding the "proof by counter-example" of the irrationality of pi, so hopefully someone who remembers how the counter-example goes can post it here.

My question is: How do we know that the value is accurate up to and including the 1.2 trillionth place that the professor claimed? Can someone comment on what the mathematical technique(s) would be used to verify the accuracy of this calculation?

Cartman

[Zaphod-AVA]

Cartman may be round, but even he had to say...

No... more... pie...

-Zaphod

Any books on Pi?

[neema]

Anyone have any recommendations for books on the current theories and the history of pi? I found comments like:

"Among the most puzzling mysteries: Mathematicians are pretty sure, but still cannot prove conclusively, that the numbers following 3.141592 occur randomly."

interesting and want to be able to read more indepth.

Re:Any books on Pi?

[mindstrm]

It's reporter speak for "statistical analysis shows the digit distribution to be completely random, but we don't have a mathematical proof for it"

and it's not just the numbers following 3.141592.. it's ALL the numbers, including those.

Re:Any books on Pi?

[swg101]

Read The Joy of Pi. (They have a website with some info)
It a a small book that give a good background on the history of pi, and much about the formulas and theories used today.

You don't understand maths, then

[smcv]

mathematicians are pretty sure, but still cannot prove conclusively

In the best case, statistical analysis could come up with something like "there is a 99% probability that the numbers occur randomly". That's not a proof, that's just quantifying "pretty sure".

Re:You don't understand maths, then

[Rhinobird]

nope, sorry, I went to a tech school to study EE instead of a real Uni and study math. I can apply it, but the deep math goes over my head. Which makes me wonder, if they can be 99% sure that those digits are random, how many 9's (99.9999999) can they be sure? Or is my question a non sequiter, like what's a mile north of the north pole?

Re:You don't understand maths, then

[jkramar]

I don't believe that your question can really be answered mathematically; it has been proven that the set of non-normal numbers (numbers whose digit aren't completely random) has measure zero. In other words, if you pick a number on the real number line in an entirely random fashion, the probability that you'll hit a non-normal number is zero. However, as we know, pi isn't just any random number. We know its digits don't repeat (it's irrational). However, your question is unanswerable in that an analysis of the already-found digits won't reveal anything about the later digits. If we just look at the set of numbers whose first trillion digits coincide with those of pi, then, as above, the probability that a randomly selected number among them is non-normal is zero. This doesn't constitute a proof, though; although the measure of the non-normals is zero, non-normals still exist. This is a bit counterintuitive; however, remember that you're multiplying 0 not by a finite number but, in fact, by C, which is a higher order of infinity than that of the naturals. Thus, the result tells us nothing and enables us to do nothing more than speculate.

Re:You don't understand maths, then

[embeesh]

When you do a statistical test, you can do it one of two ways - figure out how wrong you can be (say 5%), and figure out if your data passes this threshold. Or, you can figure out the "p-value" of your statistical analysis. This is the exact probability that you are wrong. So, conceivably, you can be realllllly (approaching 100%) but never completely sure.

Please post the findings!

[Glanz]

Would you please be so kind as to post the findings here on the boards so that interested parties may print them out? Thank you!

I wonder how much ink I'll need.......

Balancing costs and pure science

[release7]

In the New York Times article about this story, Kanada has a team of researchers who have been working on this for five years. This undertaking is very sizable in both expense and effort. But when should we end the pursuit of finding pi to the nth digit? If pi is infinite, does this mean the amount of resources needed to calculate pi as accurately as possible is also infinite?

So who sets the limits? Why didn't Kanada just let his computer algorithm run for another year or even just another few minutes to get an even more accurate number? Who decided 1.2 trillion digits was enough and why?

It's just intersting to note that the measurement objective reality is always hampered by subjective, practical matters. And it might also prove that it is impossible for man to ever know the universe---it's just too damn expensive! I'm sure someone out there has thought about this before.

Re:Balancing costs and pure science

[happyhippy]

Does he own the supercomputer? If not then I guess his time booked/rented on it came to an end.

Re:Balancing costs and pure science

[Tim C]

If pi is infinite, does this mean the amount of resources needed to calculate pi as accurately as possible is also infinite?

Unless it repeats, yes - it would take an infinite amount of time to write out an infinitely long number, let alone calculate one.

Why didn't Kanada just let his computer algorithm run for another year...

My guess would be that either Kanada, or someone higher up, set a deadline for the project. At that point, what you have, is what you have - no more calculation. After all, you have to stop at some point; if you're going to extend it by one year, why not 2, or 10?

Also, don't forget that the number has to be stored somewhere, whether in RAM or on disk; perhaps the machine simply didn't have the resources to calculate any more digits?

Re:Balancing costs and pure science

[jkramar]

Pi is irrational. Pi does not repeat.

PiFast

[fredrikj]

You could beat this record with a home computer and some time. Download PiFast and get running. The record for a home computer is 13 billion digits, in 500 hours. On a state-of-the-art overclocked P4 with 2 gigs of RAM, I'm pretty sure you could beat the 1.24 trillion world record within a nominal time. I mean, just in case you feel like you have lots of money to waste.

Re:PiFast

[fredrikj]

I'm not so sure about that. There's no reason why the CPU or any other device in a normal PC would fail. Programs like Prime 95 require over a year of CPU time for testing a single number for primality. If the algorithm and code is correct, then the only problem with calculating huge amounts of decimals of PI is that it'll require more and more disk swapping as you go along.

Actually, a distributed project for calculating PI decimals would be interesting (unfortunately not very realistic though as it's the kind of task that would be extremely hard to parallelize for separate units).

Re:PiFast

[daveed]

I don't think it's possible on normal machines, to calc the first few digits a home computer would do this real quick, the next few, you still wouldn't notice the difference, a good chunk of the 500 hours would have been spent on the 13 billionth digit, so you would have significantly longer than that processing the 13 billionth and 1st digit, then significantly longer again doing the 13... 2nd digit..... and so on.

The time it takes to process one digit is not the same as the next digit, the time increases. So when you get to the trillionth digit, you're spending a stink load of time doing that one digit.

Inversely proportional to Pooty

[japes]

These guys do not get out much do they?

But why 11?

[Hobobo]

Why would he have some prime number of fingers? 12 or 16 would make a lot more sense!

Re:But why 11?

[Kronovohr]

perhaps He, in His infinite wisdom, unzipped his pants, thereby proving (with base 11) that God isn't a woman after all.

Irrational

[LostCluster]

Is it just me, or does this seem like irrational research into an irrational number?

Re:OK, now this is overkill (error)

[wass]

no, you don't need 10^41, you'd need 41 digits of pi. (as shown in the other reply).

But it kind of blows your mind how fast power series diverge, doesn't it?

Re:You know ... you would think ...

[Zone-MR]

Ummmm NO... 22/7 is used as an APPROXIMATION for pi sometimes. And not an accurate one at that. 22/7 = 3.1428... Incorrect after the 3.14. The number is IRRATIONAL.. it cannot be expressed as a decimal nor fraction, and does not reccur.

Re:You know ... you would think ...

[iggymanz]

-1 * LOGe(-1) / i........that's simple, but not the ratio of two integers, of course.

Here's a bunch of simple fractions: (4/1) - (4/3) + (4/5) - (4/7) + (4/9) - (4/11) etc.etc. Repeat to desired precision (this is the slowest possible way to compute pi, you're taking the average of series of ratios of circumference of polygons to their altitude as number of sides increases, get a life!)

In other news

[isorox]

Kanada and a team of researchers

MPAA forces have today invaded Canada, when asked their reasons they replied:

"While we were looking through through the binary version of Pi, and one of our special forces noticed that hidden in from digit 12,166,133,883 onwards was a c source to DeCSS. Obviously these terrorists must be stopped!"

When pointing out that it was Kanada, the researcher, and not Canada the country, the Canadian government sued for trademark violation.

The case is not expected to hold up, as it is doubtful canada will be able to proove it has the computing power to calculate Pi beyond 4 decimal places - and no confusion can occur.

Re:In other news

[nsushkin]

Only Canada needs that many digits of \pi.

Like most of them

[badansible]

This doesn't show that pi is extraordinary: most of the numbers are trascendental... non trascendental numbers are roots of polynomials with rational coefficients.

Why doesn't math deal with Reality well?

[Phoenix]

I'm not trying to be flamebait here, but I'm confused on why Math doesn't deal with reality very well.

Example:

Using Standard measurements, a 10ft length can be split into three equal lengths of 3ft 4in.

Why can't that same 10' length be broken with decimal math? Why is it 3.33333333333...ad infinitum?

Also:
If I were to take a 10' length and bend it on itself so it made a circle I have a 10' circumfrence right? Then in theory I could get out my ruler and measure the radius and get a measurement that made sense. I can get real numbers by measuring, but the math doesn't agree...Why?

Re:Why doesn't math deal with Reality well?

[sopwath]

Decimal math indicates accuracy and precision at the same time. A fraction says nothing about how good your measurement tools are.

Saying 4in. when you're talking about feet is slightly more precise than saying 3/10, but less precise than 33/100.

Fractions are hard for computers to calculate correctly, which is why most programming classes tell you to use integers when you can. When you've only got 24 bits to make that fraction, theres some accuracy problems that can come up.

Re:Why doesn't math deal with Reality well?

[iggymanz]

I have good news for you, you used math to say there are three 3'- 4" sections in 10 feet. And for any precision measurement you make on a 10' length, you can already tell using math what the radius should be if length is bent into circle, without even making that second measurement! See, math deals with reality so very well we can often predict quantities before even directly measuring them. Sure there are issues of accuracy, precision, and imperfections, but even so engineering uses math to predict, measure, and plan in the real world.

Re:Why doesn't math deal with Reality well?

[Anonymous Coward]

In the former case, math does deal with it well; it's 10/3 ft. In the latter case, all measurements have a certain accuracy, and the measurement of the radius you get isn't nearly as exact as saying the radius is 5/pi (assuming, of course, the circumference is EXACTLY 10 ft.)

Re:Why doesn't math deal with Reality well?

[bbum]

Feet are counted base 10, inches base 12. So, you take your 10' length of rope cut it into three pieces of 3 and 1/3rd feet each. Since each foot is 12 inches, the 1/3rd of a foot is expressed in inches as 1/3rd of 12 inches -- or 4 inches.

In some respects, 12 is actually a much more convenient base to work with when doing things like carpentry and other work that often involves taking 1/2, 1/3, 1/4, 1/12 1/6 of the something. In all cases, 1/x where x is 2,3,4, or 6 ends up as a whole number.

With base 10, you basically have 1/2, 1/5, and 1/10th of something.

---

As far as the 10' rope turning into a 10' circle and the math not agreeing w/a radius measurement that makes sense, I honestly have no idea what you are talking about.

The math makes perfect sense in that (a) you can physically see every step in the process of going from a 10' straight line to a 10' in circumference circle to a radius whose decimal representation is slightly odd. Furthermore, it makes perfect sense in that there is a very simple formula that absolutely tells you the radius given the circumference and vice-versa.

