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Are The Digits of Pi Random?
Steve Hamlin writes "A researcher at Lawrence Berkeley National Laboratory, and his colleague at the Center for Advanced Computation at Reed College, have taken a major step toward answering the age-old question of whether the digits of pi and other math constants are "random."
In addition, a simple formula discovered makes it possible to calculate the Nth binary digit of Pi without computing any of the first N-1 digits, and do the computation with very little computing power.
"
my 6th class chemistry teacher: ... for easier calculation we'll use 2."
...doh.gov ?
"pi is 3.14... which is close to 3
Dr. Donner
(Yes, this is really true.)
BTW:
Shouldn't that be
anti
ps:
yes, I should lookup my password.
Umm, that's trivial. The string of random numbers in pi can be compressed down to Leibnitz's series or Wallis's formula, or the formula mentioned in the article, without loss. Although these formulae are all infinite series, they can be represented as a finite number of instructions to instruct a computer to "decompress" pi. finite < infinite, therefore this is pi, compressed.
Sorry, couldn't resist (:
And that, my friends, is why the RIAA is in big big trouble.
No longer are artists dependent on record company marketing and promotion. Word of mouth ain't what it used to be. It's now global, very fast, and overwhelming when it gets a good start.
If this isn't convincing, there's a Turkish lurvegod who can back the argument up too.
Offtopic, but very much on-hobbyhorse...
Your Affectionate Cousin
AC
Webster's is not a very good source for mathematical definitions. Try looking up group, ring, field, module, etc. You're not going to get the mathematical definitions of these words.
This result derives directly from the discovery of an ingenious formula for pi that Bailey, together with Canadian mathematicians Peter Borwein and Simon Plouffe, found with a computer program in 1996. Named the BBP formula for its authors...
The article simply mentions that the BBP formula helped lead to the conclusions presented here.
I would mod down the above post as Pathetic Misuse of Gematria.
here are some facts you may find more useful than jiggery-pokery in the future:
This is amazing! What this also means is that even movies not yet made are imbeded in the very structure of the universe. I'm searching for Star Wars Episode II. I think I found part of the trailer starting around the 5.32*10^907456th digit of pi, but I haven't yet located the real movie. Anyone else found it yet? BTW - if anyone finds my dissertation, let me know so I can get the graduate student counsel off my back. With all this movie hunting I haven't had time to work on...
pi = 3
IANAC(ryptographer), but this isn't quite as "simple" as you make it out, regardless.
Your first assumption is bad, to start with. Someone can always "just use more bits". Since OTP's are usually transported in whole, using a million bits for the offset isn't going to be an unduly burden. You could fit that on a 3.5" floppy with room to spare.
Now, a million bits is a lot of bits. Let's see... 2^(1024*1024) = 6.74e315653, if my calculations are correct. For reference, there are only about 1e130 atoms in the universe. Saying you could do a million comparisons per second (grab the PI subset, XOR, compare with your "known" string), you need 6.74e315647 seconds to search the keyspace (given there are no repetitions in PI). Now, IANAP(hysicist) either, but my guess is that's more time than you've got left in the universe on a good day.
So your third point isn't at all reasonable. And that merely assumes someone is using a single OTP. It also assumes they're using a simple XOR and not doing some convolution to produce multiple resulting pads, etc.
Finally, your forth point is much handwaving. Any given data might be valid. With a OTP, you can't tell! You could probably find many valid headers in even a much more limited keyspace: likely those sequence of bytes could at one offset appear to be a perfectly valid JPEG, the next a word document, the next a random stream of numbers. And who says it isn't the random number stream that's the right one?
So while this may seem like an obvious way to defeat it on the surface, it's much more time consuming and less productive than you might think, especially on arbitrary encrypted data.
IANAM(athemetician) either (I'm a programmer, dagnabit ;-)), but I think the point here is that an arbitrary "slice of PI" is a random sequence. And with the properties of a pad, you're still going to be spending a very long time trying to decrypt it.
Don't think of it as a flame---it's more like an argument that does 3d6 fire damage
I wasn't linking against libm. *sigh*
--
Okay, could someone clue me in as to where the floor() and pow() functions are, so I can compile this and try it out? According to the man pages (RedHat 7.0) they're in math.h, but they're not!
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Uh, since the article says it finds the Nth binary digit, I think it would be safe to say that the algorithm actually works in binary. That technically means you're right, since hex is a binary shorthand.
...phil
...phil
"For a list of the ways which technology has failed to improve our quality of life, press 3."
OK, convert 0.1 decimal into any base that's a power of two. Let me know when you're finished.
...phil
...phil
"For a list of the ways which technology has failed to improve our quality of life, press 3."
I don't think that's correct. Consider an irrational number whose digits after the decimal point each have a 9/10 probability of being a 0 and a 1/10 probability of being a 1. Here are some examples that satisfy this:
This is definitely random (you have no way of knowing whether the next digit will be a 0 or a 1), but it is also definitely compressable (each such number should be compressable to about 1/10th of the original size).Now, I'm not saying that PI can be compressed in this manner, but if any digit did happen to appear more than another it could be compressed while still being random. A simple Huffman coding should suffice for such cases.
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Free P2P Backup, Windows & Linux
No, that's called a uniform distribution. It's a sufficient, but not necessary condition of randomness. There are plenty of other random distributions.
you are redefining "random".
Not quite. Take at look at the second definition of "random" from dictionary.com (the one that's explicitly labeled as the mathematical definition):
And then take a look at this list of probability distributions. You will see that your "definition" of random actually only describes the uniform distribution and that there are plenty of other ways for a variable to be random.-----
Free P2P Backup, Windows & Linux
Ah, so there's the problem. You specifically said in your first post (and I quote):
That would certainly indicate that you were talking about the mathematical definition of "random" (which doesn't require "an even distribution"). I guess the Slashdot title and description didn't help matters - I don't know why they used the colloquial meaning of "random" in a context where it means something different (the mathematical context).Anyway, my original post was in response to the assertion that you can't compress "a string of random numbers". If the string were an unknown sequence of uniformly distributed random variables, then that makes sense, but that wasn't stated.
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Free P2P Backup, Windows & Linux
This guy sounds like he did something similar.
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Free P2P Backup, Windows & Linux
predictable is not the same thing as having a pattern. predictable (here) means, as someone else put it above, calculatable. just because there's a formula to predict digits down the string doesn't mean it follow any pattern OTHER THAN THAT OF THE FORMULA FROM WHICH IT IS DERIVED. pi is the output of a formula. it doesn't exist as a number in and of itself. it's derived from observation of other things - so maybe formula isn't a correct word there. but the point is, its ability to be predicted derives from its observable and formulaic nature, not from the fact that there is any INTERNAL pattern. which is what we're looking for.- --
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All that glitters has a high refractive index.
You see, without that little doohicky, the universe stops.
http://propheteer.org
...not my intent, but I suppose if the shoe fits...- -----
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All that glitters has a high refractive index.
You see, without that little doohicky, the universe stops.
http://propheteer.org
okay...interesting. by pattern, however, I mean internal to the string of digits...I still don't see that there is one. not that I wouldn't like there to be, but I don't yet see it. with your second point (one that I admittedly don't fully grok as I didn't get to read the article...damn thing is slashdotted...) If we see that there is a pattern in the binary representation of pi...doesn't that indicate a pattern nonetheless??? why are we so...so...set on wanting our patterns to all show up in base 10?
All that glitters has a high refractive index.
----------------------------------------------
You see, without that little doohicky, the universe stops.
http://propheteer.org
I was wondering the same thing...BUT. I think we're looking at this the wrong way. the number are not, and have never been, and are in no danger or question of BEING, truly random. they're there, they're set, and it's done. the question is "is there a pattern"? and so, the formula does not automatically force there to be a pattern, just forces us to realize that they're static and predictable.- ----------
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All that glitters has a high refractive index.
You see, without that little doohicky, the universe stops.
http://propheteer.org
Except some of those programs will never halt. And Turing taught us that by simply at a program, you cannot tell whether it will ever halt. You have to run it. And that may take time.
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This view is known as determinism. It is, incidently, an essentially religious view. The poster nor anyone else could possible know the truth or falsity of the above statement. Anyone who believes it (or believes it to be false) is committing himself or herself to a view with not a single shred of evidence for or against it...
--
"Convictions are more dangerous enemies of truth than lies."
Incidently, 1/3rd does not repeat infinitely if expressed in base-12, a far more logical base for numbers than 10. Just ask 13th century Polish from an alternate universe... :)
--
"Convictions are more dangerous enemies of truth than lies."
Actually, the you could make a pretty cool one-time pad for text messages. ;)
;)
Take your text and rot13 it, or do a similar transform. Then uuencode it. Remove the uuencoding headers/footers. XOR it with x number of digits of pi, from point y.
You've given the person you're communicating with a calendar with a different x and y on it every day, generated with a decent random number generator. These could be of arbitrary size.
Yes, this requires you to transfer the keys in advance, and has the obvious problem of someone stealing or copying your calendar. But your digital correspondence security is always, at it's best, only as good as your physical security. PKI methods have the same limitations. Someone gets your private key, and you're pretty much screwed. Your monitor can be read from a distance too, you know.
Of course it might not survive some good cryptanalysis, but it would be sufficient I think for casual use.
You've misunderstood something very basic. The fact that something is infinate does not guarentee that it will contain everything. There may be sequences that do not occur.
On the otherhand, once you've located a particular sequence, providing the offset and length may be a good compression algorithm, depending on how efficiently you can store the offset. ;-}
A pattern is also evidence that the compression could be better.
Or maybe just rendering 5 min of Jar-Jar galumping around...Shiver
Novel theory: Modern Man evolved from psychopath
I think someone's said it before, but, doesn't having a formula that allows calculation of arbitrary binary digit, in fact, make it NOT random? I'm just trying to grok how something can be "easily calculated" and still be truly random.
Send your friends messages of love at fuck-you.org
If this formula to calculate arbitrary binary digits is derived from being able to calculate pi as a whole, how is it proven that the original formula is valid?
The article discusses a formula now in use that allow computation with less computing power than before...Does this imply that the earlier formulae were flawed such that errors were introduced into the sequence? I'm sure someone would notice some huge error 250 digits in, but, after 250,000? 250,000,000?
Maybe my brain has discarded the answers to these questions and replaced it with PERL syntax or something. So Mathies, please, be kind.
Send your friends messages of love at fuck-you.org
Even taking your result as valid, the assumption you make is that the random numbers occur all over your possible inputs.
...
:-)
A previous poster gave a good counter example, but it is very easy to see that if, in base 10, your random numbers are generated by an algorithm which only ever produces digits in the 0-4 range, you can see that there is some scope for compression.
Perhaps it is more easily seen if in base 0xff, you only produce digits in the range 0x0 -> 0xf...so all your random numbers will be of the form:
0? 0? 0? 0?
proof that this is compressible is left as an exercise
umm the atheistic communist systems of the Soviet Union and China are clearly the most oppressive institutions.
---
anyone who believes Social Security SHOULD be a federal program instead of owned by individuals has to be the real idiot
---
Er... I think that is what I was saying...
Posted from the wireless couch.
That was my guess. Looking back at it, I probably didn't write quite as clearly as I should have... Oh, well.
Posted from the wireless couch.
I whipped up a little perl script earlier this summer that a friend and I used to informally prove to ourselves that the digits 0..9 are fairly evenly distributed in the first million digits of pi. I guess that doesn't really have anything to do with the randomness of their positions, but it didn't look like there are, for example, significantly more 5's than 8's in there...
Posted from the wireless couch.
Its got a 50% chance of being correct though and
involves coin flipping.
And I suppose it is trivial to prove that PI with an arbitrary number of digits is embedded in PI...
It's a quote by Tom Duff of Bell Labs, who knew he
was using humor to illustrate a point not about Pi, but about the problem of converting time units.
http://users.erols.com/blilly/programming/Progr
-fb Everything not expressly forbidden is now mandatory.
YOu are correct, an encrypted file is indeed not random. But one should not be able to differentiate encrpyted data from random data. The presence of a pattern, no matter how trivial, would be evidence of reduced security in the algorithm.
I have generalized here to keep this post short...
-- Minds are like parachutes... they work best when open.
http://www.lbl.gov/Science-Articles/Archive/pi-ran dom.html
That link works for me...
Thanks for the posting of the correct information.
This thread just illustrates the truth of my .sig.
Readers complain about the accuracy about /. stories when they can't even be bothered to look up the correct fucking information before posting themselves.
They're no better than the morons in Texas, Kansas, *and* Indiana that they so happily mock.
pooptruck
"It's like throwing a fair, ten-sided die forever and counting how often each side or combination of sides appears."
You know, this sounds a lot like a certain RPG I was in...
That's correct, more info here:
m l,
m l
http://www.mathsoft.com/asolve/plouffe/plouffe.ht
or for goatse.cx aware:
http://www.mathsoft.com/asolve/plouffe/plouffe.ht
Last I heard, the algorithm only worked in base 16, but that may have changed now.
:wq!
It's tough with an infinitely long number, though. There's no requirement that the compression be done on the fly - I could write a compression algorithm which wouldn't work unless I had access to the entire number.
You could say that the first N digits of pi are random or not based on the compression test, and make some sort of argument that since so far every sequence of pi we've tested was random, it's likely that the whole thing is, but that wouldn't be a very rigorous proof.