Simply because you end up with an irrational [transcendtal?] number does not mean the math doesn't make sense, it just means that our representation of numbers cannot directly represent the math that is at work.

Base 10 numbers are simply one of many possible representations for numbers. That we chose base-10 is not surprising given the number of fingers at the end of most people's arms.

I believe there were cultures-- mayan??-- that chose base-12 or base-60 numeric systems.

In some respects, we might have been better off with a base 12, base 16, base 2, or base 60 numeric systems -- while 10 seems natural to us due to the tremendous cultural history, 10 is not the most natural of bases to work with in mathematics or computing.

Re:Why doesn't math deal with Reality well?

[jkramar]

Well, how long is a 1ft length split up into 7 equal parts in your so-called "Standard measurements"? (feet, inches, etc. aren't Standard at all and aren't used very much outside the US) Right, it's 1 5/7 inches. The decimal number system wasn't designed to deal with thirds; if you know a length exactly (and it's non-integer), you should usually use fractions, because, as you've noted, 0.333333333333333... isn't very practical. Scientifically, decimal representations of non-integers makes sense only when dealing with measured quantities. (All instruments are imperfect.) If you take a 10' length and bend it on itself, then you must be in an ideal world where a 10' length is exactly 10', and the substance of this length has no imperfections. (It must bend exactly evenly to produce an exact circle, and it can't stretch one bit.) Then you could measure your diameter with your perfect ruler...
As you can see, calculating pi by physical measurement is infeasible. Most of the properties of pi that are interesting to mathematicians (its irrationality, transcendentality, relationship with e, and possible normality) can't be verified physically no matter how good an instrument you use. Mathematics in itself doesn't concern itself with the real world; it must be applied.

Re:Why doesn't math deal with Reality well?

[matrix29]

I'm not trying to be flamebait here, but I'm confused on why Math doesn't deal with reality very well.

Example:

Using Standard measurements, a 10ft length can be split into three equal lengths of 3ft 4in.

Why can't that same 10' length be broken with decimal math? Why is it 3.33333333333...ad infinitum?

Also:
If I were to take a 10' length and bend it on itself so it made a circle I have a 10' circumfrence right? Then in theory I could get out my ruler and measure the radius and get a measurement that made sense. I can get real numbers by measuring, but the math doesn't agree...Why?

Um... you do realize that the thickness of your cutting blade throws off your accuracy a TAD. Unless you're slicing by laser or you have diamond nanofiber cutting strand, your measurement will be off by what the ruler shows. In fact if you were to be REALLY picky, you'd notice that your cutting device (even with a near perfect guide) will be thrown off by irregularities in the wood and the blade itself. A laser measurement will show your cutting WAY OFF on the micrometer scales and ever more shockingly off by the nanomeasurement scale.

So if you remeasure your wood you will find that you are missing your full 10 feet if you reglue the wood back together after cutting.

Math divisions are a abstract calculation layered upon an imperfect world where perfect measurements are only an illusion at best and a pure lie at worst. You can approximate the best you can, but if you pour water out of a measuring cup some water will remain adhered to the walls of the measuring cup and some will evaporate in the process of pouring. You will never have a perfect measurement as the act of measurement is taking place in a world where imperfection is the rule.

Re:Signature of God? Probably not

[jc42]

The problem with this argument is that pi has the same value in all possible universes. So its value implies nothing at all about the existence of anything in our universe or in any other.

True, you get different digits if you use different bases. But this is also unaffected by the existence of any god or gods. In base N, you get the same sequence of digits no matter what universe you are in, regardless of whether there's a god.

There is also a conjecture, undecided as far as I know, that pi is what mathematicians call a "normal" number. (Look it up.) If this is true, then the expansion of pi in any base will turn up the pattern that Sagan described. The pattern (and all others) will turn up an infinite number of times, in a frequency distribution determines solely by the number of digits in the pattern.

Has pi been proved normal?

[jc42]

Do you have a reference for a proof of the normality of pi? The last I read, this was still at the "conjecture" stage, though there have been enticing arguments in its favor.

Of course, proof of the normality of e would also suffice, since pi and e are related by a well-known equation that has no other transcendental terms.

Re:Has pi been proved normal?

[JanneM]

No. I did write "seem" :) It does lokk like a pretty good conjecture, though, so I don't think anyone would be surprised if it it proven.

Proof that Pi is irrational

[mathematician]

Here is a proof that Pi is irrational.

On sale now!

[Anarchofascist]

Buy the complete six DVD set!

You'll need to insert all six one after the other next time you #include <math.h>

Re:On sale now!

[Anarchofascist]

Sorry, miscalculation there. 1.25 trillion digits base ten is 1.25e12*log(10)/log(256) = about 5.2e11 bytes = 520GB = EIGHTY SEVEN DVDs.

Order your 87 DVDs now!

Re:On sale now!

[Spamhead]

Yeah, but just wait. In 6 months, Kanada will issue the "collectors edition" with some bonus footage of melting CPUs.

Also included will be an exclusive behind the scenes look at him begging to the department budget coordinator to buy additional memory.

You will also receive pewter slide rules that you can use as book-ends or some such nonsence.

Just remember, this dude gets paid to "geek out" at this level. What a dream job...

Re:On sale now!

[larjon]

I'm glad I don't have to be on call for the nightly builds... ;)

/ L-G

Especially since...

[BSOD from above]

they already held the record. They calculated Pi to 200,000,000,000 places a few years ago.

Its not the size of n, its how you use it.

Thanks

[Pentagram]

Cheers for that. That will save me the trouble of actually reading the book.

there was an earlier /. article on pi

[JeanBaptiste]

that stated that somebody proved each number subset within pi appears as often as every other subset: '123' appears as often as '321' and '213' and '312' and such... it went on to state that this proves that every possible set appears somewhere, and as often as every other set...

this means that any electronic file could be represented as a start and stop position within pi if you knew the proper place to be... in other news MPAA/RIAA declare PI to be illegal...

easier definition

[wass]

Simpler way of equating pi:
pi=4*arctan(1).

Using radian units, of course!

Re:easier definition

[iggymanz]

I did that, but I also got -pi, and a bunch of other values pi radians different from each other.

It's not used for calculations

[cyberlemoor]

Obviously, there are not many calculations that we need a trillion digits for. However, we still know very little about pi, and it is hoped that by investigating more and more digits of it, we might find some kind of pattern, or something that might help us understand pi. The problem has been worked on by mathematicians for literally thousands of years. So, yes, there could be some use for a trillion digits of pi.

Full text of article:

[WilliamsDA]

3.14159265358979323846264338327950288...... ah, this is gonna take a while :-/

Re:Full text of article:

[Teach]

The scary thing is that I knew the next three digits are "419" off the top of my head. The reassuring thing is that I don't know any more digits than that.

In fact, I have this theory that the number of digits of pi you have memorized is inversely proportional to your chances of getting a date. I'm so screwed....

Re:Full text of article:

[alwsn]

Actually, when I first started dating a particular woman I had 100 digits of pi memories (I could write, recite, or type them) and she quickly learned of this fact. Rather than scare her off, she started battling me to see who could memorize more digits... I think with both stopped at 219, realizing it was stupid. Thankfully I'm down to about 60 digits memorized now, though my current girlfriend did embroider my pillow case with 50 digits of pi. So even us pi geeks can find women.

Re:Full text of article:

[Teach]

That's the most beautiful thing I've ever heard. Sound like you two aren't dating anymore... what's her phone number?

Re:Full text of article:

[Anonymous Coward]

(314) 159-2653

Re:Full text of article:

[Fr33z0r]

Very simply, "3"

For comparison...

[MarvinMouse]

A google only has 100 zeros, thus 100 places.

10^trillion is 1 followed by 1 trillions 0's... Assuming we are following the american system that would be equivalent to.

10^(10^12)

Okay... now.. let's get some interesting facts with this.

The absolutely smallest length measurable by quantum theory is the planck length which is approx 10^-34 m. Needless to say, if we have a diameter of an incredibly small perfect circle, we'll know it's circumference beyond what is possible by quantum theory (but since there are no perfect circles, and quantum theory adds probability, this doesn't mean anything really useful. :-P)

Now, since we know the smallest measurable... lets look at what the estimates for the size of the universe are. Recent estimates put it as 10 billion light years in radius source
Which works out to about... (assuming american notation on billion)

10^9 * 300,000,000 m/s* 365*24*3600 ~= 10^25 m

Okay... now if we were to measure the circumference to as accurate as allowed by quantum theory we'd have.

pi*2.10^25 ~= 6.28*10^26 10^27 with an accuracy of about 34 decimals...

So... to get perfect accuracy as allowed by quantum theory we would have at most 35 decimal places afterwards... therefore, we'd need pi with an accuracy of

~10^63...

We have pi with an accuracy of 10^(10^12) which is
63 : 10^12 ~= 1: 1.59x10^11
Way more accuracy then we really need. :-)

That's absolutely insane, but it is fun math.

Just some food for thought.

Re:For comparison...

[yusing]

10 billion light years ...
10^9 * 300,000,000 m/s* 365*24*3600 ~= 10^25 m Damn that new math. When I was a physics major we expressed 10 billion (American) as 1 * 10^10.

Re:For comparison...

[Kaz Riprock]

A google only has 100 zeros, thus 100 places

Not true. From http://www.google.com/press/facts.html:

Employees:
More than 500.

And they're not zeros, they're somebodies and they do an damn fine job at making a search engine.

Re:For comparison...

[Alsee]

he circumference [of the universe]
pi*2.10^25 ~= 6.28*10^26 10^27

Dude! The circumference of the universe is 1!

-

Your sig

[wass]

Actually, right outside my office in the physics department is a poster with a picture of Rutherford and that quote. The way it's written there,though, is (roughly, becuase i'm not there right now) "The only real science is physics. All the rest is stamp collecting."

Modern methods

[cyberlemoor]

There are many methods that can be used to calculate pi, including some listed by others in this thread, although those do not get a lot of digits "fast" enough; better methods are used today. See The Pi Pages for more information.

There is not a pattern

[cyberlemoor]

You can calculate specific digits (in hex) using the formula, without calculating previous digits. That does not mean there is a pattern.

Pi

[Alien Being]

is EXACTLY 3.

Sorry about that. I just wanted to get your attention. Glayvin!

Re:You know ... you would think ...

[SuperDuG]

I betcha if I draw a circle with pi equalling 22/7 and you use the 124 trillionth spot ... they'll look pretty similar, know why? Because calculating pi beyond 22/7 is silly.

A less subtle interpretation.

[JohnDenver]

Were you laughing at his grammar?

Below is an interpretation of the overall joke my post tried to present.

He was a bigger geek than we were. He pronounced DOS, dose, despite constant correction! His statement sounds so stupid! It was stupid! ha ha ha ha ha! It was stupid!

He should have said, "I heared of a guy who memoried pi up to the 30,000th decimal."

Not, "I heared of a guy who memoried 30,000 numbers!"

That's so stupid!!! Don't you get it??? It's so stupid!!!

He was inferior, thus we rediculed him like we were ridiculed by the football team.

This concludes the interpretation. Further layers of humor will not be interpreted...

Bah - another incomplete article

[SpinyNorman]

I can't believe you report this and don't even include the value of Pi he calculated in the article!

I guess I'll have to wait for one of the page widening trolls to post it.

*Much* bigger?

[DeathPenguin]

Sorry, but anything going beyond the hundred thousandth decimal place does not count big., at least not in my book!

No, not even if you're counting subatomic particles.

As Easy As PI

[Cokelee]

&pi;

there, I'm done, call off the research!

woh

[Anonymous Coward]

that would be a killer root pass

Yes, but...

[TomatoMan]

while it's true (I think) that any fininte sequence of digits will eventually appear in a non-repeating, infinite sequence, I think the point in the book was that the odds of our being able to find it, given the tiny tiny tiny tiny tiny tiny portion of the number space we're able to search with our extremely finite computing power, would be evidence that it was placed there if we ever did manage to find it.

Put another way, it would have to be hanging in easy reach for us to be able to find such an insanely improbable thing as (say) a 500x500 block of pre-arranged digits. In base 11, that would be 11^(25,000), a number too hideous to contemplate, and think of the size of the space you'd need to search before such a number would be found just based on probability. So if we found such a thing, we either beat bazillion^bazillion-to-one odds, or we found something that was left there for us. Interesting.

Re:Yes, but...

[JanneM]

No it's not. There are an infinite amount of normal numbers, and for a given position in the sequence, each image will be in that position for _some_ number (for an infinite amount of numbers, actually).

If we stumble upon a 500x500 base 11 encoding of a circle, it's a lucky break (assuming we care about it at all). For any place in the sequence, we have a 500x500 block of something, identifiable or not. We also have 495x495 blocks in the same position (which would be more natural, considering it's divisible by 11) which codes for some image in base 11. The same goes for any resolution, base and image. It's like the bible code - look long enough, cast a wide enough net, and you will statistically find something. it's not proof of anything.

Compare it to hands of Bridge. The chance of dealing a perfect hand is miniscule - yet it's not a message if it does happen to someone (except, possibly, "don't trust that dealer"). Fantastically improbable events happen all the time. If one of those events should happen to be finding that circle (or triangle or whatever) in the known expansion of Pi, then so be it. It shows nothing.

But hey, if you want to base your outlook on reality on a statistical coincidence (that, as you say hasn't happened yet and is unlikely to happen), then who am I to stop you?

Re:Yes, but...

[TomatoMan]

If we stumble upon a 500x500 base 11 encoding of a circle, it's a lucky break (assuming we care about it at all). For any place in the sequence, we have a 500x500 block of something, identifiable or not.

Sure. The question is: how big is the set of things we'd be able to find that would lead us to think there's something significant about it? All 25000-digit strings are obviously equally probable, so it's more a question of naming. What do you call significant?

Yes, fantastically improbable things happen all the time. Your suggestion of the perfect bridge hand as an analogue, I suspect, is off by many many many orders of magnitude. A perfect bridge hand might be a lucky break. A hundred consecutive perfect bridge hands, or a million or a billion, is almost certainly evidence that there's something else at work than just a freak series of random numbers. That, I think, is more along the lines of what Sagan was getting at.

I'd bet all the nickels in my pocket that (for example) the 1.25 trillion digits we already do have probably have some 3x3 or 5x5 "circles" in them, depending on how you interpret them and so forth. Probably quite a few patterns that we would recognize and the superstitious among us might find odd. Anything as statistically astounding as a particular 25,000 digit string that you could name? I really doubt it.

Correct my math if I'm wrong here, I may very well be. Let's use simpler numbers and say there are 1,000,000,000,000 strings of 25k digits in what we have so far. Your odds of picking one and having it be in there would be, it would seem to me, roughly 10^(25,000 - 12) to 1 against. Let's say there are a billion strings we would find significant as evidence of some kind of God voodoo. Now we're at 10^(24,979) to 1 against. Still pretty dang unlikely. A billion billion billion strings that would make us go "hmmm"? OK, make it 10^(24,961). Etc. Such an occurrence should make any scientist worth his salt go "Whoa. What the f*ck?"

This is all airy conjecture, of course. The point I'm getting at, I suppose, is that there's a level of freak randomness, many many many orders of magnitude above the numbers we toss around here, that is so hideously improbable that it's worth revisiting the notion that all freaky improbable things are equally freaky improbable. If we found a 500x500 circle in the digits of Pi, a fork lift wouldn't be able to budge my jaw off the floor.

(Then again, of course, it depends what you want to call a "circle." Chances are that any 500x500 block could be interepreted by some loony as a "circle," or something else significant. I do get that point, and at this point I surrender and acknowledge that the math is way over my head.)

Re:Yes, but...

[matrix29]

while it's true (I think) that any fininte sequence of digits will eventually appear in a non-repeating, infinite sequence, I think the point in the book was that the odds of our being able to find it, given the tiny tiny tiny tiny tiny tiny portion of the number space we're able to search with our extremely finite computing power, would be evidence that it was placed there if we ever did manage to find it.

Put another way, it would have to be hanging in easy reach for us to be able to find such an insanely improbable thing as (say) a 500x500 block of pre-arranged digits. In base 11, that would be 11^(25,000), a number too hideous to contemplate, and think of the size of the space you'd need to search before such a number would be found just based on probability. So if we found such a thing, we either beat bazillion^bazillion-to-one odds, or we found something that was left there for us. Interesting.

Actually, base converting Pi in to Base 11 is actually pretty damn EASY.

Here is the number
3.1415926535897932384626433832795
3 in Base 11 = 3
Now the rest is simple.
Multiply 0.1415926535897932384626433832795 by 11
Take the number past to the left of the decimal point and use this as your first digit of Base 11 Pi. In this case it is = 1.
Subtract that number and multiply by 11 again.
The number you get is 6.

Now if you Wash - Rinse - Repeat you'll arrive at the number in Base 11 (3.16150702865A485235215...)
Pretty simple? You can do this quickly with other bases without hitting negative powers of the base number. You can also convert a number in another base quickly using the technique from my earlier post in this Slashdot chat. The trick is to convert your target base number into the base that you're converting from. It works for all decimals just like RADIX works for all integers. Do a find for "matrix29" on this page and you'll hit my previous post right off.

You can also convert to non-integer bases (ergo Base 7.886) but the method is a tad more awkward.

Re:Yes, but...

[Rubyflame]

while it's true (I think) that any fininte sequence of digits will eventually appear in a non-repeating, infinite sequence

This is false. I will disprove it with a counterexample. Consider the number:

1.01100111000111100001111100000111111000000...

That is, n ones, n zeroes, n+1 ones, n+1 zeroes, and so on. This number is non-repeating. But the string "1010" will never appear in it.

Re:Yes, but...

[TomatoMan]

OK, right. The string "2" will never appear in your example either, since it's defined by a pattern that omits 2s. Is "transcendental" the correct term for what I was referring to, then?

Re:Yes, but...

[ChadN]

No. But possibly "normal" is.

Big deal...

[BSOD from above]

Anyone can write an iterative algorithim to produce a random, non-repeating decimal number to a nearly infinite number of decimal places. In the end this number would be about as useful as Pi to 1.2 trilion places.

You know it is true.

5 years

[alanak]

my question is how exactly does it take multiple people 5 years to create a program to calculate pi. Granted, I have no experience in doing things like this - in fact I have no idea how to go about calculating pi to 30 digits nonetheless 1.3 trillion, but maybe 5 years seems excessively long.

This was from the cnn article.

Re:5 years

[burns210]

calculating pi is not overly hard... my understanding is that it needs a custom type to be put into... integers and what not in programming are only so big, and can only hold a number that is so big... there are doubles, which can hold even larger numbers... but neither evenstart to scratch the surface of a 1.3 trillion digit number.

woo

[nomadic]

Now I finally have the measurements needed to make my cookies PERFECTLY round.

Re:woo

[DarkHelmet]

Me: *CHOMP* Good cookies...

You: Nooooooooooooooooooooo!

the real question

[aberant]

This calculating is all fine and dandy, but the real question is how many digits he can recite from memory? Recite it up to 50 digits and he can truly be king of all nerds!

Re:Signature of God? Probably not

[Fnkmaster]

The normality thing doesn't mean dick. Though true that any sequence of numbers will appear somewhere in the expansion if you had the countably infinite decimals of the expansion all at your disposal. However, as humans capable only of dealing hands-on with the finite, we can only ever obtain a finite portion of the decimal expansion, which thus represents an infinitesimal portion of the entire expansion. Therefore, only an infinitesimal portion of all possible sequences can ever be found in any finite subset of the expansion. If one such subset was so clearly representative of the formula for calculating the number itself, while it could of course be coincidence, it would be a very strange coincidence indeed (and you can calculate the odds of such a non-random ordering occuring based on its size and how far out you had to search for it in the "normal" sequence).

Your assertion that pi is the same in all possible universes seems quite silly to me. Assuming that those universes have two spatial dimensions and that the symmetry of that universe causes 360 degrees to subtend a full, symmetric rotation in those dimensions. In short, just because it represents the only kind of universe you and I can commonly conceive of doesn't mean shit, because everythin on the basis of which we conceive of that is part and parcel of the universe itself, including the laws of physics.

Re:You know ... you would think ...

[Zone-MR]

Well, yes, I agree with that. The whole thing is just a waste of time.

Well, perhaps in engineering, where it is often critical to make very exact calculations, slightly more accurate approximations than 3.14 or 22/7 need to be used.

I was just making the point, that its not a case of wanting to express something as a decimal rather than a fraction. As it cant be expressed as a fraction either if your being exact.

Windows calculator gives me "3.1415926535897932384626433832795", and noone will ever have a legitimate use for more decimal places - simply because they will be dealing with measurments that will never be that exactly done in the first place.

Pi info site

[swg101]

Here is a site that has much information on the history of pi, and how to calculate it. It includes algorithms and rudimentary programs.

See also, the first 10,000 digits!

Square root of Pi

[BSOD from above]

It has just as many places, Pi^2 would be more useful since it might take a while to calculate it if you wanted the area of a circle. Pi^3 for the volume of a sphere. It would take weeks for me to calculate the volume of the earth to 1.2 trillion places.

Harddrive Size?

[notcreative]

I calculate this to mean that you would need about 500GB of harddrive space to simply store the digits. It would costly but possible, and you'd make some really _sweet_ circles.

This isn't the biggest pi!

[magsilva]

Look at http://www.cecm.sfu.ca/projects/pihex/index.html, that effort is resposible for the biggest Pi ever ccalculated! The head news, from 2000 I think, is:

The Quadrillionth Bit of Pi is '0'!

I'm sure that this is much more than this pity 1 trillion digits... pihex has got more than 1 trillion bits before 1999, for God sake.

sequence arguement is false

[Anonymous Coward]

01 001 0001 00001 000001 ... is an infinite, non-repeating sequence of "whole numbers." Find me the subsequence with a "2" in it.