Remember: it's a "Microsoft virus", not an "email virus",
Your right to not believe: Americans United for Separation of Church and
Not at all. These are fractions, not plain integers. For instance 0.1 in base 10 is a never ending sequence in binary (and hex) notation.
Two notes:
1. Assume that your sequence appears somewhere in the decimal sequence of PI. Then surely you can predict the next digit using the knowledge of where in the sequence you are.
2. Assume that you have a subset of the decimal sequence of PI and don't know where it belongs. Then there is no way to predict the next digit, since it will occur at infinitely many places.
It it in this sense that PI is random.
Well, I don't read Hebrew, so I can't check on what he is saying. However even the appologist didn't translate it into a mathematical formula. He just did a bit of gematria and waved his hands. Now it's clearly true that the Babylonians at that time had a good understanding of PI (relatively), and that they did use the kind of fraction that you are implying the Hebrews used. And they were hired contractors on the job that is being written of. But even putting this all together, I can't make it say that the biblical scribes were able to cope with the Babylonian knowledge. The do seem to have caught on that certain numbers were important (the Babylonians started doing their math with balances, so they were whizes with rational fractions), but I can't see, from what has been said, that the Hebrews knew why the numbers were important, or even which number divided which (or what division meant).
Now clearly, a part of my ignorance is due to my not reading Hebrew. If I did, then I would have a better idea of whether they were able to copy the specs from their contractors without error. But the evidence provided so far doesn't show that.
Caution: Now approaching the (technological) singularity.
I think we've pushed this "anyone can grow up to be president" thing too far.
OK. Use 0x00 or 0xff. Now it's a real world case.
Caution: Now approaching the (technological) singularity.
I think we've pushed this "anyone can grow up to be president" thing too far.
Perhaps. But it's a nice tool to beef up your rot-13 substitution cypher.
:-)
What you want is, say, a 1 MB block of the digits of PI expressed in base 256. Then you pick a starting position, and for each byte that you transfer, you rot it by the value of the corresponding byte.
P.S.:
Caution: Now approaching the (technological) singularity.
I think we've pushed this "anyone can grow up to be president" thing too far.
Huh... I thought pi was a movie
Only when pi is written in the US. Now its got a sequel - American Pi^2!!
Donte Alistair Anderson Roberts - hi son!
Karma: Chameleon
I think Granny Weatherwax was beaten to the idea by Douglas Adams (RIP) in Hitch Hikers Guide To The Galaxy. The Infinite Improbability Drive worked on the exact same principle.
(Actually I think my sig is derived from some philosopher or other too).
Donte Alistair Anderson Roberts - hi son!
Karma: Chameleon
I've also done some testing with other transcendental numbers, such as e (2.718281828...), and they all seem to show great randomness properties, in the information-theoretic sense at least. However, I have a feeling to "trust" Pi more than e, given that you can write e in form of continued fractions with repeating patterns, and nobody has yet found a pattern in the continued fractions of Pi.
There's nothing about transcendental numbers that makes their digits random; pi and e happen to be special. For instance, the first number proven to be transcendental was (some variation on) 0.101001000100001000001...; that sequence isn't random by any definition.
As for my pseudo-random library project, my programming skills are quite bad, but if you have some knowledge of scientific computing (multiplication algorithms using FFTs, for example), you can contact me and I might revive the idea.
I'd recommend to anyone interested in projects like this to look at George Marsaglia's page; his tester may help you avoid releasing crap. You can also search Usenet archives (i.e., Google) for some generator source code he's posted. Knuth also has a very detailed treatment of pseudo-random number generation in TAOCP vol. 2.
It might be better suited as just a commonly known hash. Given some piece of data, such as your login password, reduce it to a number and index into pi that many bits in, and generate N bits of hash. Somehow I doubt it will be as fast (especially for large numbers) as say MD5. But it could be interesting because it can be extended quite easily to as many bits as you want.
Now if that algorithm were faster than MD5 for indexes through 2^128, or faster than SHA1 and RIPEMD160 for indexes through 2^160, then I think we might have a winner.
now we need to go OSS in diesel cars
Digits of contants are nice and easy source of bits to test for randomness, but what about random orderings of a set of integers (1 .. n)? I've found at least one source that might be interesting, but I can't figure out how to test it reliably; to do that, I'd need to convert the set to bitstream. How do I do that? I mean, it's obvious I can get a list of random numbers, first one in range [1,n], second in [1,n-1] and so on, but that's n numbers, each with different range.
P.S. I doubt anyone reads this. I just rant to myself, helps to clarify to myself what I'm thinking. This story is, like, couple of hours old already!
- Kaatunut
You know, instead of repeating that "absence of proof" phrase again, I'll do the slightly rares version of the same idea. Here's a simple pseudo-perl code to see if your argument (usually one defending religion) is fallacious in one specific type:
Set appropriate values to variables and run the above code. Does the argument it outputs make sense?
Since the religious type is known to occasionally act in slightly irrational way, here's a simple example:
$_=<<EOT
# [appropriate substitutions]
print;
That's how you probably sound to non-believers. See also Invisible Pink Unicorn.
- Kaatunut
People think that randomness is this impersonal force that makes things happen for no reason at all.
What it really is, is an explanation when the factors involved in the outcome are too complicated to grasp.
Nope,
there's just a difference between deterministical chaos and randomness.
That doesn't mean the the latter doesn't exist.
Deterministical Chaos is exactly that what you described, i.e. the outcome of a process depends in a non-continous manner from the input data.
prominent examples are
- three body problem
- weather (all see Lorentz (sp?) equations)
- mandelbrodt set
etc.
Alle this problems have in common that a very small modification to the input may lead to drastic changes on the output.
For instance the three body problem descibes the motion of three bodies (sic!) under mutual gravitational force. Algorithms which want to "predict" the motion of these three bodies have to be very precise, i.e. you have to calculate with something like 100 decimal digits.
Now to the question of randomness. Well, it really isn't easy to find and your believe that it doesn't exist was actually shared by Laplace (mathematican/physician) end of 19th century (I believe). He said something like:
"Give me the state/position of all particles in the universe at one time and I can calculate the state/position at any time in the future or past."
But - later on quantum mechanics was discovered/developed, and still today every scientist belives that there lies real randomness.
This is for instance deductible from Heisenbergs uncertainty principle (sp?).
It's fundamentally _impossible_ to precisely predict speed and position of a given electron. It's fundamentelly impossible to predict when and in which direction an alpha particle will be emitted from a collapsing atom nucleus.
That's why there is the notion of half-life period, thats just the time when the probability is 50% that the alpha particle has been emitted.
But will it always be impossible? Or do we just think it is impossible because we lack information and understanding?
0 00 /posts/topic37229.shtm
;), this is an area which seems to have a strong attractivity for, hmm, exotic people and ideas.
This is always the question with scientific theories. I used some sort of shortcut with the word fundamental, read this as "fundamental following the state of todays physics".
Your "lack of information" is called hidden variables and is in itself subject to some theories. And a general consensus is that, yes, that "fundamental" is fundamental.
I did a quick search on google for:
hidden variables quantum mechanics
and found the following nice and short thread which deals with this theme:
http://www2.abc.net.au/science/k2/stn/february2
If you want to read more, just peruse google, there are a lot of nice essays out there - but don't believe everything you read one the internet
Oh, btw., in the light of this laplace citatiton, when I think of the consequences this would have on such nice ideas as "free will", I'm really positive that I like the uncertainty of quantum mechanics more.
The string 55378008 was found at position 623901 counting from the first digit after the decimal point. The 3. is not counted.
-- atomly
A lot of people are playing fast and loose with the word 'random' today. The value of Pi, in whole or of any one digit, isn't random at all. It's entirely deterministic, defined rigidly by a simple formula. No matter how many times or ways that formula is interpreted, the value of Pi is the same, and not random.
What can be said to be 'random' (really pseudorandom or, in the parlance of mathematicians, 'random enough') is an arbitrary digit or sequence of digits from pi, given that the starting decimal place N is also random, or at least non-repeating. The randomness of pi is that each succeeding digit of Pi has no correlation to the preceding digit.
Of course we all know this inherently, but it wouldn't hurt to be a little clearer in these posts about exactly what is random (or not) about Pi.
Kevin Fox
--
Kevin Fox
If you want to see a guy who spends way too much time with PI, check out Mike Keith. There is some other, truly amazing stuff on his home page as well.
Rudy Rucker once wrote a story where a shell was discovered with strange markings on it. Those marking turned out to by a coded message about an alien civilization, sort of like we sent in the Voyager probes. The story gives lots of details about those aliens, their culture and something that they referred to as the Joke. It seems that this wondrous coding scheme for all of their accomplishments and history was equal to pi.
I memorized pi to 50 places the summer between grade 10 and 11 just so I could be a smartass in math and physics classes the next year.
It's actually not hard to do. Break it up into groups of 5 digits, and memorize 5 digits a day and keep practicing. A few minutes a day, and you can be a loser^H^H^H^H^H, er, cool geek like me too.
I wish I could get into the real story, but it's slashdotted, so I'm posting pointless crap like this. Geez.
Torrey Hoffman (Azog)
Torrey Hoffman (Azog)
"HTML needs a rant tag" - Alan Cox
Pi will not, unfortunately, give you an arithmetical method of producing random digits.
If you pick digits from pi's decimal expansion with some deterministic method, say, every third digit, the sequence will be the same each and every time you run it. What you do get from pi are non-repeating pseudorandom numbers: you can eg. pick every nth digit where n is your seed (cf. usual (pseudo)random number generators)
To get truly random numbers from pi, you need pick the digits randomly... for which you of course need a random number generator...
[ Antti Rasinen ]
While I find your post interesting and stuff, I'd just like to jump in here and point out that you can send me a tarabyte of data that indicates that pi is random and it doesn't make one lick of difference. Show me the *proof*!
--
Talking to you, girl, is like long division.
Old 97's
Slashdot 's editors are dickheads
in vi
:%s/666/666/g
Vi will find and replace all occurances of 666 with 666 and give you the number of replacements.
$sig=$1 if($brain =~
function nthDigitOfPi(long n) {
:)
return rnd(10);
}
I am curious what language this would be?
In any case
print nthDigitOfPi(1);
oh ow
If you had read H2G2 you would have known the real message from GOD is :
WE APOLOGISE FOR THE INCONVENIENCE
;-)
I don't get it. What does "115" have to do with a spooky coincidence?
www.timcoleman.com is a total waste of your time. Never go there.
Yes, that's a good philosophical position..
.|` Clouds cross the black moonlight,
I'm just wondering, if there's a "formula" for the n'th bit of the thing, it *can't* be random, can it?
For values of `random' that mean `uncompressible' of course, it can probably rate pretty highly.
~Tim
--
~Tim
--
Rushing on down to the circle of the turn
According to algorithmic information theory, a bit sequence is random if the binary representation of every program that outputs the bit sequence is longer than the bit sequence itself. By this criterion, the first N digits of PI are highly non-random because there is exists a short program to compute all of them. Another way of looking at this is that the sequence of the first N digits of PI are highly compressible -- you can compress the whole thing down to the binary representation of a small program to compute those digits, and the size of this program grows at most logarithmically with N (const + representation of N).
Also, the 13th occurrance of 222222 is at 13371013!
Too spooky for me, as I was born on 2/22, and 13 used to be my lucky number.
-- Len
Wow we can do a Pi@home!
;)
I'll take apple myself
make Linux, not Microsoft. sin(beast) = -0.809016994374947424102293417182819
(I'm a geographic group theorist, not a number theorist :)
Aren't 'trancendental' and 'non-algebraic' the same thing? I thought they both mean 'not a root of any polynomial with integer coefficients'?
-- Help Digitise the Public Domain at DP.
Don't lots of people go insane thinking about the how's and why's of pi?
"I don't think it's selfish, to eat defenseless shellfish." -NOFX
Huh? For the sequence to be random, each subsequent outcome must have an equal probability of occurring. That is, each subsequent digit must have an equal likelihood of being a zero or a one.
Look at it this way. Let's say you have a conversation between two people. One person (A) is the "generator" of the sequence. Now you seem to have the point of view of (B) where the digits are allegedly "random". But (A) is simply generating digits with a 90% chance of a 0, 10% chance of 1. Is that random?? Of course not. Your definition of random is confusing at best, and let's be straight -- you are redefining "random".
Any mathematically random data cannot be compressed.
You said something about Pi, that if one number had a higher frequency, it could be compressed while still being random. Well, the sense of random that we were using is a random probability distribution. The digits of Pi are not random in the sense that they are calculated by some random means. They can be calculated. We were using the sense of random that they had an even distribution.
If indeed one number had a higher frequency (leading to compression), Pi would cease being "random" in the sense that we have been using it.
That's wrong. A truely random system has a probabilty that the compression would work. It could genenrate a set of numbers where all are about the same. Probability is low, but the chance is there.
Compressing an infinite set would be pointless, since their is always more data and you don't gain anything. These sets are FINITE. Besides. What exactly have to do with compression. Compession is taking into account "patterns" and compressing that way. If a set of random numbers are drawn from a hat, the probablity that a number is repeated is small, but there. And the probablity that a number drawn from set A is in set B is not zero, but 1/x as x->inifity. Being zero, and being almost zero are two different things.
Someone else may better speak to this.
Randomness is a funny thing. What I mean is that we use the term in several circumstances and we confuse ourselves. In this situation there isn't anything truly random in the sense we normally think of it; the digits are already determined, we just need to look them up (using the formula discussed). This contrasts with what we normally think of as a truly random process such as flipping a coin, waiting for an atomic nucleus to decay, or using a Pop-O-Matic Bubble.