Re:sequence arguement is false

[cybercuzco]

but the sequence isnt random, pi's sequence is.

Re:sequence arguement is false

[Lionel Hutts]

No, it isn't. Pi's digits are the same every time I look.

What exactly do you think "random" means? If "random" means "normal," noticing that random numbers are normal isn't much of a shocker.

not normal

[swg101]

The sequence you propose does not fit the definition of "normal". A normal sequence is one in which all 10 digits appear with equal distribution.

Re:not normal

[khslinky]

True. And whether or not pi is normal is still an open question. The problem is that many posters here don't take that into account: they seem to assume that the digits of pi must contain every possible sequence of digits merely because it is infinite and doesn't repeat the same pattern over and over.

A trillion places?

[pkplex]

I can represent pi exactly, right here: 22/7.

No need for a stupid computer because you can work things out on paper using fractions, and you dont need to round any decimal places, because there are none, and results are exact.

Re:A trillion places?

[Detritus]

355/113 is much better, plus it has a nice structure.

Re:A trillion places?

[dlakelan]

Whoops, not even close to exact.

Pi is "irrational" meaning it is not the ratio of any two integers (such as 22 and 7).

22/7 is about 3.1429 pi is closer to 3.14159

Back to the drawing board.

What if...

[RobinH]

What if we used a beowulf cluster of toothpicks?

It's called Buffon's Needle

[Flamesplash]

You could at least give credit where due ;)

Here's one of the nicer sites I've seen that has a java applet to simulate this.

Ick!

[Jugalator]

The number is the subject of numerous books -- from "The Joy of Pi" to "Sir Cumference and the Dragon of Pi: A Math Adventure" -- and has fascinated and confounded mathematicians for centuries.

Sir Cum-ference? Ewww... I wonder why the weird mathematicans got "fascinated" :-P

Comment removed

[account_deleted]

Comment removed based on user account deletion

Google nitpick

[Anonymous Coward]

"Google" - search engine.
"Googol" - 10^100.

Re:Google nitpick

[quintessent]

Are you sure? I don't remember ever seeing it spelled that way.

Re:Google nitpick

[sameb]

Just ask google's history page.

Damn Kanada!

[edgezone]

Aaargh! I have to take High School geometry all over again after my math teacher retroactively re-graded for accuracy based on the new value of pi.

Thank a lot!

IS there a proof of that?

[mindstrm]

I mean, I hear this repeated a lot. but just because something is infinite and nonrepeating does not mean that every possible combination exists.

Re:IS there a proof of that?

[AmishSlayer]

You have a weak understanding of probability. If there was an infinite sequence of random numbers, then yes every combination will appear. Here is the math

inf * 0.000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 00000000000000000000000000000001 = inf

Re:IS there a proof of that?

[JanneM]

If Pi is "normal" (approx.: infinite, nonrepeating, and each digit occurrs with equal probability) then yes, it is true. The question is of course whether Pi fulfils those criteria. There is a conjecture that it does, and there's quite a bit of circumstantial indications that it is true. To my knowledge it is not yet proven, though.

Re:IS there a proof of that?

[mindstrm]

Right... in other words, if it is truly normal, which is still just conjecture. I understand that well.

I guess I'm trying to point out that many people seem to think that just because something is infinite and non-repeating, that every combination therefore exists, sort of like when you were a kid and fantasized about how, if the universe was infinite, somewhere there was a planet just like this one, except the Canucks were a winning team, and Bush lost...

Reminds me of that commercial...

[weave]

There is a U.S. cable net commercial where the guy is sitting at his computer and all of a sudden a dialog box comes up and says "You've reached the end of the Internet, there are no more pages left to see." and the guy says "Woah, honey, come here..."

Imagine this program screaming along calculating a few more trillion places when all of a sudden it stops. Pi is NOT infinite after all.

Imagine the hiliarity that would ensue (oops, wrong web site...)

Re:Reminds me of that commercial...

[FTL]

>You've reached the end of the Internet, there are no more pages left to see.

Dmoz maintains a list of dead-end pages.

Re:Reminds me of that commercial...

[EaTiN cOfFeE bEaNs]

Correct me if I'm wrong, but didn't DirecTV run an ad for their DSL service with that same principal?

Re:Reminds me of that commercial...

[Kunta Kinte]

There is a U.S. cable net commercial where the guy is sitting at his computer...

It's a DirecTv DSL commercial.

It ends with him going back to his wife's chair, noticeably stunned. Wife: "I thought you where surfing the internet", Man ( still in disbelieve ): "Yeah, but I finished it...", Wife looks like him like he's a nut.

Re:Reminds me of that commercial...

[Soul-Burn666]

Don't wanna be redundant, but Pi was proven to be not rational... any finite sequence of numbers is rational... (the digits after the zero divided by the 10^(num of digits)... like 0.44 = 44/100)

Re:Reminds me of that commercial...

[weave]

You're right, it was DirectTV. OK, so I guess I'm a marketeers worse nightmare. I remembered the ad, but associated it with the competition.

Re:Reminds me of that commercial...

[Kunta Kinte]

You're right, it was DirectTV. OK, so I guess I'm a marketeers worse nightmare.

I only remembered it was DirecTv because I use their DSL. For some reason, I don't think the ad was very good at pushing the brand, it's a funny ad but the brand does not stick.

They need a mascot or a spokesperson. Maybe a talking modem or something :)

Re:Reminds me of that commercial...

[fferreres]

Mh, what do you mean? It can't stop, all it can do is start to repeat a pattern (whatever the pattern is, it is everything except ramdomness).

download pi

[Anonymous Coward]

according to a quick calculation, downloading pi to this many decimal places would cost $7,810.15 (cdn) in over-your-bandwidth charges if you are connected through bell sympatico DSL.

long live pi. down with bell.

Re:This is a waste of time and money

[scotch]

Much like your participation on slashdot? Seriously.

Re:Signature of God? Probably not

[jc42]

Your assertion that pi is the same in all possible universes seems quite silly to me. ... because everythin on the basis of which we conceive of that is part and parcel of the universe itself, including the laws of physics.

Sorry, you're dead wrong here. First, pi and circles have nothing to do with physics. There are no circles (as mathematicians define them) in our universe. Pi is an abstract concept, not a physical object. We can conceive of them nonetheless. The human mind is hardly limited by the physics of our universe. Suggesting that it is is, well, silly, and flatly contradicted by watching an hour or so of Saturday-morning cartoons. I can conceive of things that don't exist in our universe, and so can you.

It's possible that another intelligent species might not conceive of pi. But any that do will come up with the same value (though they may represent it in a different base). Or they may use circumference / radius, giving a value of 2*pi, but that doesn't affect the discussion.

Pi's value is what it is. It has nothing to do with anything in any physical reality. It's a pure mathematical concept, and as such, will have the same value for anyone who conceives of it.

This is really no different that observing that 1 and 2 have the same value in all possible universes. You may name and write them differently, but that doesn't affect their values. Pi is merely another (somewhat more compicated) number. Not even a god can change its value. They can define another value, but it won't be pi.

--

lucky for them.

[kyjello]

Awesome... now those junior high kids can do their math properly for a change.

One problem

[Tablizer]

What if God or Mother Nature decides to change the constant for whatever reason? Kanada will be really pissed because he has to start over.

(I know, it is a definition, not a physical constant, but I can still play what-if.)

Stupidity

[bigox]

The local AM radio station (cough ... 780) said that the exact value of pi is not currently known. Hell, maybe they'll know that when we have the exact value of sqrt(2). I guess old stuffy AM radio farts are just that.

I happen to know

[Ethelred Unraed]

Just how is Pi calculated?

As a matter of fact, I happen to know that this system used a cunning mechanism containing a Canadian-built robotic arm, a No. 10 coffee can, a piece of string and a ruler. The machine measured the circumference and diameter of the can over and over again, and then sort of calculated the margin of error (correlated against 22/7) over and over again. And voila! It was discovered that pi is in fact 3.142857143...

Mind you, the article said they calculated pi to over a trillion places. They didn't say it was *accurate*.

Cheers,

Ethelred

In other news...

[hansroy]

I counted the number of ceiling tiles in the office yesterday while trying to avoid working at 4:45 PM. 77.

Wrong - Try this

[spineboy]

Divide 555,000,000,000,000 by 555 and
you get 1,000,000,000,000
I didn't want to type out a number 500 digits long, but you get the point

A scene from the climactic final battle...

[Mark Garrett]

MPAA forces have today invaded Canada...
When pointing out that it was Kanada, the researcher, and not Canada the country...

Jack Valenti: KA-NA-DA!!!
Kanada: VA-LEN-TI!!!
Valenti: KA-NA-DAAAA!!!!!

Sorry. Apologies to Katsuhiro Otomo.

Yes, half of everything...

[BSOD from above]

1.2 trillion places may seem like a lot, but it is still roughly 0% of the total number. Given that Pi repeats infinitely, Kanada et al. have calculated about the same percentage of Pis total digits as my calculator uses.

Nope.

[starsong]

Area of a circle is

(pi)*( radius^2 ), not (pi*radius)^2.

And, the volume of a sphere is

(4/3)*(pi)*( radius^3 ), again not involving pi^3.

Also, 'cause the Earth isn't a perfect sphere you'd have to do a LOT of measurement. :)

Base Pi

[orcaaa]

Its so much simple to give the answer in base Pi rather that base 10. In base Pi, Pi = 10 :-}

Cheating

[mthed]

What stops someone from stealing the latest pi calculation, tacking on like 3 trillion random digits, then making some phony explaination about how it was calculated. Who checks these things :)

Can DMCA or copyright law be beaten with pi?

[dstone]

Okay, maybe this is a stretch, but hear me out. I believe pi is considered to be normal. See here and here for background on what "normal" means. Essentially, it says the digits are equally distributed over the long run. I believe then, that you can also prove that by exploring sufficiently deep within pi, you will find every conceivable string of digits (ie, in any order you desire and of any length). I think my math is reasonably correct here, but feel free to put me back on track.

Anyways, if this is the case, all digital works are already rendered in pi. All past and future audio master recordings are already in pi. All source and binary distributions of all software are already dumped in pi. Etc.

So the implication is: Am I breaking simple copyright law or the DMCA by computing pi? Am I a criminal for posessing a sufficiently large dump of pi's digits? If I find the rip of a new audio CD in pi, can I keep it?

PI = 3

[PrimeNumber]

Only in Indiana

Better Approximation

[Anonymous Coward]

You can approximate pi as a fraction, which is what projects like this do. (pi is approximately equal to 31/10, or 314/100, or 31416/1000, or ... but these are just approximations; 22/7 is a good enough approximation a lot of the time, but that's just an approximation too)

My favorite fractional approximation of Pi is 355/113 which is (to 14 places) 3.14159292035398.
This differs from the true value 3.14159265358979 by less than 0.00001% while 22/7 has an error of 0.04%.
It is also easy to remember:
start with 113355 (first three odd digits repeated)
break it up with a / : 113/355 and

invert 355/113

I figured out what Pi is!

[aliens]

It's the US national Debt, give or take a few billion.

A very good approximation

[laing]

sqrt(sqrt(2143/22))
I remember reading this in an old Scientific American over 20 years ago. It's accurate to 9 digits which is 3 times better than 22/7.

PI time wasting links

[bedessen]

If you're in a time wasting mood, you can try these:

Search for a string of numbers in the first 100 million decimal digits of pi. Try your birthday, or whatever.

Search for a char or hex string in the binary representation of pi. Find your name in pi, woohoo!

More pi time wasting stuff.

Science or politics?

[MarkusQ]

The plots are given high levels of water, heat, carbon dioxide and nitrogen in different combinations to simulate predicted global climate change in the next hundred years.

Unless there is something more solid that they aren't reporting, this looks more like politics than science. At least, the way they report the findings sounds very skewed:

"The three-factor combination of increased temperature, precipitation and nitrogen deposition produced the largest stimulation [an 84 percent increase], but adding carbon dioxide reduced this to 40 percent," Shaw and her colleagues wrote.

In other words, they are saying that high Co2 levels increased plant growth 40%, but because of their agenda they are reporting this effect as a reduction because it is less than they would have seen if they'd done something else.

A more likely/solid conclusion might be: if the climate changes plants in a given area might not be as well adapted to the new conditions as they were to the old.

And this is news...how?

-- MarkusQ

I don't know

[GunFodder]

I don't know why over a trillion digits of pi would be useful, but I have seen something similar. There was a book I once saw that contained nothing but pages and pages of random decimal digits. I imagine that pi would be quite suitable for whatever purpose this first book was for.

Looks like there is some competition in the random number book business :)

OBLIGATORY SIMPSONS QUOTE

[Cyno01]

Prof. Frink: Pi is exactly 3!*other scientists gasp and snap to attention*

Because of the hidden meaning of Pi.

[wackybrit]

Many believe Pi is some sort of message sent to us by our creators/aliens/Gods/etc. One day we might be able to decode Pi into a meaningful message, somewhat like they did in 'Contact'.

Imagine if we got to the 200 billionth digit and it was like 'So now connect the red wire to the.' and you'd be hitting yourself for not decoding a few more digits.

Re:Because of the hidden meaning of Pi.

[SpaceRook]

Imagine if we got to the 200 billionth digit and it was like 'So now connect the red wire to the.' and you'd be hitting yourself for not decoding a few more digits.

We're an order of magnitude above the 200 billionth digit. Still no message from aliens. Sorry to disappoint you.

Re:Because of the hidden meaning of Pi.

[egreB]

Do we know this? Has anyone actually translated this to ASCII or something? Had been great fun to run it against a dictionary to check for words.

However, I'm fairly sure the number 42 has a meaning inside Pi..

this is such a waste of time.

[supernova87a]

Didn't the Alabama department of education already calculate the value of pi out to a trillion places? 3.000000... and I think their trillionth digit is zero too...

What a timesaver for the kids!

Information theory

[stud9920]

You, Sir, despite your low member number, would get an F- for information theory at the university I was tought and now teach.

There is nothing that compresses to one bit. There is such thing as a most efficient way of encoding any message. Counted in bits. and no, not just one bit. One bit would just contain enough information to say "Pi" or "Not Pi". "Not Pi" would according to my intuition not be an acceptable answer, you also have to say "What kind of 'Not Pi'". And that takes bits. You forgot that your algorithm is supposed to possibly generate all possible messages, or else it's "not fair".

Pi would not compress at all, given it's an infinitely long number. (To be precise, it's length would be reduced from inf to inf/(alphabet entropy) which is still inf, although a "smaller" inf). If you are content with a finite number of digits, its length would be reduced by about a little more than three bits per decimal (because log2(10)=3.???) with any decent entropy encoder. You could try to reduce this further by taking two decimal digits at once, but unfortunately it would not work, as not only are Pi's digits uniformly distributed from 0 to 9, pairs of digits are also distributed uniformly from 0-99, so you would remain with 6.???? bits (log2(100)) per decimal digits pair.

Another approach you might take, if you want infinite precision (silly on a finite machine), or more generally random precision, is to write a code in a predetermined programming language, in this case a series developement, or whatever the number thorists use nowadays to calculate pi, and decide that the "decompression algorithm" is a compiler (that is perfectly legal, as any finite message can be passed that way, eg "#include <iostream> int main(){cout << "The message";}").

My idea is that the c compression algorithm would be beat by a perl compression. Maybe try in BrainFuck, it might beat perl, but BF sucks at multiplications.

Anyway, the most optimal compression for pi is probably saying "Pi" by itself. Any decent geek knows at least one way to calculate that/ find it on project gutenberg/whatever. But don't ever think that you could compress it to two bytes or less : you gotta be sure that I will not understand "the string of decimal digits a.k.a. Pi, do write it in numbers when decompressing", not just "mu turned over", "Pi the string" or "Private investigator". This certainty takes bytes.

Another example is : "you cannot encode '3 4 8 15 3.141592653 78 54' as '3 4 8 15 pi 78 54', because that would increase the number of symbols in the alphabet, and all the other symbols would have to contain more bits as a result, so the compressed message length would suffer- hope there are a lot of 'pi' in the compresed message".

I must leave now, gotta go bowling with friends. Start your flames, I can see blatant holes in my reasonments. Hope you get the point. Mailing a link to the message to my signal theory professor (formally one of my bosses), so I will suffer if I told bullshit.

Re:Information theory

[Jerf]

A compression function is a mapping from input to output. A decompression function maps from all possible outputs of the compression function, back to all possible inputs (though there may be some illegal input to the decompression function). As long as decode(code(x)) = x for any x in the domain, it's a "compression" function, even if possibly a really bad one. There's an infinite number of such functions but most of them are terribly uninteresting. For instance, a particular 'code' might repeat x twice and one of its corresponding 'decode's might cut the input in half again; it meets the definition but we'd never be interested in that.

Different functions perform better or worse in different domains, which is why we have "zip", "gzip", "bz2", "shl" or whatever the lossless audio encoder is, and all kinds of other compressions.

It is trivial to define a function that maps one bit to pi, even if pi is defined as some infinite sequence, instead of a finite symbol representing the infinite concept. You just do it.

Where all numbers are in binary:

decompress(x) = { (the infinite binary encode of pi) if x == 1
what gunzip would do if x != 1 }

Perfectly permissible since "1" isn't a legit gunzip file.

compress(x) = { 1 if x == (the infinite binary encoding of pi)
what gzip would do if x != pi }

For your choice of binary encodings of real numbers that makes sense in this domain.

You seem to have neglected that strings have length, and that just because a given thing compresses down to one bit, does not mean that all things the compression scheme produces will be one bit. In fact, that's impossible for obvious reasons.

There's a perfectly well defined mapping that exists. Of course you can't implement this directly since x can be infinite in this case, and would thence take an infinite amount of time to check if x is pi for the compression case, but it's the same kinda thing as "you can't implement a Turing Machine because you can't have an infinite tape." The function itself, like Turing Machines, is perfectly well defined.

There's nothing unrealistic about this, either; the same principles underly the proof that no compression algorithm can compress all input. You forget that there is no "one true representation" of anything; we can define symbols to mean whatever the hell we want.

(This assumes gzip is defined for infinite input, which IIRC it is, since it's a stream-based compressor; conceptually, there's no reason that gunzip won't perfectly happily run forever on an infinite input, giving perfectly well-defined output, as long as the machine in question has infinite memory.)

Pi would not compress at all, given it's an infinitely long number.

Trivially wrong anyhow, even with your misunderstandings. The people in the article who generated over a trillion digits of pi did not pull them out of their ass; there's a mathematical procedure that produces the digits of pi, as many as you have time to compute. Realistically, that means that pi is compressed as the Turing Machine that spits these digits out, and this Turing Machine is fed to the Universal Turing Machine, which "decrypts" (normally we wouldn't use that word, but a UTM fits into the definition of a decryption function, mapping input to output) the output into the string of numbers. The Pi TM is finite, the output is not. Again, you can't run in finite time, but conceptually, the TM represents all of Pi, given enough time. (It "limits" to it, if you like, as time goes to infinity.)

(The corresponding encryption routine for UTM as a decryption routine is much, much tougher, beyond human capability to perform optimally, and often at all; many interesting things about that have been proven.)

A friend of mine has toyed with a theory of "computable" numbers, lying somewhere between the reals and the rationals. A "computable" number is one where there exists a Turing Machine that will output it, as time goes to infinity. Since there are fewer TMs then real numbers, it's clearly smaller then the set of reals, yet equally clearly, it's larger then the rationals, since it includes things like Pi, e, and, most interestingly, any number we could ever conceivably communicate to each other in such a way that we could construct it. That's the most interesting part of it; it's not the full reals, yet you can't point to a real number or reference one that is not in this "computable" set. Not directly germane, but perhaps interesting to anybody following the posts this deeply.

Anyway, the most optimal compression for pi is probably saying "Pi" by itself.

Ironically, you further demonstrate a decompression algorithm ("simplifying an expression into its decimal equivalent according to the corpus of human mathematical knowlege") that decompresses the sixteen-bit phrase "Pi" into the infinite decimal sequence.

My idea is that the c compression algorithm would be beat by a perl compression.

And what is that supposed to mean, anyhow? Algorithms exist independently of their implementation in a given language!

Your understanding of information theory is skin deep; you recall some of the results but you do not understand the deeper logic. I'm not an expert but I'm pretty confident that this post is accurate enough for Slashdot. (I'd be a bit more careful with definitions and domain specifications for a class assignment, but this isn't, and it's long enough.) The exactly compressions techniques you learned are just a special case that happens to be useful in the real world, not the be-all end-all of compression.

Re:Information theory

[SN74S181]

One of the best 'lossy compression' algorhythms for pi is the expression 355/113, which is accurate to 8 places (it's 3.141592920...). When I discovered how close to pi 355/113 was (with a program I wrote for my SR-56 programmable calculator back in about 1978) I recognized it for what it is: a value for pi that is significantly more accurate than any but the most extremely precise measuring devices for use in the real world. For almost all practical purposes 355/113 is pi. Also, there isn't any other common factor that results in ~pi to a greater precision until you get up into much larger integers. I confess that like probably many others particpating in this discussion I am a pi geek as well. I wrote programs to calculate pi for my SR-56, and it's been one of my 'benchmark' programs to cobble together for all new programmable calculators I've had. Back in the day I was willing to run down the batteries (between tether points where I could plug in the power pack) carrying that SR-56 around in my pocket calculating pi.

Where but on slashdot could we discuss such things? It's good to be home.

Re:Information theory

[stud9920]

3145926/1000000 is more precise and easier to understand

Re:Information theory

[isorox]

Where but on slashdot could we discuss such things? It's good to be home.