So what do we mean? Somehow we mean that it appears random---that we couldn't just guess and get the next digit.
The way that Baily and friends have formalized this is to say that in the limit all consecutive sequences of length k occur with equal frequency. So, for example, if I look at the first 6 digits of the decimal part, .141592, the sequences of length 3 are 141, 415, 159, 592. These occur with probability 1/4, and the other 996 occur with probablility 0. But, as we look at more and more digits they all become equally frequent, occuring with probability 1/1000. The same happens for all length sequences if we look out far enough.
In computer science we often discuss pseudo-randomness. In this case we start with some number of truly random digits, and then have an algorithm that generates more digits. If no bounded time algorithm can determine that these aren't random, then they are pseudo-random.
Now, whether there's really any randomness in the world or not, that's metaphysics. (Editorial: And metaphysics is crap!)
They're not random, they're tied directly to the rules of our universe. And if you look in the right way, they may contain a message.
But Ellie already found it in that book, so this news story's a little old, eh?
I think that by random, they mean that the Nth digit has no correlation with any other digit in the sequence.
Regardless of the existence of a pi formula, pi is not random any more than the constant e. Afterall, pi always starts with 3.
I don't practice what I preach because I'm not the kind of person that I'm preaching to.
However, I have a feeling to "trust" Pi more than e, given that you can write e in form of continued fractions with repeating patterns, and nobody has yet found a pattern in the continued fractions of Pi.
I though that you could construct Taylor series for functions like arctan(x) and arctan(1) = 1/4*pi, so pi = 4*(ArcTanTaylorSeries(1)).
It is predictable to anyone who is knowledgeable about these things, that 666 should appear at position 2440, since the world will end in that year!
lol
I seem to remember reading somewhere (I'll dig out a pencil & paper sometime I'm more bored than I am now) that 140 ish digits is enought to calculate the circumfrence (sp?) of the observable universe to within the diameter of a hydrogen atom...
There's a reason its called "Weed" college among the locals...
Now, when someone will show how good he is in maths, I will tell him about that. Chances are that he'd never have heard about this discovery, and he will shut up.
-- Pure FTP server - Upgrade your FTP server to something simple and secure.
{{.sig}}
Its compression not as in winzip, but
1,2,3,4,5,6,7,8,.......
is N(x+1)= N(x) + 1
Live today. Tomorrow will cost a lot more!
if it is truly a random number generator, you will not be able to predict the next number in the sequence.
If you can, then it is not a random number generator!
Live today. Tomorrow will cost a lot more!
I think all the compression people are missing the point.
If we can predict a previously unknown nth decimal of pi, then pi cannot be random.
This is where compression kicks in.... we have compressed pi to the formulae that we use to predict the value of the nth decimal.
I say lets use this algorithm to predict the 2000 billionth decimal, and then get a few beowulf clusters working... if the two values match, then pi cannot be random
Just my £0.02 worth
Live today. Tomorrow will cost a lot more!
the set is descrete. they are a string of integers.
-- MartinG To mail me: echo kewyjlcxyzvjfxbqwh | tr bcefhjklqvwxyz
Great commentary on BattleBots the other night:
(introducing a returning champion)
"With just one more execution, he'll be eligible for the governorship of Texas!"
- - - - -
Napster-to-go says "Fill and refill your compatible MP3 player", which is a lie. It's not MP3. It's WMA with DRM.
"With just one more execution, he'll be eligible for the governorship of Texas!"
Moderation Totals:Troll=1, Total=1.
This moderation brought to you by the Friends of George (FOG)
- - - - -
Napster-to-go says "Fill and refill your compatible MP3 player", which is a lie. It's not MP3. It's WMA with DRM.
This of course assumes the Church-Turing thesis. I think that Kolmogorov complexity is still a valuable concept, however, given stuff like Chaitin's Omega, which would appear to have an undefined K. entropy, it is difficult to give priority to computation. I think that in the next hundred years we may see Church-Turing disproved.
Pi being impossible to calculate (decimals go on forever) means that no circle that exists anywhere in nature is perfect. Of every circle from the electrons to galaxy clusters, not one is perfect. They are all "askew". This blew my mind a bit back in the day when I had to care about pi (math class).
;-)
Maybe pi is not calculable because we can't percieve enough dimensions? In a cartoon a sphere is a circle...
Maybe circles are perfect after all! Our established math knowledge is horribly flawed! (As my test scores often indicated...)
Maybe we would understand if we were born with fingers on each hand!
Anyway, after leaving the academic field Pi has become more "great movie!" than "what is reality?". Now where's my beer...
Many posts are talking about Pi being random.
Well, Pi is not random: it's Pi!
Also, the digits of Pi are not "random". It I want to compute the 1024th digit of Pi, I can. So, it's not a random number.
The word "random" might not be the most appropriate here. The question is whether any "substring" of Pi (expressed in any base) appears exactly as frequently as any other substring of the same length.
Another good reference is in Chapter 17 of "A History of PI" by Petr Beckman (ISBN 0-88029-418-3). Beckman goes so far as to include a reproduction of the bill.
The bill doesn't actually give any values for PI directly. Rather, it supposes to provide simple formulas for computing the value of PI based on enclosing squares. Here's part of section 1:
"It has been found that the cirular area is to the quadrant of the circumference, as the area of an equilateral rectable is to the square on one side."
And later in section 2:
"By taking the quadrant of the circle's circumference for the linear unit, we fulfill the requirements of both quadrature and rectification of the circle's circumference. Furthermore, it has revealed the ratio of the chord and arc of ninety degrees, which is as seven to eight, and also the ratio of the diagonal and one side of a square, which is as ten to seven, disclosing the fourth important fact, that the ratio of the diameter and circumference is as five-fourths to four, and because of these facts and the further fact that the fule in present use fails to work both ways mathematically, it should be discarded as wholly wanting and misleading in practical applications."
Now, there are about three different "values" for PI in there. Dr. Edwin Goodman, who came up with this tripe, was a "circle-squarer", one who thought PI could be computed, well, easily.
It's scary that the guy was a doctor given his profound misunderstanding of simple geometry, and seeming inability to do the simple measurements which would prove his theories wrong.
But this bill is much bigger in folklore than in real life. Some facts regarding it:
1. It was never passed in to law. Ever. It passed unanimously in the House, and was tabled in the Senate, and never voted on there.
2. It doesn't say that PI=3, PI=3.14, or that PI is equal to any other simple number. Rather, it gives a number of methods (all of which are grossly incorrect) that could be used to very easily calculate PI.
3. It never made it into any textbooks that I know of. That was the goal of Dr. Goodman.
4. It did happen in Indiana, 1897, House Bill No. 246
5. Dr. Goodman, in section 3 of the bill, also claims to have previously trisected an angle. The guy was either a fruitcake or a charlatan.
Michael
Do you have ESP?
The formula sums up the expression in the square brackets multiplied with 1/16^k - so that means (if I am not completely mistaken) that that expression gives you the k-th *hex* digit of pi. You just have to use the right number system to make the problem easy ;-) (not very useful if you want decimal digits, though)
EagerEyes.org: Visualization and Visual Communication
No. The true, message of god is "all your base are belong to us".
Either ally with Him or don't. But don't think that you'll win a fight with God.
Actually, the point of the research is not to caculate additional digits of Pi, but to understand the mathematical nature of Pi. And such inquiries about mathematics have been show to be immensely useful in all sorts of real world applications. Take the whole "quantum physics" thing. One could ask, "who really cares how a quark behaves on the sub-atomic scale?" Today, it has been found that a significant fraction of the US economy is based on the application of quantum physics.
A deep unwavering belief is a sure sign you're missing something...
Perhaps my point was obscured by my vitriol. Let me try again without the ad hominem.
The upshot of the article at ldolphin.org is that 1 Kings 7:23 predicts a value of 333/106 for pi. The author arrives at this value by extracting the numbers 3, 106, 111, and combining them as 3*111/106. Surely you can see how this might seem contrived to a skeptic. Since these numbers are relatively small, it should not be difficult to find them in a given text. A numerologist of superior skill could produce the ratio 355/113 from the same text.
Now, 333/106 is a pretty good approximation to pi, but it is not remarkable. The reason that it is close to pi is that it is a so-called "continued fraction convergent" to pi. (See this page for an introduction to continued fractions.) The first few convergents to pi are 3, 22/7, 333/106, 355/113, and 103993/33102. Every irrational number can be represented by a continued fraction, and the convergents are used to find rational approximations. (The approximation 355/113 was known to the Chinese in the 5th century AD)
By the way, I think it is silly to call 1 Kings 7:23 a biblical contradiction. The Bible is not an engineering manual, and surely the measurements given are accurate enough for their purpose. On the other hand, I see no difference between numerology and Alex Chiu. (oops, that just slipped out)
Numerology can be used to prove anything whatsoever. The fact that some idiot or liar can manufacture the ratio 333/106 from a random biblical verse is not particularly surprising or compelling. Also, one wonders why God in his infinite wisdom didn't encode 355/113, which is a far better approximation to pi.
Incidentally, the website you mentioned (ldolphin.org) is a goldmine for skeptics who wish to discredit Christianity. It is filled with some of the most credulous pseudoscientific bilge that I have ever encountered. He makes Art Bell look like James Randi.
The corrected URL:
& startpos=242423
http://www.angio.net/pi/bigpi.cgi?UsrQuery=424242
Searching for a 6-digit, not an 8-digit, string...
.... um, i lost you after "0110100001101001".
Sorry to quote John Boatwright... Never heard of him. I just picked the first article I could find on Google that essentially made the point about handsbreadth rim arguement that I'd seen elsewhere before. I sincerely doubt it originated with Boatwright. I vaguely recall reading that some Jewish rabbi first pointed it out back, pre-1000 AD, but I'll do a followup here if I find a source that's better or the references the original source of the arguement.
--LP
Technically, it's not conjectured to be "random"; it's conjectured to be "normal", which means that there's an even distribution of the digits 0-9 (in base 10) or 0 and 1 (in base 2).
--LP
And I also found a brief article (from a non-religious website) describing how a Jewish rabbi named Nehemiah in ~150 AD first made the argument that the diameter of the tub was 10 cubits from outer rim to outer rim, whereas the 30 cubit circumference was measured around the inner rim.
I wouldn't consider these "proofs," just provocative re-examinations.
--LP
If the circumference measurement is from inside the brim (or something like that), you get a value for pi that is 0.073% accurate, well within the significant figures used by Hebrews for measuring at that time.
Not that the bible is a Mathematics text...
--LP
There was a distributed computing project called PiHex that lasted several years for computing the five trillionth, 40 trillionth, and the quadrillioth bit of Pi, using a variant of the Plouffe discovery, Bellard's formula.
A proof that digits of Pi are random would indeed be news, albeit not exactly a surprise; I'd comment on it but the article's link seems bad or swamped at the moment.
--LP
P.S. Google has a nice list of Pi links.
Consider the infinite, non-repeating binary number 0.1101001000100001000001...... At nowhere in the sequence does the pattern 10101 occur, nor 111, nor 101101, nor...
This post expresses my opinion, not that of my employer. And yes, IAAL.
Creationist make the same claim. Except they say it's proof of creation.
Before you embarrass yourself again by verbally masturbating in front of the whole /. audience, you may want to investigate the claims of your opposition. At least just a little.
Oh, but why bother. They're all "simpleminded illiterate tribesmen" anyway. Feh.
this reminds me of carl sagan's contact. hi says that pi is a proof of the existance of God. according to the book, if you go on finding the value of pi in a certain base, you will there will be a string of 0s and 1s, which when arranged in a square matrix will represent a circle :-)
wonder if it would be easier to try it now... crazy idea...
Don't Panic
The use of the word random when referring to Pi may be somewhat of a misnomer. It's not that the number Pi itself is random... it's the ratio of a circle's circumference to its diameter. The randomness (or random-like behavior) is in the digits, specifically, predicting the next one. There seems to be no pattern, much less a repeated pattern in the digits of Pi that is consistent throughout the number.
"The best laid plans of mice and men gang oft agley..." - ROBERT BURNS
And the question would be What do you find at position 242424 of pi?
Say no to software patents.
In Europe, we put the day of the month before the month. I.e. this would be the 11th of May.
Say no to software patents.
For an even more spooky coincidence, click twice on Find Next, and carefully note the 3 last digits of the error message (start position...).
Say no to software patents.
Just wondering, but did that come from the book Python Standard Library?I was just reading it, and that quote at the beginning of the Random chapter stood out.
No - Not BS.
The size of the decompressor has to be included in the compression calcs.
In addition splitting on a predfined series of bytes is consdered to be a "trick" as it merely offloads the data to the filesystem involved.
Remember kids! Guns don't kill people - Americans kill people.
Maybe in Japanese, it doesn't rhyme, and yours does! :D
"Slow down cowboy!
Slashdot requires you to wait 20 seconds between hitting reply on comments.pl and submitting a comment.
It's been 12 seconds since you hit 'reply'!"
Okay, screw you, I type fast and I'm on a high-speed connection. Dipshit.
--TheOrangeSquid Is it any wonder things seem so awry? We swim in a sea of confusion and don't have to think to survive
Still, the *ratio* is the same, no matter how long the ruler.
You think that's bad? Here's a guy who juggles while reciting pi:
http://www.cs.rpi.edu/~moorthy/moorthyjug.html
The string 666 was found at position 2440 counting from the first digit after the decimal point. The 3. is not counted.
I *knew* PI was the tool of the Devil.. it just had to be. I'll be we can even find out the answer to who shot JFK in PI if we search hard enough...
perl -le '$_="6110>374086;2064208213:90<307;55";tr[0- >][ LEOR!AUBGNSTY];print'
Polymorphism -- It's what you make of it.
The meaning of randomness has to do with Kolmogorov complexity. While Kolmogorov was primarily a statistician, Kolmogorov complexity could actually be considered a topic in theoretical computer science.
Let's say you have a string s of length |s|. If the smallest possible Turing machine that can output s has size > |s|, then s is a random string.
On the other hand, if the smallest Turing machine that can output s has size some Turing machine exists that can write down s and yet is smaller than |s|. Thus pi isn't random.
NB: for the other poster who was trying to use information theory to prove pi isn't random, please not that the probability that the nth digit takes on a certain value is now known to be a deterministic function... this may change your results.
Just remember what Asimov said:
...'"
"The most exciting phrase to hear in science, the one that heralds new discoveries, is not 'Eureka!' (I found it!) but 'That's funny
Unix: Where
Disclaimer: I am a high school student who just finished up Algebra II
You mean, not everyone on Slashdot is a rocket scientest with 14 degrees who holds a prestigious position in a large company?! Oh, the shock!
What are you talking about?
111/106 = 1.04716...
It's not even close to Pi. Are you paying any attention whatsoever to what you're writing? Are you a complete moron?
It's not accurate at all!!
Think before you post, and at least have the minimal, reptilian, common sense to check your fucking numbers.
If you look long enough, you should be able to find all the Metallica songs, every work of literature, decss and every movie ever made in pi. As such, this work should be suppressed as a DMCA violation!
I'm trying to teach myself to set people on fire with my mind... Is it hot in here?
in 1897 Representative T.I. Record introduced House Bill 246 suggesting three values for pi: 3.2, 4, and ~3.23. These three figures were based on the work of an amateur mathematician Edward Goodwin. The bill was quickly forwarded to the Committee on Swamp Lands (of course), which then forwarded it to the Committee on Education. This committee gave it a pass, where the House approved it unanimously. The bill made it to the Senate.
Before the Senate could make asses of themselves as well, a professor of mathematics at Purdue named C.A. Waldo, intervened, and it died an embarrassing death.
For a more humorous account, read Cecil Adam's account of this at the Straight Dope.
Insert simplistic political, ideological, or personal proselytization here.
See:a l.html
http://www.utm.edu/research/primes/glossary/Illeg
Short version:
This guy took gzipped decss code, and found that it could be expressed as a prime number. So there is an illegal prime. Considering that pi is infinate, it also exists in pi, so some part of pi is also illegal.
Doh.
-Dan
Pseudorandom numbers are often used in place of true random numbers, because usually what is needed is a set of numbers with certain properties common to random numnbers, e.g. uniform distribution. Note that for cryptography, pseudorandomness is often not sufficient, and truly random numbers are needed. These are usually generated by sensing the physical world in some way, where, we assume that the combination of chaotic processes and quantum effects makes the incoming values truly unpredictable.
It has also been a popular quote for cookie files for years.
Q:How many libertarians does it take to stop a Panzer division? A:None. Obviously market forces will take care of it.
There has existed an algorithm to find the nth hexadecimal digit of PI for a couple of years now. It seems to me, going from hex to binary is trivial.
More info can be found http://www.mathsoft.com/asolve/plouffe/scimath.txt - there.
This may be the first time they've EVER had significant traffic on their servers. How often do YOU look for cool, interesting articles on the DOE's website. Not very often, I suspect. =)
And they'll be hearing from God's lawyers. This "formula" is clearly a circunvention device.
--
--
Stay tuned for some shock and awe coming right up after this messages!
Hey I once derived the quadratic formula while sitting in calc @ class simply because the lecture was so boring, that the derivation was more interesting than anything els i could do at the time ( i had already gotten bored with doodling ;-)
in base pi, pi= 10. not exactly random. Therefore, pi is not random in all bases.
You can't convert any base to any other base "just like that"! After the decimal place you end up dealing with fractions that are sometimes impossible to properly convert to fractions of another base.
Here's an easy proof: Try converting 0.025 into a clean non-repeating binary fraction.
Don't waste your time. You can't.
You mean the so-called missing links? Well, they're not missing, at least there are examples for the fish->reptile transition, for the reptile->bird transition (archeopteryx is most famous) and, IIRC, for the reptile->mammal transition. These are not transitions from a species to another (the species is the only real biological indicator of differences, because it forms the border of which animals can breed/have fertile descendants; all the other like race, family, ... are artificial) but from different orders of animals. If such a transition is possible, why shouldn't the transition from species to species of the same family/order be possible?
Human history has shown some transistions from species to species, eg. wolf->dog, wild boar->domestic pig, etc. What more prove do you need?
Plus, the anthropology (Is that the right word? I'm not a native English speaker, and I'm also stoned right now.) of the human species is so well documented; the adaption of a new species should be obvious!
Free Manning, jail Obama.
This assumption is valid since you're dealing with copyright materials which are not infinite in length. Mind you, you never know about the source code for windows XP 2005 - when the source code becomes infinitely long, it, most likely, will not be found as a subsequence of pi. Never underestimate the power of MS code bloat!
If Pi was "random," as apparently the poster of this story is far from anything you would consider a Mathematician or even a "Math intelligent" person, then each time you derived Pi the numbers would be different. e.g. 3.14159... 3.204845... 2.09284...1.38485... You have confused the word "Random" with something it truly is not. The question you are looking for is each time I derive a "new" digit of Pi, will it be predictable, will it be cyclic? That has nothing to do with Pi being random. Because each time you want to derive that certain Nth digit of Pi, it will be the same. That proves Pi is not random, just not cyclic, as of yet.
Bottom line, philosophically at least, Pi cannot be random because it can be completed defined in a very small number of characters (e.g., the ratio of the circumference to the diameter of a circle expressed in a flat (Euclidean) space. Even better, as the solution to a simple infinite series.) This argument also applies to e, and some other transcendental numbers.
Of course, authors like the ones quoted in the root story tend to confuse statistical tests with fact ...
XOR the input data with your representation of pi then compress the result with a conventional compression algorithm.
This algorithm will be quite bad with conventional input (text files etc) but will compress pi quite effectively.
The digits of pi are no more "random" than the digits of 1/3. There is a pattern to the digits of pi because there is an algorithm for calculating them. We just don't see the pattern a first glance.
You could say that pi is more random than 1/3 because of the even distribution of the digits, pairs of digits, etc. Is there a measure of the "randomness" of a sequence of digits? Isn't "randomness" in this sense as subjective as "compressibility".
A final thought, would we consider the digits of 1 random if our number system was base pi?
Here is a java applet that will convert a decimal number to any base larger than 1 (even fractional bases). Source code is also available from this page although with a tiny bit of thought you could probably code it yourself.
In the "1,2,3,4" case you are using information about the likely source of the digits to predict the next one (would you still say 5 if you knew the digits were generated by rolling a die?)
All sequences provide no information about the next digit in the sequence unless you know something about the likely method used to generate that sequence.
if they have found a:
"simple formula discovered makes it possible to calculate the Nth binary digit of Pi without computing any of the first N-1 digits, and do the computation with very little computing power."
Doesn't this make it possible to assign an extremely large number to N - say, a billion times bigger than we've ever calculated pi out to before - and see if that resolves to a zero? And, if so, proving that pi eventually ends? and working backwards from this theoretical point, couldn't you quite easily find the last digit of pi basically using a binary tree method?
---- keep it simple.
Sheesh, I love it when people pass around old barbs that are obviously false.
Try this link for the real deal on "legislature makes pi == 3".
Did a state legislature once pass a law saying pi equals 3?
or for the goat-fearful
http://www.straightdope.com/classics/a3_341.html
Come on down here and say that, boy.
pi = 4 * (1 - (1/3) + (1/5) - (1/7) + (1/9) - (1/11) ... ) doesn't count?
--
Please consider making an automatic monthly recurring donation to the EFF
Does a base have to be an integer? If not, pi in base(pi) is 10 exactly.
<grub> Reading
Suppose there is some base b such that the digits of repeat. Then Pi * b * m = n where m and n is some integer. And so we would have Pi = n / b *m. But m and n are integers, as is b. So you've just shown that Pi is a rational number. It is not. Hence, no such base exists.
What have we been telling people? It's all about Base 5!
Ewige Blumenkraft!
-shpoffo
it was /.ed - also try his home page for more links and such.
Algorithmic information theory defines the amount of information in a string as the length of the (shortest, I would presume--you can always pad code) program that generates it. A random sequence is one that's uncompressible--the best you can do for a program to emit it is to have a copy of the sequence itself in initialized data and spit it out.
Now...if there's an algorithm to generate an arbitrary digit of pi, obviously you can use it to write a function to generate all of them (eventually, in the sense that for any fixed N, you'll only have to wait a finite amount of time for the Nth digit to come out). That seems pretty darned compressible to me, so how the heck can the digits of pi be random? Is my understanding totally off here, or do counterintuitive things happen for infinite strings?
The value of the starting position for any useful work would be such a large number that simply expressing the number would in almost all cases take far more bytes than to store the work itself (barring some infinitesimally unlikely occurrence, which is just as unlikely as finding the work randomly occurring in any other form in nature, such as twigs falling from a tree and arranging, just by random luck, into letters and spelling a message). In fact, since the stream of digits of pi is infinitely long, not only can every work that has ever, does, and will ever exist be found in it, but every single work must lie buried an infinite number of times! For any work, you could come up with as many numbers as you want to describe its position.
Small numbers indicating a quantity cannot be copyrighted, but the numbers necessary to express these digit positions would be far beyond any useful quantity imaginable. Just as all digital data is reducible to a long number, these digit positions are encoded data, not useful quantities, and therefore would be protected by copyright as just a representation of the works they are "pointing" to in the digits of pi.
the natural logarithm of 2, often written "log(2)"
Isn't that supposed to be ln(2)?
--
Something very similar has already been done. Search for the "GNU superoptimizer" : "GSO is a function sequence generator that uses an exhaustive generate-and-test approach to find the shortest instruction sequence for a given function."
doublea,b=4,c;main(){for(;++a<2e6;c-=(b=- b)/a++);printf("%f\n",c);}
This is something I never understood. Why is it better to be a slave to randomness than a slave to the universe before you? When you really get down to it, that's just the problem (as I see it): nobody can come up with even a vague definition of "free will". If it's randomness, then a geiger counter hooked up to the control circuitry of a robot has free will. If it's deterministic, then the crudest pseudo-random number generator has "free will". (At what point is there sufficient complexity to call it "free will"?)
It can't be nothing more than randomness, because clearly there is no "will" there at all. There is no rhyme or reason, and thus no meaning, and I can't call that "free will". There must be some order to bring meaning to the decisions we make. But order is the result of determinism! But simple determinism gives me no power that randomness didn't. It still leaves all power in the hands of those before me. So what is it? What is free will? A combination of the two? But that still leaves me utterly powerless. Obviously if I sit at home drinking beer all day, I will go nowhere, I must get off my butt to have a fruitful life. But how is this any more than a mere illusion of freedom?
That's what I hate about philosophy: there's no way out!! ;)
PS -- I'm no physicist, but I've taken a few intro quantum classes and things. And I gotta say, I still don't quite buy the notion of nondeterminism. But who knows, and in the end, does it really matter? I just wish scientists would be more clear about the fact that everything they say is nothing but hypothesis, with a dash of evidence thrown in. I don't know that the sun will rise tomorrow. An infinite number of things could stop that from happening. When I hear scientists spewing their drivel about what they "know" it annoys me so much... (I think the key to the universe is simplicity, and it seems all these guys want to do is make the theories more complex to fit their crappy experimental results. "What?!? That's not what was supposed to happen. I know: there must be a brand new particle that we don't know about, that can do anything, and that is impossible to detect. HAH! Our theory is complete! And look, our experiment even backs it up!") Ok, enough of my ranting...
The streets shall flow with the blood of the Guberminky.
I think the more generalized algorithm (which might be what they are talking about) is the PSLQ, which is only like a year and a half old. They talk a little about it at http://www.nersc.gov/news/bailey1-20-00.html, but the PSLQ link seems to have been removed due to a copyright lawsuit.
Actually, "We apologise for the inconvenience."
Vintage computer games and RPG books available. Email me if you're interested.
Mmmmmm, pi
PHC
This Wiki Feeds You TV and Anime - vidwiki.org
So, anyone can calculate nth digit right? nth digit doesn't require n-1 digits, right? Am I the only one thinking "THIS WOULD MAKE A GREAT DISTRIBUTED CLIENT!!!"?
Peace,
Amit
ICQ 77863057
[o]_O
I couldn't get the link in the story to work, and found this while searching for the story.
The Economics of Website Security
As a current Reedie I get a big kick out of everytime I hear The Center for Advanced Computation At Reed College. Basically a renovated two-bedroom residential house that was annexed because it was too close to the campus this unassuming building is even unknown to most Reed community members. It's absolutely hilarious when the media come by looking for The Center for Advanced Computing and we point them to a small house behind some trees. As far as I know Crandall is the only faculty there.
Crandall is a wonderful lecturer whose seminar talks never fail to befuddle the rest of the faculty (let alone the undergraduates). Personally, I have made a solemn vow stay away from the areas of physics that will likely degenerate into number theory. Regardless of my preferences, his class in scientific computing is more than worthwhile.
e = 2.7 something (irrationnal number)
"I remember Y1K, every abacus had to get another bead"
That would be pointless.