Sorry dude, theres geeky, theres nerdy, theres even obsessivly geekishly nerdy, but that pales in comparrrison to someone that spent their youth calculating pi ;)

Re:Information theory

[stud9920]

isn't 355/113 by any chance just a partial sum of the taylor series that converges to pi ? No wonder it's so close...

Re:Information theory

[stud9920]

Where all numbers are in binary: decompress(x) = { (the infinite binary encode of pi) if x == 1 what gunzip would do if x != 1 } Perfectly permissible since "1" isn't a legit gunzip file. compress(x) = { 1 if x == (the infinite binary encoding of pi) what gzip would do if x != pi }

What will gzip-the-decompressor do if it encounters a 1 in the middle of the message ? Was the 1 generated by a "Pi" in the source, or is it part of a non-pi-containing source, that happens to compress to something else than "00000000000000000....".

What will gzip-the-compressor do when it encounters the following as a source : "generic-genreic-pi-pi" ? Which 1s came from gzip-the-generic-compressor ? Which 1s came from "pi" ?

My idea is that the c compression algorithm would be beat by a perl compression.

And what is that supposed to mean, anyhow? Algorithms exist independently of their implementation in a given language!

Even just the same algorithm can be coded purely differently in c an in pl. I expect the c version to be longer bytewise (it's english), shorter once compressed (must have approximately the same entropy per char as english) and faster (it always is). I expect the pl version to be shorter (It's like written chinese--to me anyway), longer once compressed (it's nothing but s or @s or &s) and slower (yes I now it got better). As for the BF version, it will be much longer (go forward by one, go fwd by one, inc 1, inc 1), much shorter once compressed (only 5 chars --> low entropy, many repeats --> RLE). Yes, I forgot to point out, nothing forbis us to add a generic entropy encoder in the codec system.

For your choice of binary encodings of real numbers that makes sense in this domain.

I never chose binary encoding, I barely quantified my encoding in binary, because that's what's been done sonce 1948. I speak nowhere of binary encoding, I only count in bits. But if you realy want a suggestion, just leave the digits the way they are. I do suppose if you take zillions of decimals, you do take the care to store everything as a huge binary number (that's what electronic computers are good at). Decimal, be it BCD or ASCII, would be too much a overhead (BCD is throwing away 0.68bits per decimal-hey, see, a decimal doesn't look BINARY to me, I used the term thoroughly in my parent post-, ASCII is throwing away 4.68bits per char). Don't even think about huffmann, distribution is uniform, huffmann is not good at that

Re:Information theory

[Old Wolf]

Your love of the formulated text has clearly slowed your mind.

One bit is sufficient to encode pi, consider this algorithm (let's code it as a program called "punzip"), that works on a series of bits with a length:

1) if the first bit is 1, return pi [include some small algorithm in the decompression program to generate pi to the desired fixed length]

2) if the first bit is 0, discard it and send the rest of the input to "gunzip".

I'm sure you and I both could write this program in C (or even in shell script), as well as the "pzip".

Re:Information theory

[Old Wolf]

Actually it isn't, for example:

Pi: 3.141592653....
355/113: 3.141592920....

a difference of less than 4 parts in 10 million, or 0.00004%

But:

Pi: 3.141592653...
3145926/1000000: 3.145926000....

a difference of less than 5 parts in 1000, or 0.5%.

So 355/113 is way more accurate than your representation. But in either case, you are missing the point: 355/113 is useful because it is easier to remember, and also importantly, the numbers are small so it is easy to use in calculations where you don't have a calculator.

Re:Information theory

[Cryogenes]

A friend of mine has toyed with a theory of "computable" numbers, lying somewhere between the reals and the rationals. A "computable" number is one where there exists a Turing Machine that will output it, as time goes to infinity. Since there are fewer TMs then real numbers, it's clearly smaller then the set of reals, yet equally clearly, it's larger then the rationals, since it includes things like Pi, e, and, most interestingly, any number we could ever conceivably communicate to each other in such a way that we could construct it. That's the most interesting part of it; it's not the full reals, yet you can't point to a real number or reference one that is not in this "computable" set. Not directly germane, but perhaps interesting to anybody following the posts this deeply.

This is not quite correct. I can communicate a number to you without being able to compute it. One way to construct such a number is via the Turing halting problem. For example, enumerate the set of all Turing machines (there are standard ways of doing this, pick any you like). Then define r to be the real number between 0 and 1 whose nth digit (in binary) is 0 if the nth Turing machine halts and 1 if it doesn't.

This is a well-defined real number. It cannot be computed by a Turing machine because that would solve the halting problem.

I am not an expert in the field, but I believe there is a whole hierarchy of sets of definable numbers, depending on the language you permit to be used in definitions. And, like with Gödel, no matter what language you take, there will be some definable numbers that escape you.

On a related note, consider the following delicious paradox.

Let M be the set of all natural numbers that can be defined in less than 300 bytes. Let n be the smallest natural number not in this set.

Re:Information theory

[Migraineman]

Your assertion that "nothing compresses down to one bit" is inaccurate. A variable-length coding scheme will compress down to one bit, and is simple to demonstrate. For the following compression scheme, I'm going to define a simple set of code words (I don't feel like typing a lengthy dissertation here, and the point should be sufficiently obvious):

1=123456
01=116
001=97985
0001=abcde

So, for an input of "11697985", the compressed output is "01001". For an input of "123456", the compressed output is "1" - a single bit.

While this system does not result in a finite number of code words, and does not necessarily result in a smaller "encoded" output, it is still mathematically valid. In signal processing, it is common to change a calculation from the time-domain to the frequency-domain. The main purpose behind this translation is to allow a finite calculation on what would otherwise require an infinite amount of computation. Just because the calculation may involve "infinite series" in one domain doesn't require that it be declared "impossible." A simple, finite frequency in the frequency-domain has an infinite expansion in the time domain. A finite event in the time domain has an infinite expansion in the frequency domain. Find a domain where an event compresses, for lack of a better term, down to a finite element. We just haven't found the correct domain to translate Pi down to yet (outside of the base-Pi domain, which is just a different flavor of "not useful.")

Now if you want to revise your position from "theoretically impossible" to "not terribly useful," that's a whole different discussion.

Re:Information theory

[jomagam]

Information theory is about streams of data that you do not know before compression. You can compress anything into 1 bit like this:

compress(x) = if(x==PI) {1}
else { "0x" }

Re:Information theory

[swillden]

A friend of mine has toyed with a theory of "computable" numbers, lying somewhere between the reals and the rationals. A "computable" number is one where there exists a Turing Machine that will output it, as time goes to infinity. Since there are fewer TMs then real numbers, it's clearly smaller then the set of reals, yet equally clearly, it's larger then the rationals, since it includes things like Pi, e, and, most interestingly, any number we could ever conceivably communicate to each other in such a way that we could construct it. That's the most interesting part of it; it's not the full reals, yet you can't point to a real number or reference one that is not in this "computable" set. Not directly germane, but perhaps interesting to anybody following the posts this deeply.

First, your friend's idea is either fatally flawed, or he has made a breakthrough of fantastic proportions, because this set of "computable" numbers would have a cardinality between that of aleph_0 and c, violating the continuum hypothesis. It's probably a reasonable working assumption that he has not, in fact, revolutionized set theory.

The first flaw that jumps out at me arises from the fact that TMs need input, which means that the set of possible TM outputs is the same cardinality as the power set of Z, which can be shown to be c.

You can eliminate that flaw by restricting the input to a fixed value (say, the null string). On the surface this looks good, but it sounds like you and your friend are assuming a result that is not true. In particular:

That's the most interesting part of it; it's not the full reals, yet you can't point to a real number or reference one that is not in this "computable" set.

This is a strong statement, one that must be proven. How do you know that you can't find a real that is not computable? The fact that Pi and e are computable means nothing, it's easy to construct a mapping from the integers to the integers plus a countably infinite number of additional numbers (i.e. all the irrationals you can name). And that is precisely what you have done if you fix the input to the TMs: constructed a particular enumeration of the integers plus a countable set of irrationals.

And, as I said above, if you don't fix the inputs, and consider the results from all possible inputs to all possible TMs, then you've defined a mapping that covers the reals.

Re:Information theory

[Jerf]

Touche. I was too glib. Good catch.

Re:Information theory

[fferreres]

Pi would not compress at all, given it's an infinitely long number.

Thus, reproducing the original only takes an infinite number of known operations. Gratend, you can't compress pi. Or you can say you can, but what you can never do is see finish uncompressing it in finite time.

This brings us back to patterns or celular automata or the like. Can complexity arise from patterns? I mean, does pattern X originate from a finite set of rules, or is just noise?

You can't really compress white (for example) noise, but you can compress pi. In fact, the math we know is already that, a compression mechanism that encapsulates the actual irrational number, though everybody knows how to turn Pi into a supposedly infinite string of decimals.

Re:Information theory

[Jerf]

What will gzip-the-decompressor do if it encounters a 1 in the middle of the message ?

This is a stupid question in two ways:

1. 1 only decompresses to pi if the message is of length one, and == 1. Only if this is not true do I invoke gzip. gzip under my definition is exactly the same as the gzip on your hard drive (except running on an infinite memory machine).

2. You clearly don't understand anything about how gzip works. gzip-compressed text is not even remotely substring-invarient... a 1101110 string in one part of the compressed file may mean "Hello!", and in another part of the compressed file may actually be parts of three tokens, or merely a part of a larger token. Thus, your question What will gzip-the-decompressor do if it encounters a 1 in the middle of the message ? is just about meaningless; the answer totally depends on the context it is encountered in, since on average it's encountered roughly half the time. The question of "What would gunzip do if it encountered a (anything here)?" is a valid one and had to be answered before it could be written! Well-defined answers exist.

As for the rest of your "What Would Gunzip Do?" questions, I suggest you run the program and find out for yourself. Alternatively, consult RFC 1951 and RFC 1952.

Even just the same algorithm can be coded purely differently in c an in pl.

Oh, so you mean compressing source code. The sentence was incredibly ambiguous: My idea is that the c compression algorithm would be beat by a perl compression. sounds like "c compression algorithm" is a compression algorithm written in C.

Somebody took something similar to that idea and ran with it: You may want to look in Google for some programming comparisions based on taking a benchmark task in many different languages, gzipping the code, and comparing that size, instead of the raw text size. The idea being that the gzip would tend to factor out the verbosity differences and touch on the actual complexity (though of course it's far from a perfect match, and it's hard to even define the relatively complexity of two implementations in two languages in a way that captures everything we intuitively mean, if you think about it). Interesting results.

I never chose binary encoding

"Your choice of encoding" means here that it's true for all encodings of binary real numbers that are reasonable for infinite-length numbers. Sorry, didn't mean you personally.

Re:Information theory

[Jerf]

First, your friend's idea is either fatally flawed, or he has made a breakthrough of fantastic proportions, because this set of "computable" numbers would have a cardinality between that of aleph_0 and c, violating the continuum hypothesis.