:) Hey, you know the odds of a certain signal having various frequency components! And, you know the total amount of retained frequency in that block. Put the two together... you have very accurate probabilities :) You might get an extra 5% out of sound and 15% out of images (and even more out of video if it uses 3d transforms instead of layered 2d transforms).
To do valid compression with huffman coding, you first need to determine frequency of occurance. But, the frequency of occurance is all you want in this case - why finish the job of building the tree?
Actually, arithmatic compression is much better. If you're not familiar with it, it is a probabilistic way of storing bits, where, the better you know the probabilities, the better you can do. Worst case, its compression is equivalent to huffman coding. Best case, the sky is the limit, it depends on your knowledge. I have an interesting application I want to try some time using huffman coding to store MDCT blocks
Any signal compression algorithms already use arithmatic encoding? Anyone know?
-= rei =-
"Well, then fire it up and show me what this..." (sigh)
I think maybe you missed the point. If you try to measure by putting your elbow to the ground 30 times and eye-balling it, and you have a systematic error of one inch each time you do that, it's 30 inches of error by the time you're done. Given those circumstances and no knowledge of higher math, 3 isn't such a bad estimate.
For all intensive purposes, "whom" is no longer a word. That begs the question, "who cares"?
Also the range of characters used is often very limited - simply decoding a few characters with every sequence, and see whether it would give you any characters outside the expected set would let you throw away a huge number of starting points very quickly.
This problem is easy enough to solve. Just gzip before you encrypt. Then they might look for the gzip magic number, but you can exclude that too. The only real problem would be if gzip uses a "dictionary". They might be able to pick words out of that. So, gzip might not be the ideal choice but I'm sure there must be some compression method that produces statisticly random results, even in the early part of the file.
For all intensive purposes, "whom" is no longer a word. That begs the question, "who cares"?
Well, if you're going to be using "cubits", it's not like precision is really a concern to begin with. IIRC, The cubit was the distance from the elbow to the tip of the longest finger. Whoever was ruler at the time set the standard. It would be interesting to see how close we could come doing it by hand, or with a "cubit-stick".
For all intensive purposes, "whom" is no longer a word. That begs the question, "who cares"?
Function: adjective
Date: 1565
1 a : lacking a definite plan, purpose, or pattern b : made, done, or chosen at random <read random passages from the book>
2 a : relating to, having, or being elements or events with definite probability of occurrence <random processes> b : being or relating to a set or to an element of a set each of whose elements has equal probability of occurrence <a random sample>; also : characterized by procedures designed to obtain such sets or elements <random sampling>
---
So, pi probably has no plan or pattern, but arguably does have a definite purpose. It wasn't probably made, done or chosen at random, though it's hard to know.
I don't know if we can talk about probabilities together with pi, more than "if we pick a 5 from the decimal representation, which is the probability of the next digit being 8".
Because the probability of a random number being equal to some predetermined value when the set of possible random numbers is not discrete is exactly zero. Perhaps that's why.
Sure, this is a good way of being sure some number is not random. But it doesn't work the other way round. You can't compress already compressed files or encrypted files (well, from good encryption and compression programs), yet they're not random.
_No_ compression algorithm ever will compress purely random numbers.
Or to be more precise, no compression algorithm ever will compress more than 50% of all possible inputs of the size n or less for any given n. Proof left as an exercise (it's really simple).
Mathematically, given two infinite sets A and B such that A is continuous and B a discrete subset of A, the probability of a randomly picked number from set A belonging to set B is not only very small (i.e. low probability), but zero. Now A is the group of all "numbers" and B is the group of all compressible numbers. Ergo, the chance is not there.
t=pi*t_e, where
t = the time required to finish a project, and
t_e = the estimated time required to finish a project (which also happens to be equal to d-dt, where d is the deadline and dt is the current time).
How about - pi?
Ok, it depends on how we define randomness. You're right on that.
Ok, perhaps better for me to not argue about something I don't understand too well myself either. I'll try to find one of those books for reference if you're interested in the higher mathematics.
And about compressing an infinite set, you're right. I should probably have said "a random number of n bits for a very large n". I'm not that much mathematician yet, though :-)
you really are an idiot
Thank you, sir, for pointing that out.
Irrational Pi,
Use the magic formula.
This post is worthless!
Comment forecast: Bits of genius surrounded by a sea of mediocrity.
Since pi predates the DMCA and everything 'protected' under it, pi must count as 'prior art', invalidating ALL copyright and patent claims.
Hmm, I always thought there were 10 digits in pi. 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
ok then your [sic] infringing on my copyright! Could you as [sic] me next time before STEALING my comments for your own?
You'd probably ask for 2 * 1 / 3.14159... tickets then? (pi being base and you wanting less than 1 unit)
--
I like paying taxes. With them I buy civilization -- Oliver Wendell Holmes
pi = 3
And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it round about.
1 Kings 7:23
Sorry. You have incompletely analysed the verse. Try this: http://www.yfiles.com/pi.html
*** On the Internet, no one knows you're using a VIC-20
Ok, last post; I'm getting sick of this.
Trolls aren't supposed to get sick of trolling!
Slashdot is jumping the shark. I'm just driving the boat.
Doh!
That should be:
"each n-bit sequence occurs 1/(2^n) of all n-bit sequences".
Doh!
The other notable advantage is that your encryption/decryption code is both widely available and doesn't appear to be cryptographic software. This is a huge advantage to those people who can be convicted/shot simply for having the ability to encrypt and decrypt information.
If it is possible to calculate digits of Pi starting at any point, then you could easily use Pi as a pseudo-random pad.
Once you know the starting digit location, you can easily decrypt something that has been XOR'd with the sequence from that point onward. But - given that each n-bit sequence occurs 1/n of all n-bit sequences, there are essentially an infinite number of options facing the code-breaker - even after each successful step!
If you are feeling particularly vicious that day, encrypt with two XOR sequences, based on two difference starting points.
I evidently should have read the comments before posting this. Quite obviously, I and others were wrong.
"If you think education is expensive, try ignorance" - Derek Bok
Yes, I have too much spare time.
"If you think education is expensive, try ignorance" - Derek Bok
I thought this was not only extremely difficult, but actually impossible. Pi is proven to be irrational many years ago now. I thought it had also been proven (also many years ago) that you cannot ever know the Nth digit of a irrational number without computing the N-1 first. I also remember an earlier slashdot discussion when some 20 year old had calculated the trillionth digit of pi using some distributed scheme. Naturally, some poster wanted to know whether he had calculated the trillion-1 digits before that one also, and he got instaniously drenched by replys claiming and proving that such a feat is impossible, so at least I'm not the only one having heard this.
Could some math grad with knowledge about this help clear it up?
"If you think education is expensive, try ignorance" - Derek Bok
Using the 'random' digits in pi to calculate pi ... ain't math great? :-)
Actually, that's truer than you might think.
If pi is truly random, then any arbitrary sequence of digits (of any length) is present in it somewhere, at some position. Choose a suitable encoding for the information you're looking for - any encoding will do, e.g. three digit groupings for ASCII numbers - and then just search through the digits until you find that particular sequence. It will be in there somewhere, but quite possibly at an unbelievably astronomical position. In fact, the longer the sequence, the higher the probability that you find it later in the sequence.
BUT WHAT DOES IT ALL MEAN?
Just that it takes all the information in the universe to make it all the way around a circle.
Somewhere, right now, Archimedes is laughing.
DISCLAIMER: the contents of this message are also encoded into the digits of pi. However, the author is not at liberty to discuss the start position.
-----
Free yourself. Everything else will follow.
Well the site has been slashdoted so can someone tell me, did they crack pie or didnt they? the description seems to suggest they did but i cant help but think there is a catch.
I propose we all switch now. Using whole numbers for our bases is childish, and makes dealing with the natural universe so much more awkward than it needs it be.
If pushed, I'd consider 'base e' instead. Can't wait to argue about the next bill in a restaurant. Bistromathics here we come.
> But don't think that you'll win a fight with God
;)
I can't win with God? Guess I'll just have to fight without him, then
OtakuBooty.com: Smart, funny, sexy nerds.
function nthDigitOfPi(long n) {
return rnd(10);
}
Have fun: Join D.N.A. (National Dyslexics Association)
A 64-bit floating number (double) representation of Pi by division of two 32bit integers is:
245850922 / 78256779 = 3.141592653589793e+000
..or even just different coloured guys with different and intresting cultural backgrounds. Any appeal to genetic breeding for races was scientifically scotched donkeys years ago. There has been so much interbreeding between the various 'races' over the past 10k years that it's kinda meaningless outside some obvious phenotypical differences.
That's not to deny various out of africa versus other theorys on anthropological evolution, but it's not something that makes much difference. evolutionary selection in homo sapien sapiens has been pretty much limited to pigment adaptions and a few things.
Excuse the Unicode crap in my posts. That's an apostrophe, and slashdot is busted.
This is good :)
:)
Because we can now skip forward in pi orders of magnitude further than before, we could (if we wanted) use pi (with a random and gigantic start point or seed) as an xor source for cheap and nasty encryption
HelpGeeks - don't bother visiting, it's not worth it! Really!
Any (irrational, nonrepeating) number which we can get the nth digit of by a straightforward, direct formula, that exhibits close-to-randomness would work for the purposes of encryption...
:)
:)
:) Rubbing it in their faces that it's just xor'd with pi would just be icing on the cake :)
But...
With using pi, the fun part comes when you tell someone that the encrypted data they're trying to break has just been xor'd with pi at different starting points a few times, and that you don't feel like telling them the starting points
Even better - the starting points could be taken as a integer equivalent of say an MD5 hash of each of three thirds or four quarters of the password you used
I'd personally hate to be the cryptographer trying to break into your data when the keyspace is potentially infinite depending on password length
...All I want for Christmass (3042) is the look on their face when they hit the last 2048bit key and still can't get to my pr0n...
HelpGeeks - don't bother visiting, it's not worth it! Really!
not pi in the sky
I beg forgiveness, for I
ate all the pi, *sigh*
-- www.globaltics.net
Political discussion for a new world
The most obvious application is Xor encryption. A one time pad (series of bytes used only once) was the first uncrackable encryption method.
An easily computable stream that looks random, and cannot be prestored in its entirety seems tough to crack to me. And there's no difficulty in having everyone you want to read your message all having the same pad. Transmitting the starting position to be used is the only difficulty.
What you're assuming is equivalent to what these people are trying to prove. In short, given a sequence of random digits the likelihood of finding a given specific digit in the sequence is 1 - (9/10)^N, where N is the number of digits in the sequence. Hence as N --> infinity, this likelihood approaches 1. This is painting in broad strokes, admittedly. Now further, given a random sequence of m-digit numbers and a specific m-digit number, the likelihood of finding out desired member in the sequence is 1 - ((10^m - 1)/10^m)^N for an N digit sequence. Thus as N --> infinity, once again our chances of finding our desired number approach 1, for any specific value of m. So your intuition is pretty good, in that if the digits of pi are "random" in their distribution, then your chance of finding a given sequence within them is 1.
cheers, Scott
There is actually a theory that Pi is one of those special numbers (whose name escapes me - making this post a little less useful) which is random in all bases.
I guess that refutes the assertion that the size of the compressed data plus the size of the decompressor is always larger than the size of the uncompressed data.....
Two negatives make a positive, but two positives don't make a negative.
If you disagree with me on social issues, then it's pretty clear that you are a narrow-minded bigot.
There you go, infinite digits in a small, finite space. You don't get better compression than that.
I think that is more a rule of thumb than anything else.
Remember, the formula applies to pi and pi only, so it's not general purpose.
Also, it's easy enough to come up with other cases that refute always. You have a file that's 5 megabytes of the letter 'a', and you encode it using RLE. If the decompressor is >5 Mb, you are a hell of a sloppy programmer. This isn't a real world case, but it's enough to shoot "always" down in flames.
Sorry for the inconvenience
I demand a million helicopters and a DOLLAR!
I'm fairly positive there's no such thing as Base 9.5, just base 9 and base 10 (for example) so I don't think Base (pi) would work either.
pi, during the time period in question, was intellectual property ROTFLMAO, thats hillarious what else did they do patent smells and not let them watch movies they bought? Oh wait, we do that. *Ashamed look*
Anyone who wants a good source of xor'ing digits for their encryption program. Just carve them into groups of 8 (or 16 or whatever you need).
TWW
"Encyclopedia" is to "Wikipedia" what "Library" is to "Some people at a bus stop"
TWW
"Encyclopedia" is to "Wikipedia" what "Library" is to "Some people at a bus stop"
Ah, well, in that case it's all subjective. I think it's cool and you don't. Personally, I don't like Mozart but plently of people do. That's the trouble with sociology (and the other soft-sciences): no one is ever wrong (or right).
TWW
"Encyclopedia" is to "Wikipedia" what "Library" is to "Some people at a bus stop"
A few years ago, the Texas state legislature official rounded Pi to 3.14 because it was "easier". The have since undone that but just think about what that says about the man leading my country. I didn't vote for him.
---
Heh. "Pi"-racy.
--
I feel fantastic, and I'm still alive.
I seem to remember reading somewhere that 140 ish digits is enought to calculate the circumfrence of the observable universe to within the diameter of a hydrogen atom...
Interesting. I seem to remember reading somewhere that 40 digits is enough to calculate the circumference of the observable universe to within the diameter of a proton. Methinks an urban myth is circulating.
There is a base 10 version of the algorithm available too..
-ShunScene
p.s. Pi has been proven irrational, transcendental and non-algebraic.
However, some posters are assuming that every (finite) number sequence occurs starting at some location in Pi. In chaos terms, this is equivalent to saying there exists a dense orbit in phase space (a sufficient condition for chaos to occur). This has certainly been conjectured, but I have not (yet) seen a proof, and have been led to believe it is independant of claims about the digits of Pi having "normal" auto-correlation co-efficients. (c.f. Polya's constant )
Well, Yes and No...
Over the field of real numbers, transcendental is indeed equivalent to non-algebraic.
However, I don't think they are necessarily equivalent over other fields.
-ShunScene
Can someone tell me some down to earth, real reasons that anyone should care what the 12,345th digit of Pi is? I mean really, who cares?Well for most general engineering purposes 5 to 10 places is enough. How many car parts are manufactured to a milli millimeter spec, for example? and to tell the truth, once you hit the quantum level further precision can get a little silly.
"It is a greater offense to steal men's labor, than their clothes"
there's also the problems that in order to store one bit of information you need a bit to store it in. so in order to store all the information in the universe, your computer has to be the universe. reading the results from this program would not only be impossible (since you'd have to also be part of the computer/universe) but would also not be very useful, since what you were trying to predict would aredy have happened.
... and I don't think it some random digit he was putting in the pie, if you know what I mean.
Now what does this tell us for Pi? Well, there are many small programs that compute its continuation. While those programs are probably not the shortets possible, they are much shorter than pi.
I.e. Pi is not truely random, not even close.
would recognize the sequence and reduce the file to "p256M" :)
A few years ago, the Texas state legislature official rounded Pi to 3.14 because it was "easier".
;-)
Lol. Football Phisics 101 classes often round gravity to 10 m/s for the same reason.
If the only tool you have is a hammer, you tend to see every problem as a nail.
Blarf.
*Physics
If the only tool you have is a hammer, you tend to see every problem as a nail.
Blarf.
Is that really the point? This is pure science for the science's sake. What's wrong with that? Besides, one day this knowledge may become useful, who knows? Perhaps, by studying the digits of pi, we'll be able to come up with theories about it's nature. Or, perhaps the pursuit of simplified equations/methods for calculating the digits of pi will lead to other mathematical revelations (just look at this situation... no one thought an equation like the one mentioned in this article was possible). There's no way to tell. I mean, think of all the things people have researched without thinking of practical applications ahead of time. Quantum mechanics, atomic theory, relativity, the myriad forms of pure mathematics such as number theory... and, I'll bet you, all along, people were saying "What's the point of all this? Who cares?" But, because of quantum mechanics, we have computers... because of number theory, we have encryption. So, please, think twice before making comments like this... you never know, one day, the theory behind the nature of Pi may drive the random number generator you use to encrypt your email.
When doing math, it is generally assumed that "log" indicates the natural logarithm, as base 10 holds no special meaning to mathmaticians. Because using both log and ln would be redundant, "ln" is never used. Only log(x) (to indicate "log base e") and "log base R(x)" where R is some number, are used.
"Easy as pi" takes on new meaning...
You see? You see? Your stupid minds! Stupid! Stupid!
A friend of mine once derived the Cosine law in the middle of his Grade 12 Math exam... he could remember the derivation, but not the law.
"Free beer tends to lead to free speech"
I can't exactly remember where, but this is old from at least 3 years.
The C Program implementing this algorithm has been noded on Everything 2.
--
Trolling using another account since 2005.
Incase that link goes down, here is a copy of that article. BERKELEY, CA -- David H. Bailey, chief technologist of the Department of Energy's National Energy Research Scientific Computing Center (NERSC) at Lawrence Berkeley National Laboratory, and his colleague Richard Crandall, director of the Center for Advanced Computation at Reed College, Portland, Oregon, have taken a major step toward answering the age-old question of whether the digits of pi and other math constants are "random." Their results are reported in the Summer 2001 issue of Experimental Mathematics. Pi, the ubiquitous number whose first few digits are 3.14159, is irrational, which means that its digits run on forever (by now they have been calculated to billions of places) and never repeat in a cyclical fashion. Numbers like pi are also thought to be "normal," which means that their digits are random in a certain statistical sense. Describing the normality property, Bailey explains that "in the familiar base 10 decimal number system, any single digit of a normal number occurs one tenth of the time, any two-digit combination occurs one one-hundredth of the time, and so on. It's like throwing a fair, ten-sided die forever and counting how often each side or combination of sides appears." Pi certainly seems to behave this way. In the first six billion decimal places of pi, each of the digits from 0 through 9 shows up about six hundred million times. Yet such results, conceivably accidental, do not prove normality even in base 10, much less normality in other number bases. In fact, not a single naturally occurring math constant has been proved normal in even one number base, to the chagrin of mathematicians. While many constants are believed to be normal -- including pi, the square root of 2, and the natural logarithm of 2, often written "log(2)" -- there are no proofs. The determined attacks of Bailey and Crandall are beginning to illuminate this classic problem. Their results indicate that the normality of certain math constants is a consequence of a plausible conjecture in the field of chaotic dynamics, which states that sequences of a particular kind, as Bailey puts it, "uniformly dance in the limit between 0 and 1" -- a conjecture that he and Crandall refer to as "Hypothesis A." "If even one particular instance of Hypothesis A could be established," Bailey remarks, "the consequences would be remarkable" -- for the normality (in base 2) of pi and log(2) and many other mathematical constants would follow. This result derives directly from the discovery of an ingenious formula for pi that Bailey, together with Canadian mathematicians Peter Borwein and Simon Plouffe, found with a computer program in 1996. Named the BBP formula for its authors, it has the remarkable property that it permits one to calculate an arbitrary digit in the binary expansion of pi without needing to calculate any of the preceding digits. Prior to 1996, mathematicians did not believe this could be done. The digit-calculation algorithm of the BBP formula yields just the kind of chaotic sequences described in Hypothesis A. Says Bailey, "These constant formulas give rise to sequences that we conjecture are uniformly distributed between 0 and 1 -- and if so, the constants are normal." Bailey emphasizes that the new result he and Crandall have obtained does not constitute a proof that pi or log(2) is normal (since this is predicated on the unproven Hypothesis A). "What we have done is translate a heretofore unapproachable problem, namely the normality of pi and other constants, to a more tractable question in the field of chaotic processes." He adds that "at the very least, we have shown why the digits of pi and log(2) appear to be random: because they are closely approximated by a type of generator associated with the field of chaotic dynamics." For the two mathematicians, the path to their result has been a long one. Bailey memorized pi to more than 300 digits "as a diversion between classroom lectures" while still a graduate student at Stanford. In 1985 he tested NASA's new Cray-2 supercomputer by computing the first 29 million digits of pi. The program found bugs in the Cray-2 hardware, "much to the consternation of Seymour Cray." Crandall, who researches scientific applications of computation, suggested the possible link between the digits of pi and the theory of chaotic dynamic sequences. While other prominent mathematicians in the field fear that the crucial Hypothesis A may be too hard to prove, Bailey and Crandall remain sanguine. Crandall quotes the eminent mathematician Carl Ludwig Siegel: "One cannot guess the real difficulties of a problem before having solved it." Among the numerous connections of Bailey's and Crandall's work with other areas of research is in the field of pseudorandom number generators, which has applications in cryptography. "The connection to pseudorandom number generators is likely the best route to making further progress," Bailey adds. "Richard and I are pursuing this angle even as we speak." For more about the normality of pi and other constants, visit David Bailey's website. The BBP algorithm for calculating binary digits of pi was found using the PSLQ algorithm developed by Bailey and mathematician-sculptor Helaman Ferguson; it is discussed at Bailey's website and also in the Fall 2000 issue of Berkeley Lab Highlights. The Berkeley Lab is a U.S. Department of Energy national laboratory located in Berkeley, California. It conducts unclassified scientific research and is managed by the University of California. Contact information: Scientific queries can be addressed to David Bailey at dhbailey@lbl.gov
Lawrence Berkley lab has the orignial story their website
Does it make you happy to crap on other people's faith? I love science and all the evidence in the world points to evolution, but I still believe in creationism. Are you telling me that there is not the slightest chance that creationism is the truth? Faith tells us that God is all knowing and all powerful. He could have made the universe 5 minutes ago and created you in it with all the memories you have. You think you can out think the Lord. How naive. In your opinion I am simple minded. I cannot imagine a more narrowminded egocentric view than to think that you can disprove all of Chritianity with a, "Uh-ahh"
In other words, all strings of random numbers have entropy of 1? Nope. Explain to me why I can get this out of a "perfect" random number generator:
000000000000000000000000....
Now, granted, the probability of that is *low*, but it's there just the same.
Now, your statement would work just fine if you were talking about the *complete* digits of PI. In fact, if you give me a stack of disks with a complete listing of all of the digits of PI, I'd be happy to compress it for you.
Actually i think the three bodies problem is how to convince ones girlfriend to include the third body in that mutual gravitational field
-Ted
-=-=- Quantum physics - the dreams stuff are made of.
No-one did. They just rounded up his number of votes. It was easier that way.
While the article title and the intro at the top both use the word "random", it becomes clear after the dateline that what they should be using is "normal". The question is whether the digits of pi, and also other similar natural constants, have a normal distribution of digits (in base 10 or any other base).
As plenty of other people are pointing out, the use of "random" suggests that the digits shouldn't be fixed, which is a bit of a problematic concept when you're talking about a constant... ;-)
Because they don't subscribe to the myth of evolution (a theory growing weaker and weaker by "conventional" science), doesn't make it crack-pot science
Oh PLEASE.
Evolution is a fact. Pure and simple, and as well established a fact as virtually any other scientific principle. DNA and the human genome is pretty rock solid proof of it. Now, HOW evolution works is still a theory, and there are many theories that try and explain it. That's an entirely different issue.
But ANYONE who believes some 'god' just created earth and then poof, man and all the animals "just appeared", is a simple-minded moron. The Bible is a book of MYTHOLOGY and FABLE. A learning text for simpleminded illiterate tribesmen. Adam and Eve is an interesting little fable, but there is no way any rational intelligent human could believe that that cute story is REAL DOCUMENTED HISTORY unless they had been brainwashed by a religious cult all their lives, and are incapable of escaping those mental chains.
- Spryguy
- Spryguy
There are three kinds of people in this world: those that can count and those that can't
i'd hazard a guess that pi, in base pi, is 1.
I guess you think that means that ten in base ten is "1", eh?
- Spryguy
- Spryguy
There are three kinds of people in this world: those that can count and those that can't
Infinity is not a number, so the phrase 'infinityth digit' is meaningless.
- Spryguy
- Spryguy
There are three kinds of people in this world: those that can count and those that can't
If you still believe in creationism then you are a fool. I've read up on it, and it's so much pseudo-scientific gibberish. There is NO evidence that the world is only 6000 years old, and NO evidence that all species were created more or less simultaneously, and haven't changed since.
I treat kooks and morons that believe in creationism with the same exact level of ridicule that I treat kooks and morons who believe the "world is flat". Because they are exctly the same ilk... brainwashed, intellectually dishonest, gullible fools.
Yes, I am telling you there is not the slightest chance that creationism is the truth. How any intelligent thinking being could even imagine that it is The Truth is beyond me. Tell me, is the tooth-fairy The Truth? Santa Clause? A flat earth? An earth-centered solar system?
And you call ME naive? You *are* simple minded, given this reply.
- Spryguy
- Spryguy
There are three kinds of people in this world: those that can count and those that can't
Oh yeah. And even the Pope belives in evolution. "Christianity" does not require the believe in "creationism", at least as literally described in Genisis. There is no fundamental incompatibility with belief in the Christian God and/or Jesus, and the theory of evolution. The only incompatibility is if you believe the story of Adam and Eve is LITERAL, rather than allegory. And most intelligent people know it as allegory. A story. A simplified version. MOST Christians I know do not believe in 'creationism'. So debunking the inanity of creationism does not in any way "disprove all of Christianity".
- Spryguy
- Spryguy
There are three kinds of people in this world: those that can count and those that can't
Nothing I say will convince you of anything.
... it's not weak at all, but I'll certainly agree it's incomplete since we don't understand the MECHANICS of it all and still have a lot to learn ... but it's a far cry better than the evidence for Creationism, which is nearly zero. The only 'evidence' there is is the book of Genisis. Hardly compelling in any way, scientifically. There is NO other evidence supporting a 6000 year old earth with static species.
True enough, due to the fact that you're trying to argue SCIENCE by using FAITH. The two are incompatable at a fundamental level. You cannot ever PROVE anything by using faith-based reasoning, by definition.
I only ask you to fully examine the evidence supporting your beliefs. Personally, I find it weak and incomplete.
Oh, I've fully examined them, and it all makes perfect sense. I've also read up on Creationism, and it makes no sense what-so-ever. You may think the evidence is 'weak and incomplete'
You might want to investigate DNS studies, especially mitochondrial DNA. You might want to investigate embryo development, where our common roots with other species are far more evident (the residual tail that forms in early stages of development, which then disappears... same with gill structures). You might want to map DNA across species and see how the mutations and changes of occured over time. Never mind the fossil record, the geological record, etc. Hell, the way the HIV virus mutates so rapidly is an excellent example of very rapid small-time-scale evolution at work.