No. A computable number is defined as having a TM that will output it, though possibly in infinite time. Thus they have the same cardinality as the set of TMs, which is the same as the set of integers. They are interesting only because they seem to give us all the practical (and I can't emphasize that strongly enough) usefulness of the reals while technically only having the same cardinality as the ints.

You can eliminate that flaw by restricting the input to a fixed value (say, the null string).

You can do that "without loss of generality", to use the math phrase. Figuring out the transform for "TM + input" -> "TM" is left as an exercise for the reader.

We do this all the time in proving things about computability; since we can just suck the input into the TM, it removes one (useless!) variable from the proof, which makes them that much cleaner.

This is a strong statement, one that must be proven.

Well, yes and no. It's an English statement, not a math statement, so proof would tend to look like proof by definition. It would basically run as "By the act of pointing to a claimed incomputable number, you are either showing me how to compute it, or you are not pointing at a unique, well-defined number. A sibling to your post did construct a unique, well-defined real number that is not in the computable set to my satisfaction, so to the extent that my phrase had any mathematical meaning, it has already been contradicted. However, that was my error, not my friend's.

BTW, note that nobody claims these "computable numbers" are good for anything; it's mostly a thought experiment. I tend to see it as a nifty demonstration that the integers are more flexible then many people give them credit for.

Re:Information theory

[dvdeug]

For almost all practical purposes 355/113 is pi. Also, there isn't any other common factor that results in ~pi to a greater precision until you get up into much larger integers.

I'm not near my computer, so I can't find the exact numbers, but I don't remember that being true. I wrote a BASIC program that showed me, for every integer denominator, the fraction closest to pi with that denominator, if it was more accurate than the last fraction shown. 22/7 is amazingly accurate - there's no fraction more accurate than 22/7 before 355/113. But 355/113 is followed by a string of fractions each with a slightly larger denominator and slighly more accurate.

Re:Information theory

[swillden]

Thus they have the same cardinality as the set of TMs, which is the same as the set of integers.

This statement contradicts what you said in your earlier post:

Since there are fewer TMs then real numbers, it's clearly smaller then the set of reals, yet equally clearly, it's larger then the rationals...

The above statement, of course, is incorrect (since the cardinality of all three sets [TMs, rationals, integers] is the same: aleph_0). Whether you changed your mind or were simply more precise the second time around, you appear to have it right now.

Re:Information theory

[BMagneton]

Curious.

It must be said, IANAM(athematician).

(Commence massive hand waving) The more I think about it, the more intuitive the connection becomes between discrete states of Turing machines and being restricted to the realm of the countably infinite.

If I'm thinking about it correctly, the idea of a distinct class of countably infinite computable numbers is fascinating. With them, you can partition out the computer output of your choice from what the computer is trying to produce or represent. Pseudo-random number generators and simulations of physical phenomena are two things I'm concerned with in particular, and they both try to represent something in the uncountably infinite domain with something that isn't.

Thanks for the thought-provoking post.

BMagneton

Re:Information theory

[SN74S181]

Pardon me, but you've wandered into the wrong discussion to make fun of pi fanatics. Tread lightly. ;)

Again?

[Felipe Hoffa]

Are we starting this story again? The fact is that there are a lot of files that CAN NOT be compressed. Period.

See the rationale about it, thanks to the guys at news:comp.compression. There you will find the story behind some scams involving 'infinite' compression or 'universal' compressors.

Fh

Uh Oh!

[Bald Wookie]

They're gonna get my PIN number for the ATM machine...

It's the last four digits of Pi.

Re:Signature of God? Probably not

[trolleri]

You're the one wrong here.
The universal math that we are observing can only be used to describe relations within its scoop - they must be suitable for its nature.
And that, in it self, does not conclude that there is only one kind of math with one specific nature(nature as in "nature of math").

Ex:
The values 1, 2, PI and so forth that you're talking about are possible to describe with our math, since it's suitable for its nature.
But with a different type of math with a different kind of nature, it might be impossible to describe them, or describe them as a concept even (=> the void math for instance, where no relations can exist)

'nuff said

Why not

[daemonboy]

The practical answer is that it can be used to verify the accuracy of the supercomputer it is run on.

The other answer is why not. Its a challenge just like any other. The mathematics behind the formulas that are used to generate thse digits are exremely elegant and can be very interesting. Many of the most startling discoveries of formulas to compute the digits of pi were made by the indian mathemetican Ramanujan who died in his twenties and had no formal mathematical training.

A good start at exploring why this can be interesting can be found at The PI Pages

Finite vs Infinite

[belphegore]

The whole point of the monkeys/typewriters/Hamlet thing is as you state -- an infinite number of monkeys typing would *definitely* create Hamlet. The link you point to misstates the issue as being "enough monkeys" and "enough time" -- both of which imply a large but finite number of monkeys and time. The entire point of infinity as a concept is that your rational ideas of what constitute sound integer arithmetic break down somewhat when you talk about infinity.

In fact, if you had an infinite number of monkeys typing, you'll in fact have an infinite number of them typing out Hamlet, as well as an infinite number of them typing out every other thing which has ever been written. An infinite number of them will also be typing out Hamlet exactly right, except they are all misspelling "Ophelia" as "Ophilia". Another infinite number of them are writing a treatise on inifinity, etc.

Infinity is not merely a really big number. It's special.

Re:Finite vs Infinite

[Teach]

You are correct. "God's Signature", the string of 500x500 mostly zeroes, definitely appears within the infinite digits of full-blown pi. In fact, it does so many times.

I just worry when people who don't understand the difference between really big numbers and infinity try to apply similar logic to real-world events. They'll say, "Oh, an infinite number of monkeys will eventually produce Hamlet! Well, the universe is very big and has been around a very long time, so it seems probable that X would eventually happen by sheer chance."

Quoting from the Monkeys article linked in one of the parent posts, even if we assume "17 billion galaxies, each containing 17 billion habitable planets, each planet with 17 billion monkeys each typing away [on a simplified typewriter containing only 32 keys] and producing one line per second for 17 billion years, the chances are 99.999999999995% that the phrase 'TO BE OR NOT TO BE, THAT IS THE QUESTION.' will not appear in the output."

You see, for most X, the probabilities are so slim as to be miraculous. I just hate seeing normally intelligent people who know something about infinity unwittingly believing that such miraculous things are probable while at the same time calling others naive or gullible for believing different miraculous things.

Re:Finite vs Infinite

[IHateUniqueNicks]

Sorry dude, but you forgot to check out the type writers. They've only got one key.

Re:Finite vs Infinite

[Idarubicin]

You are correct. "God's Signature", the string of 500x500 mostly zeroes, definitely appears within the infinite digits of full-blown pi. In fact, it does so many times.

Actually, if pi is normal, then the string we're looking for will appear an infinite number of times.

Staggering, isn't it?

How do they know they got it right?

[RodeoBoy]

I guess it doesn't matter though because nobody can prove them wrong and there is no practical use for this.
Now can we get back to useful thing?

addendum on "larger then the rationals"

[Jerf]

BTW, I use "larger" in human intuitive sense in that case: The computable numbers is larger then the rationals because the computable numbers contains all rationals, plus more numbers.

Of course mathematically, both sets are the same size, the cardinality of the set of integers; we can talk of Turing Machines running forever but not of "infinitely long" Turing Machines, which is counter to the definition.

(Which highlights the interesting point of that idea, that all the numbers we ever use are still just the integers in a very real sense, even when we talk about "pi" or "e". Not necessarily groundbreaking stuff, but interesting to some of us math wonks.)

I post this in an effort to forstall the inevitable "correction"... ;-)

Re:addendum on "larger then the rationals"

[Jerf]

Mathematically, and in a very provable sense, the cardinality of the reals is greater than the cardinality of the integers.

Yes, I know. You miss the point: All computable numbers, by definition, map to integers, because the TMs do. Thus, there are countably infinite computable numbers, despite the fact that those computable numbers include reals, transcendentals, etc.

Not all reals are computable; in fact uncountably many of them are not, because there are more reals then integers in every mathematical sense.

How do you know those calculations are correct?

[daemonboy]

One of the ways is to use otherwise seemingly useless calculations to verify the given supercomputer is calulating correctly. And calculating the digits of Pi is one that can be verified relatively easily by other algorithms or computers.

If the calculation of the model simulation for that cure for cancer is off in the 1.4trillionth digit calculated, compound that millions of times and your cancer drug could be the next thalidomide.

Re:How do you know those calculations are correct?

[sopwath]

Good point, but it sounds like they used that 400 hours in between the other work they were doing, so they obviously trust the machine enough.

Re:Precision

[pediddle]

60 decimal places of what unit? Meters? If so, that's fine. But if your unit is meters*10E82, then 60 decimal places is not all that much.

Pi is equal to an even 3 apparently

[Madman27]

I can't remember the details exactly, but a while back, a senator tried to have Pi officially changed to 3 because that was what was stated in the Bible.

Just be glad he wasn't a math teacher.

Re:Pi is equal to an even 3 apparently

[DaCool42]

I'm pretty sure the Bible doesn't say anything about pi. In anchient times, however, it was widely accepted to be exactly 3.

Re:Pi is equal to an even 3 apparently

[Madman27]

This is from the Old Testament. (found in my History of Pi book).

"Also, he made a molten sea of ten cubits from brim to brim, round in compass, and five cubits the height thereof; and a line of thirty cubits did compass it round about."

Egyptians and Babylonians had a much better approximation of Pi, long before the Bible was written. The Babylonians calculated Pi to be 3 1/8. The Egyptions had it at 4 * (8/9) ^ 2.

I'm not saying that the Bible was making an absolute claim that Pi was 3, as they're attempting a general description of an object in the quote. But my main point was that this senator took it literally, and I'm sure he had a large backing of complete idiots.

Re:Pi is equal to an even 3 apparently

[Ziviyr]

Nifty, if you walk the line inbetween the Gypts and Babs its only off from the real Pi by ~.001154! More accurate than 292/93 (~.0018077 differential).

Neither of them are easier or more accurate than remembering 3.141 though...

Oh! 355/113 is only off by ~.00000027, competing with 3.1415926.

Do I get a cookie?

Re:Precision

[Deadstick]

Well, um, yes...so what does that have to do with pi? Pi is not defined in terms of any physical measurement, except maybe in shop class.

rj

If you don't think Pie recurrs...

[telstar]

aparently you've never eaten Thanksgiving dinner at my place. Give it an hour or two, and you're bound to see it make a reappearance...

Re:Signature of God? Nah....just randomness

[efuseekay]

At first when I read Sagan's book at the impressionable age of 15, I was dumbfounded by this idea.

Now I am older and more cynical, I became somewhat disappointed that good old Carl himself have fallen into his own trap of "hiding signatures" in randomness. Basically, if you look hard and long enough into a series of random numbers, you might find an apparently "unrandom" event, perhaps the 1432323th decimal place of PI spelling out "God is Here". He had himself written about this in his book The Demon Haunted World.

In science, you only cares about experiments that are repeatable, or at least statistically sound if not repeatable (e.g. The Big Bang happens only once but...). Finding a circle in PI is exactly the kind of unrepeatable, unpredictable idea that is beyond the realms of science.