The evidence for creationism? Your belief because someone told you so. Hrm. Not exactly compelling, and it certainly doesn't hold up under scientific scrutiny.
- Spryguy
- Spryguy
There are three kinds of people in this world: those that can count and those that can't
"Does it make you happy to crap on other people's faith?"
Do you believe in enabling other people's ignorance, just because they cloak that ignorance in the name of 'faith'? What about people claiming the earth is flat? Their faith tells them so. Does that make them any more right? Or that the earth is the center of the solar system. Or that 2+2=5?
If some mental midget wants to believe utter lies and crap, they will. But I feel no particular compulsion to enable them in their self-delusion.
Besides, there are tons of people who simply dont' understand evolution. Evolution is the fact we observe. Darwin's Theory of Natural Selection was but one early theory trying to explain evolution. We don't know exactly how it works. Maybe it works entirely by devine intervention, who knows. We just know it DOES happen.
Creationism is just pure bunk, through-and-through. It's using pseudo-science and hand-waving in order to try and 'scientifically' rationalize the LITERAL interpretation of creation in the bible.
Which of course doesn't (and can't) work.
But more than that, it is an amazingly stupid argument to begin with, totally ignoring the fact that virtually every religion out there has its own creation myth. What makes the christian one "right"? Why isn't the Norse myth of creation the right one? Why aren't we teaching THAT in schools? Or the Shinto version? Or the Hindu version? Or any of the other 4,000+ creation myths out there?
- Spryguy
- Spryguy
There are three kinds of people in this world: those that can count and those that can't
here's a simple test... try to compress the "random" string of numbers; if you can compress a string of random numbers, it isn't
Oh god, that woman is John Romero!
Indeed God is all knowing and all powerful (and also benevolent). That implies the knows what is best, and has the ability to do the best. Nothing that God does is "not the best". Hence when he created the world he made it perfect and there is no need for miracles. There can be no observable evidence that God exists - or his creation would be imperfect.
Given that God is benevolent, then he will not be so decieptful as to plant false evidence. A God who treats us as a game is not a God but a devil.
Any creationist either has not thought it through or they are certifiably insane. Scientology makes more sense...
Zero Sum (don't amount to much). [root@localhost]
Please present yourself to the federal police for arrest...
Zero Sum (don't amount to much). [root@localhost]
Any God who would do that *IS* a devil....
Zero Sum (don't amount to much). [root@localhost]
the more interesting question in terms of crypto is: if you know the value of the nth bit of pi, but not n (ie, you know the number but not the place), can you determine the value at the (n+1)th bit? if you think about it for a second (informally of course) you'll see that you can't. in a true random stream the odds of any given number following any other given number are evenly distributed, which in binary means, even if you know if bit n is 0, the odds of bit n+1 being 0 are 50%. therefore, you can use the bits of pi as a stream cipher. pick a starting offset (bit n of pi), calculate the successive bits of pi equal to the length of the message, xor it with the message, and transmit. the starting offset then becomes the key to the message, and if you transmit that securely (always a big if) then the protection is as good as a one time pad. this works as long as either a) pi is not cyclic or b) the cycle (if any) of pi is longer than the length of the message. the ability to calculate the n'th digit of pi does not necessarily imply that pi is cyclic, although it might. personally, i think this is hella cool. i might just have to write a /dev/pirandom driver; seed yourself at boot time from the {u,}random entropy pool, use that as the initial offset. it'd take some doing, bignum arithmetic and all that, but definitely doable.
i'd hazard a guess that pi, in base pi, is 1.
probably a number that's irrational in a number system with a rational base will continue to be irrational in all number systems with rational bases. (this is in fact by definition; "irrational" means "not producable by dividing one integer by another", and bases are just notational division.)
if you really care to think about irrational number bases (and the associated notational headaches), then assuredly there is some number base wherein pi is rational (an "integer" ratio of the base). which isn't terribly useful information, unless you have a way of specifying irrational numbers to infinite precision, a way of notating them, and the mental flexibility to think in them. we wish you luck.
The scary part about this approximation is that 2pi, a very important number in trigonometry, is 666/106. Now THAT'S scary. (But on the plus side, that means that even if 666 is the sign of the beast, the sine of the sign of the beast is zero.)
And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it round about.
The question whether the decimals of pi are random has more to do with our definition of random than pi itself. Of course we all agree that pi remains the same forever. On the other hand, if you want series of random integers, why not start at some decimal in pi?
So, are the decimals random? The answer is not more interesting than that to the question whether a number can be "probable prime" (either it is prime, or it is not, so how could it probably be prime, the probaility is definately 0 or 1).
In addition, a simple formula discovered makes it possible to calculate the Nth binary digit of Pi without computing any of the first N-1 digits, and do the computation with very little computing power.
In light of this article, the obvious method is now:
srand(time); #random enough, thank you
$nth_digit = random();
Duh!
"And like that
Pi is not random, because there is an algorithm that computes its digits. It only "seems" like random.
Omega is a number that is well defined and has a value, but we are unable to compute it.
Omega is not random. It's a constant. But If someone (God?) gives us the value of Omega and the value of any random number, and we have to choose wich one is Omega, we have no logical way to decide.
For more information see : The "Omega Number" & Foundations of Math
---
MOD THE CHILD UP!
And today, thanks to the hard work of a pair of students at Carnegie Mellon University, we can read that language.
And without further ado, here is the hidden message starting at the 74088 digit:
In addition, a simple formula discovered makes it possible to calculate the Nth binary digit of Pi without computing any of the first N-1 digits, and do the computation with very little computing power.
sure, they can get the Nth binary digit down, but that says nothing about the last digit of binary pi. After years of calculation, estimation, and bullshitization, my team of expert physicists, mathematicians, and old-wise-men-of-the-village, have come to the ultimate conclusion that the last digit of pi in binary is undoubtedly equal to 1.
to which I said "thats non-sense. write down pi, then add a 0 to the end. is that not the same value?"
After many more years of careful study of the texts, bringing in ancient greek and latin scholars, and a couple ancient arameic into the mix, my team once again came up with a solution. The last siginificant digit of pi in binary is undoubtedly equal to 1.
I thought long and hard about this and realized it was certainly true. Then, when I thought a bit more, I realized we found something even more significant. Knowing that any number with a fractional part in binary must, when converted to decimal, end with 5, I therefore reasoned the last significant digit of pi equals 5. Looking at my printout of decimal pi, I then resolved that when speaking in sigificant terms, pi does in fact equal 3.1415. I had always laughed at my engineering professors who seemed to use such silly approximations, but now I know for a fact 3.1415 is for all intents and purposes equal to pi.
this is bound to be a joke
Nope, that was Kansas. And once they were quietly taken aside and had it explained to them by a math professor, they withdrew the bill.
--
"Open source is good." - Steve Jobs
"Open source is evil." - Microsoft
How are you going to reduce it to a number? By adding all the bits or something? Well, there would be many very similar strings that would add up to that number. For example:
"The dog was red" would be the same hash as "Teh dog was red".
Things like CRC (haha), MD5, and SHA take a more statistical approach. The two sentences above will most likely not generate the same hash.
Except that there is a good chance that the index to pi requires more bits to represent than the length of the origional file. It has been mathematically proven that a compression algorithm cannot be created which compresses every file. (Note: I guess this topic comes up so often that the comp.compression FAQ contains a version of this proof.)
Come test your mettle in the world of Alter Aeon!
Or even better:
:)
Use *base pi*. Why didn't someone think of this before? It would save a LOT of effort... those silly guys!
Now, all we have to do is find out what pi^1 is... oh, wait.
Since the website seems to be /.ed, I give you the article:
===
Are the Digits of Pi Random? A Berkeley Lab Researcher May Hold the Key
A researcher at the Department of Energy's National Energy Research Scientific Computing Center (NERSC) at Lawrence Berkeley National Laboratory, and his colleague at the Center for Advanced Computation at Reed College, have taken a major step toward answering the age-old question of whether the digits of pi and other math constants are "random." Their results are reported in the Summer 2001 issue of Experimental Mathematics.
July 26--Pi, the ubiquitous number whose first few digits are 3.14159, is irrational, which means that its digits run on forever (by now they have been calculated to billions of places) and never repeat in a cyclical fashion. Numbers like pi are also thought to be "normal," which means that their digits are random in a certain statistical sense.
David Bailey
Describing the normality property, David H. Bailey, chief technologist at NERSC, explains that "in the familiar base 10 decimal number system, any single digit of a normal number occurs one tenth of the time, any two-digit combination occurs one one-hundredth of the time, and so on. It's like throwing a fair, ten-sided die forever and counting how often each side or combination of sides appears."
Pi certainly seems to behave this way. In the first six billion decimal places of pi, each of the digits from 0 through 9 shows up about six hundred million times. Yet such results, conceivably accidental, do not prove normality even in base 10, much less normality in other number bases.
In fact, not a single naturally occurring math constant has been proved normal in even one number base, to the chagrin of mathematicians. While many constants are believed to be normal--including pi, the square root of 2, and the natural logarithm of 2, often written "log(2)"--there are no proofs.
The determined attacks of Bailey and his colleague Richard Crandall, director of the Center for Advanced Computation at Reed College, Portland, Oregon, are beginning to illuminate this classic problem. Their results indicate that the normality of certain math constants is a consequence of a plausible conjecture in the field of chaotic dynamics, which states that sequences of a particular kind, as Bailey puts it, "uniformly dance in the limit between 0 and 1"--a conjecture that he and Crandall refer to as "Hypothesis A."
"If even one particular instance of Hypothesis A could be established," Bailey remarks, "the consequences would be remarkable"--for the normality (in base 2) of pi and log(2) and many other mathematical constants would follow.
A simple formula discovered with the integer-relation algorithm dubbed PSLQ makes it possible to calculate the Nth binary digit of Pi without computing any of the first N-1 digits, and do the computation with very little computing power.
This result derives directly from the discovery of an ingenious formula for pi that Bailey, together with Canadian mathematicians Peter Borwein and Simon Plouffe, found with a computer program in 1996. Named the BBP formula for its authors, it has the remarkable property that it permits one to calculate an arbitrary digit in the binary expansion of pi without needing to calculate any of the preceding digits. Prior to 1996, mathematicians did not believe this could be done.
The digit-calculation algorithm of the BBP formula yields just the kind of chaotic sequences described in Hypothesis A. Says Bailey, "These constant formulas give rise to sequences that we conjecture are uniformly distributed between 0 and 1--and if so, the constants are normal."
Bailey emphasizes that the new result he and Crandall have obtained does not constitute a proof that pi or log(2) is normal (since this is predicated on the unproven Hypothesis A). "What we have done is translate a heretofore unapproachable problem, namely the normality of pi and other constants, to a more tractable question in the field of chaotic processes."
He adds that "at the very least, we have shown why the digits of pi and log(2) appear to be random: because they are closely approximated by a type of generator associated with the field of chaotic dynamics."
For the two mathematicians, the path to their result has been a long one. Bailey memorized pi to more than 300 digits "as a diversion between classroom lectures" while still a graduate student at Stanford. In 1985 he tested NASA's new Cray-2 supercomputer by computing the first 29 million digits of pi. The program found bugs in the Cray-2 hardware, "much to the consternation of Seymour Cray."
Crandall, who researches scientific applications of computation, suggested the possible link between the digits of pi and the theory of chaotic dynamic sequences.
While other prominent mathematicians in the field fear that the crucial Hypothesis A may be too hard to prove, Bailey and Crandall remain sanguine. Crandall quotes the eminent mathematician Carl Ludwig Siegel: "One cannot guess the real difficulties of a problem before having solved it."
Among the numerous connections of Bailey's and Crandall's work with other areas of research is in the field of pseudorandom number generators, which has applications in cryptography.
"The connection to pseudorandom number generators is likely the best route to making further progress," Bailey adds. "Richard and I are pursuing this angle even as we speak."--by Paul Preuss
===
Enjoy.
-- russ
Natural != (nontoxic || beneficial)
From the discoverer of the formula:
"This formula permits one to compute the n-th binary or hexadecimal digit of pi, without computing the first n-1 digits, by means of a simple scheme that requires very little memory and no multiple precision software. Here are various items that may be of interest: "
Yes, that's right, binary or hexadecimal. In other words, you can forget about doing "cool stuff" with it. Who cares what pi is in binary?
--
I don't think anything is ever "random".
Everything has its logical base... but alot of things can be way beyond our comprehension and thus can be considered "random".
Too bad I can't get to the article to see how they are defining random. I have studied random numbers quite a bit, and have worked on the assumption that any thing that can be calculated is not truely random. So under that definition no, it isn't random, and neither are any of the random number generator algorithms.
The comon test for randomness is the chi squared test which actually tests for dispersion of numbers. That is are number occuring in 'equal' frequencies in an order that isn't too similar to the order in other sections of the sample. Failing the chi squared tests shows you aren't 'Pseudo Random' passing it only proves your numbers are dispersed not random
As x approaches total apathy I couldn't care less.
Not sure where I first saw it, but I think it was in a quotes section when I was digging for blowfish code.
+++ UGUCAUCGUAUUUCU
"Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin."
(John Von Neumann, 1951 )
+++ UGUCAUCGUAUUUCU
So, before we all go off building what ever you build out of a PI digit generator, check the code carefully. And test it. Looks like it may have some precision problems with large digit indices.
StoneWolf
Or is it slashdotted already??
There is no spork.