So that's too bad.

BTW, Sagan could not have used the motivation from String Theory since at that time he wrote the book 11-D ST has not been invented yet. He probably used base 11 because you can paint an ASCII picture with 0 and 1. (Base 2 is the other common example, wonder why he didn't use it.)

Think about it more...

[efuseekay]

Dude, they measure it to 1.24 Trillion, not 10^(Trillion).Someone had pointed that out, but...

If you think about it, you could not have fitted the entire observable universe with enough paper to record (even if you write in very very very very small fonts) the number of decimals if you know PI to 10^(Trillion).

In fact the entire observable universe had about 10^120 atoms. So you are out of luck very soon. (You can imagine packing more atoms, but then the universe will become too dense and collapse on herself so fast you won't have time to expand to her current volume).

Re:Think about it more...

[Anonymous Coward]

Think about it again yourself...

I was saying that there are as many zeros in 10^trillion as there are decimal points in pi.

My calculations work.

Re:Think about it more...

[efuseekay]

Which is also wrong of course, since pi is an irrational number,i.e. it has formally infinite number of decimal points.

Re:Think about it more...

[Guppy]

"In fact the entire observable universe had about 10^120 atoms. So you are out of luck very soon."

No problem. I just whip out my handy dandy quark notcher, punch a whole, and double my capacity!

What about e?

[kazad]

Has e been found to more decimal places? Pi and e are so related [Euler's equation, e^(i * pi) + 1 = 0], I wonder if precision in one will lead to precision in the other.

Re:Signature of God? Probably not

[rmdyer]

I don't think you are correct. There are no other possible "kind(s) of nature". There are only those that are logically consistent. There are no black white things. There are no infinitely heavy stones that can be lifted by an infinite power. Only those things which are logical are possible. The universe is seen as a giant quantum mechanical machine that is working out every conceivably possible logical outcome. Whenever the possiblity exists that there can be two outcomes of a situation, the universe bifurcates and continues to follow both logical paths. Math is universal because it is based on logic even if it is only a tool. All thinking beings that logically exist will observe the same quantity of PI no matter what base. Beings that don't logically exist, like tall short people don't think.

More than 'nuff said.

Re:Signature of God? Nah....just randomness

[rmdyer]

This was a fictional story. I'm sure Carl didn't really think that any signature existed in that sequence. You might not recall, but he was heavily influenced by his wife at the time. You know how irrational women can be.

You are right. The entire bible and koran could be spelled out in the sequence, but so can the ingredients on cereal boxes.

Interesting stats

[JoeRobe]

I just pulled up Mathematica and ran some amusing stats:

Assuming that 2000 characters can fit on a 8.25 x 11 inch page, you can print 10 pages/second, a page is 1 micrometer thick, you can print 2000 pages/toner cartridge, and you can speak 2 numbers per second...

Printed pages: 6.2 x 10^8 pages (620 million)
Printing time: 117.96 years (excluding leap years)
Stack of printed paper: 62 km high
Toner cartridges: 310,000 cartridges
Time to speak the entire number: 19,660 years
Length of a continuous-page printout (ala dot matrix): 170,500 km, which could go around the earth 4.25 time, or get us halfway to the moon.

Feel free to check my work, or to add stats to this:?)

Utility is not the importance

[Flamesplash]

I completely agree that this is not a useful method for calculating pi.

However; I think it is very very important in the shear fact that it exists. Mathematics is represented all around us in nature in ways we simply don't see. This is one of them, and I think people should realize that there is a logical underpinning of how nature works, even if we will never comprehend it

Post went to wrong article

[MarkusQ]

This post was supposed to go here.

-- MarkusQ

Re:Precision

[Old Wolf]

He clearly meant 60 significant figures, which is just as precise regardless of what units you choose.

Re:Signature of God? Probably not

[forgotmypassword]

You talking is full of jibberish.

Here is the raw deal.

Math exists without need for our physical laws. There are plenty of branches of mathematics with little to no relation to our real world.

The math that does apply to our real world in the form of physical laws, are only mathematical models. None of physics is exact, and it has never been.

For example: Pi cannot be found exactly in this universe.
Pi does not exist in our world embeded in spheres. You cannot construct a perfect sphere in the real world because once you hit the plank scale you can no longer have perfect classical smoothness.
Pi does not exist in our world embeded in oscillators, for much the same reason.

Pi only exists in math, in idea. The fact that we can use pi and math to model the universe to some degree of approximation is only a testiment to how ordered and logical the universe is.

So though you may have not realized it, the poster had an important point. Pi and math would be the same in all universes, because Pi doesn't exist in the universe any how, they only exist in our minds.

Not only pi is interesting, also different sqrt's

[alfaiomega]

I once found a website (it was at least 5 years ago, I can't find it now -- if anyone knows the URL, please post a link) from which I could download (and I've downloaded -- it took me some time on my 14.4kb/s modem) 500000 (if I remember correctly -- or was it a million?) decimal places of the square root of 9. Too bad it was uncompressed text file, because -- which is quite an interesting property of this value -- it can be compressed somehow better than pi, which I found out later (but I can't find my benchmarks now).

no

[forgotmypassword]

If you want the best fraction, then use some finite terms of the best rational approximation.

I believe the first few approximations go like

22/7 Archimedes
333/106
335/113 Tsu Chung-chi
103993/33102

Interesting enough Tsu called 22/7 the inaccurate value and his value the accurate value.

Re:Not only pi is interesting, also different sqrt

[JoeRobe]

Square root of 9 to 500000 places? wouldn't that just be 3. with a string of 500000 zeros after it?

If that's the case, it can definitely be compressed!

A comparison

[the_truk_stop]

If only the editors of /. could emulate the properties of pi...you know, not repeating, and all that.

Re:Signature of God? Nah....just randomness

[ChadN]

Right. If we actually found such a "signature" in Pi, even if we could prove Pi was normal, and that we should find any length sequence of digits in it, I would be astounded enough to have my faith in Atheism shaken (back to agnostic probably).

Knowing that the sequence does exist in Pi, doesn't change the fact that actually FINDING such a long sequence would be remarkable. We have to deal with the physical limitations of exanding Pi, after all.

So, at least as a literary device, I don't think it is invalid (but while perhaps suggestive, it isn't "proof of a creator" by any means)

I think the movie "Pi" had similar issues, BTW.

Re:Not only pi is interesting, also different sqrt

[alfaiomega]

Square root of 9 to 500000 places?

Yes.

wouldn't that just be 3. with a string of 500000 zeros after it?

Yes.

If that's the case, it can definitely be compressed!

Yes. That's what I said. What's your point?

Re:Signature of God? Probably not

[trolleri]

I understand what you're saying, but I'm not sure you're understanding me.
Perhaps I was confusing you when talking about the "nature of math", what I was addressing was the essence of math, e.g. the logic behind it. Sorry for that.

If you still think I'm wrong I'd like to see the proof, it would be earth shacking news, somewhat like "Scientist inside a box proves there's nothing outside the box!".
You could of course try using paradoxes but I doubt you'd get far with that. Somewhere you'd probably end up with an assumption, like "There's only one kind of true logic, because that is the only logical way" or something similar to the trapped scientist.

--

Re:Signature of God? Probably not

[trolleri]

> You talking is full of jibberish. Because you can't look it up in your textbook? I read your comment and you're missing my point.

Re: accuracy

[Grizzlysmit]

not to mention security, imagine the harm if this guy got a life , why terrorists could know more didgits of pi than we do!!!, I think we need legislation against people like this ever getting a life, in the interests of international security & the war on terrorism (of course) :-)#

Erm,

[Kanasta]

How do you actually calculate PI?

I thot it was an irrational number, that could not be represented fully in any form. Thus you wouldn't be able to 'calculate' it?

Or, don't tell me they drew a really good circle and got to measuring it's circumference etc.....

I'm sure someone here must know how it's done.?

Useful in real world!

[mattr]

1. Promote interest in mathematics
2. Provide unassailable code publication anywhere in the galaxy (works on Earth too).

Okay, take your decss or whatever and gzip it. What are the odds that this archive exists in the teradigit string (probability indexed by archive length please)?

Obviously you just need to provide the offset in the teradigit string which ought to be available online somewhere.

But even if it isn't publically available, since (thanks to Zapman (2662) 's link) you can get any digit of pi without calculating the whole thing, you can resurrect the archive easily.

If SETI incorporated this kind of analysis we might even have a free distributed client..

Why? For the same reason.....

[linxman]

...a dog will lick it's balls.

Because it can.

Website, Kolmogorov Complexity, Question

[cancerward]

You can download billions of places from Kanada's website.

The question about compression everybody is asking - ie how far can it be compressed - is a question about the "Kolmogorov Complexity" of pi, that is, what is the smallest program which can produce 1.24*10^12 digits.

Finally I have a question, though it's probably too late to be answered. I can see how 206.158 billion is arrived at (Borwein algorithm, 3*2^36). But how is 1.2411*10^12 arrived at? Is it a multiple of a power of 2?

Re:Couldn't over-look the implications...

[ShavenYak]

Dude, we've waited long enough for Doom 3 already, don't give them any more ideas!

Re:Math != Reality cuz math is abstract

[Phoenix]

You know, that was the most sense of an answer I've ever gotten. I never figured it to be a matter of Base-n before.

Thanks

Okay, let's put it into a tried-and-true size form

[Ignorant Aardvark]

How many LoC's is this latest version of Pi? I don't care about how many gigabytes of data this would be. I just want to know how many Library of Congresses I could fill by writing out every single digit in neatly ordered books. By the way, isn't it conceivable that maybe somewhere along the way they got a digit wrong? Computers aren't perfect. Magnetism can and does cause things to go awry. Imagine if one of the digits is slightly off, and it threw off scientific progress for centuries. Yeah, that would suck, we'd aim at Alpha Centauri and miss because the 682nd billionth digit is off by one.

Article facts wrong

[deblau]

From the article:

Among the most puzzling mysteries: Mathematicians are pretty sure, but still cannot prove conclusively, that the numbers following 3.141592 occur randomly.

The word random has a very specific mathematical and information-theoretical meaning. In brief: a number, as represented by a sequence of symbols (digits), is random if it is incompressible; that is, if there is no algorithm, expressed using symbols which define a Turing-complete language, which can generate said number using fewer symbols than the number they generate. In other words, it takes fewer characters to write down the number itself than it does to "generate" the number using an algorithm. This is most certainly not the case with pi, as there are many finitely-expressible algorithms out there which generate it.

Re:Not only pi is interesting, also different sqrt

[Christopher Doopov]

I don't know the URL of this website, however I know the program to generate this number. You can download it here and this is how you must run it:

perl -leprint\"3.\",0\ x\ number

Where number is the number of decimal places you want it to compute. I hope it helps.

Re:Not only pi is interesting, also different sqrt

[alfaiomega]

perl -leprint\"3.\",0\ x\ number

Actually, it was more like:
(echo 3.;yes $?)|tr -d '\n'