If you're really curious there is a nice 189pg book you can read. It's called "A History of Pi" by Petr Beckman. It's published by Barnes & Nobles Books. I read it a year or so ago, and found it very interesting and at some points downright entertaining.
--Random Fortune--
Ha! I laugh and that it is all a joke.
The RFC is quite flawed as it requires the server to listen on port 314159, whereas the current TCP ports are limited to 64k.
My religion forbids the use of sigs.
So the real question is, will this make an RFC 3091 protocol happen more rapidly?
Sig: Tell all your friends NOT to download the Advanced Ebook Processor:
LedgerSMB: Open source Accounting/ERP
One has to be careful with such humor on /. Many will fail to catch it and think you are serious.
As to RFC 1149, Maximum trasmission retry for TCP is 120 sec. So RFC 1149 is nearly useless except for some ICMP and UDP...
Sig: Tell all your friends NOT to download the Advanced Ebook Processor:
LedgerSMB: Open source Accounting/ERP
How about depressing para-techno bands?
Emacs: for people who just never know when to
Consider:
Therefore, somewhere in the digits of Pi is a string of digits which, when transformed into binary, form the code to decrypt CCS on a Linux box. All the scientists have to do is find the correct starting position and how may digits need to be calculated. The resulting information could be spread throughout the internet and used to decrypt protected content.
Further investgation into the true nature of Pi is a violation of the DMCA! This must stop at once!
or... Holy moley! Taking that same argument, one could reason that every movie ever made, or that ever could be made, is buried digitally in Pi somewhere! Piracy is built in to the very structure of the universe!!!!
Tatsujin
I think it was Faraday who, when asked of what use was his mucking around with electricty, replied "Of what use is a new-born child?"
Enough said.
One day I feel I'm ahead of the wheel / the next it's rolling over me / I can get back on / I can get back on
As an engineer, I'm dissapointed they haven't released a similar study on the exponential constant, e. After all, e is probably even more fascinating since e^(j*pi)= -1
--z
In Soviet Russia, the Beowulf cluster imagines you!
(and nobody has yet found a pattern in the continued fractions of Pi) Actually, if I understand you correctly, they have: Pi = 4 * (1/1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 ...)
If you adapt this scenario to something that generates purely random real numbers, the principles are the same. The probability of *predicting* the outcome is 0. But once you run the generator and get a number, the probability of having gotten what you got is 1.
I think its nice to asume that we could use logic to calculate with precicsion that they do hold a real value.
But i dont think so. Numbers are abstract, they are simply a way of approximating reality.
I think think pi could possibly be realized without a constant. That number 3.1415926... will go on forever.
It must be random because with a ratio that has an infiniate number of variables, there is no perfect answer. Each additional number seems arbitrary.
Who knows, anyones guess is as good as mine, this is the first time i ever thought about it as random, but visually, it seems to make sense.
oh well.
--phinn
"Sorry for the inconvenience"
q:]
MadCow
I used to have a sig, but I set it free and it never came back.
Whithout being a flaming asshole, what applications are there for knowing if the digits of PI are random or not?
Also, since Pi is a ratio that we 'choose' to express in a base10 numerical system, would the fact that the digits are random in a decimal system mean that they would be random if we expressed Pi in a hexidecimal or octal system?
The next Slashdot story will be ready soon, but subscribers can beat the rush and slashdot the links early!
The definition of random is that the value can not be predicted, and of course we can predict any digit of PI. I would say that PI is no different than a long sequence of 1's in respect to being random. It is like a tree falling in the forest, does it make a sound? You could argue about that, but it would just be the definition of the word "sound" that you were discussing.
And does anyone know if that link is incorrect in some way? My DNS can't resolve it.
OK,
- B
--
http://www.bradheintz.com/
- updated
"It is a profound and necessary truth that the deep things in science are not found because they are useful; they are found because it was possible to find them." -Robert Oppenheimer
Robert Moody from the Department Mathematical Sciences, University of Alberta illustrates the importance of curiosity based research in his paper using lasers as an example of why curiosity based research is necessary.
Carl Sagan in his book, The Demon Haunted World, also stresses the importance of curiosity based research using James Clark Maxell's discoveries as an example of how it effects our lives today by providing the necessary building blocks for radio, television, computers, lasers, etc.
It may be a while before we can find any spectacular applications with this new knowledge of pi, or we may not find any spectacular applications before we dissapear in the cosmos.
The point is: We'll never know if there are any spectacular or even merely useful applications if it isn't shared, discussed and debated throughout the community.
"Communism is like having one [local] phone company " - Lenny Bruce
I had this feeling as well. That's why my initial library project had two requirements:
1. small seeds (perhaps some 256 bits would do) of a truly random source.
2. a way of computing square roots and logarithms to arbitrary precision.
The idea was to XOR at least two irrational and/or transcendental numbers, such as square roots (of non-squared numbers, of course!), pi, e, logarithms, etc. It'd be necessary to use some of the random bits to choose which of the numbers will be calculated (pick from one of the four choices above, or others that might be implemented). Then, should we choose a square root or logarithm, we'd need more random bits to pick whether it'd be the square root/logarithm of 32857 or 18764874, say. The rest of the random bits would be used to pick a starting bit in the calculation stream, inside each of the calculated numbers. At last, all streams would be XORed, and that would be the output.
Although it seems like a lot of calculation, remember that computers are getting more powerful everyday, and calculating hundreds of thousands of digits takes a second or so in gigahertz computers; and a truly random source of data (such as timing the decay of radioactive atoms) is slow at generating large streams, but can perfectly cope with the small seeds we need for this library.
Join the NFSNET. Our prime goal is making little numbers out of big ones. http://www.nfsnet.org/
What John Walker's program refers to as optimum compression is Huffman coding, IIRC.
Join the NFSNET. Our prime goal is making little numbers out of big ones. http://www.nfsnet.org/
I should have put it more clearly -- there's nothing special about transcendental numbers in general, indeed. It's just that I've analyzed two of them, Pi and e, and both have uniform statistical distribution and so forth. However, each new addition to this hypothetical library would have to go through the same extensive testing I've applied here, being transcendental or irrational or whatever.
I'd recommend to anyone interested in projects like this to look at George Marsaglia's page; his tester may help you avoid releasing crap.
I have done some testing with DIEHARD as well -- all randomness analyzers seem to back my ideas (:
Join the NFSNET. Our prime goal is making little numbers out of big ones. http://www.nfsnet.org/
Also, there are other uses for random numbers outside of generating cryptographic keys; for example, Monte Carlo simulations.
Join the NFSNET. Our prime goal is making little numbers out of big ones. http://www.nfsnet.org/
Mathematicians usually write logarithms in the base e as simply log. While it is usually taught that log assumes base 10, the base-e logarithm is much more important in math than any other logarithm. If you had been in a calculus course, you'd know that.
Join the NFSNET. Our prime goal is making little numbers out of big ones. http://www.nfsnet.org/
According to Wolfram Research's Mathematica, Log[x] is base e, while Log[10, x] is base 10. Of course, your TI-89 knows better, right? Perhaps you can take the time to ask a mathematician -- a real mathematician, not a high school teacher.
Join the NFSNET. Our prime goal is making little numbers out of big ones. http://www.nfsnet.org/
Look at the original post for stats on the first 512 megabits of Pi.
Also, I had made slight modifications to the ent program (used to generate the stats above), in order to treat the input as a stream of 16-bit values, not 8-bit as done in the above stats. Here are the 512 Mb and 1 Gb stats output by this modified version:
Join the NFSNET. Our prime goal is making little numbers out of big ones. http://www.nfsnet.org/
Sorry, that's not a continued fraction.
.5*(sqrt(5) + 1). Number theory is fascinating. But I'm straying out of my way now. Try to write the fraction on paper, and you'll understand what continued fractions are.
I can't find an expansion for Pi, but for the golden ratio (1.6180339...) they go like this:
Phi = 1+1/(1+1/(1+1/(1+1/(1+1/(1+...)))))
What's interesting, Phi can be wrote as
Join the NFSNET. Our prime goal is making little numbers out of big ones. http://www.nfsnet.org/
Here's the output of John Walker's ent program for 512 megabits of Pi:
For the entropy test, a completely random sample would have an entropy of 8.0 bits per byte, and the ideal Chi Square distribution would be 256.0 (considering there are 256 degrees of freedom in an 8-bit data structure, or 2**8 possibilities.) As you can see, that's about as random as you can get. And the larger the samples you feed it, the more it converges to the ideal values.I've also done some testing with other transcendental numbers, such as e (2.718281828...), and they all seem to show great randomness properties, in the information-theoretic sense at least. However, I have a feeling to "trust" Pi more than e, given that you can write e in form of continued fractions with repeating patterns, and nobody has yet found a pattern in the continued fractions of Pi.
As for my pseudo-random library project, my programming skills are quite bad, but if you have some knowledge of scientific computing (multiplication algorithms using FFTs, for example), you can contact me and I might revive the idea.
Join the NFSNET. Our prime goal is making little numbers out of big ones. http://www.nfsnet.org/
The government survives Code Red, but can't withstand a single Slashdot link...
Liberty in your lifetime
Yep, and wouldn't go through right anyway: I didn't type the 16-bit (or whatever) code for ; I typed the &entity; for it (π). If I cut and paste the then pass it through the comment preview it gets mucked up too.
Liberty in your lifetime
Uh huh, and here's another, with 50% better compression:
Liberty in your lifetime
It may seem like that, but then you never know until you discover something. Electricity was "just a toy" when it was first being researched. It turned out pretty well. No, I don't know what calculating pi to this extent could possibly produce that would be worth all the effort, and it probably won't be, but then again . . . .
--- Don't be a player hater: I meta-mod ALL negative mods as Unfair.
There are pi seconds in a nanocentury.
Ask me about my vow of silence!
Jorge Luis Borges wrote a story called "The Library of Babylon" about an extremely large, but finite, library of random books. Every possible book of a certain length, in a certain character set. In those books can be found all human knowledge, almost infinitely repeated if you consider whitespace and non-essential typos. Also, all human fallacies, the complete works of Shakespeare, including those that were lost to posterity, and works that purport to be the Bard's but are not... etc. The only complication is in finding anything. Same goes for the digits of pi. Infinite sequences are counter-intuitive in lots of ways.
Get your teeth into a small slice: the cake of liberty
You wrote; "And 355/113 is even easy to remember. One One Three over Three Five Five..."
Shouldn't you have said Three Five Five over One One Three??
Just my $0.02 (Canadian, before taxes)
Quite possibly one of the most significant articles to ever slap slashdot across the face (possibly- i havent read the article yet), and already with a mere 11 comments posted, it is SLASHDOTTED.....
ZOUNDS!!@$!$
description of thought.
contradiction or deviation from thought of article.
semi-insightful, semi-obvious, somewhat-karma-whoring conclusion.
(posted after seeing a X != Y on every story for the past day)
> Explain to me why I can get this out of a
> "perfect" random number generator:
>
> 000000000000000000000000....
Actually, you can't. The probability of getting any particular string of digits out of a perfect random number generator is 0.
Yet if you ran such a generator to generate one number, you would get a number whose probability was zero.
Therefore truly random number generators do not exist. QED, there is something behind Quantum Mechanics' random wave functions.
So tired, must sleep...
I am for the complete Trantorization of Earth.
Unless, of course, QM physics pulls randomness from a pool of random numbers transfinitely larger than the set of reals.
Which could indicate building QM on a real-number based spacial mathematics could be wrong...
So very, very, very tired...
I am for the complete Trantorization of Earth.
State legislators do have an interest in defining a legal definition for PI (and other constants) for use in contracts.
If someone only wants to sell you enough dough for pies 3.1 x D, where D is specified by you, you're gonna get pretty upset if you manufacture pies by the millions. Now you're in court, and the judge says, tough luck, there was no PI specified in the contract, and there was no legal definition of accuracy...
I am for the complete Trantorization of Earth.
Is there a closed form to be had from that formula?
- jonese (http://farmaccidentdigest.com)
--
Caveat Emptor is not a business model.
You can however not write a generic compression algorithm that can compress any and all inputs without further information (as this would allow you to pass the output back in to that algorithm and repeat, until it has been compressed to nothing - which is a clear paradox).
"random" in the context of the article should rather be "unpredictable based on previous digits". What that means is that it should be impossible to predict the next digit og Pi based on the previous digits without resorting to other information about Pi (such as how to recognize Pi, and find a formula to calculate it) - the number itself doesn't appear to include any discernable information that allows you to accurately predict the next digit.
For an example, consider the sequence "1,2,3,4". Most people would predict the next digit to be "5". And most likely you'd be correct, because you'd assume you were looking at a part of a sequence increasing with one per digit - there's a relationship between the digits that can be easily calculated.
Noone has found that with Pi, and this article is about trying to prove that no such relationship can be found.
Or for the benefit of non-mathematicians: It's "random".
(disclaimer: I'm certainly no mathematician, and I'm sure someone can find some silly flaw in the above :-)
--
Remove Trash+ to reach my actual inbox
What in the Universe is actually random anyway?
The fact that you yourself do not know the next digit or outcome does not mean that it is also unknown to some other.
A number of posters have made the insipid insight that pi is not random.
/. poster - the article itself takes pains to qualify their use of the word random:
It only appears in the wording of the original
"Numbers like pi are also thought to be "normal," which means that their digits are random in a certain statistical sense."
Thus, the authors of the article are well aware that a. that the digits of pi are not literally random - it has a constant value, and the individual digits of its infinite expansion will always be the same, and b. that pi is not random in the information theoretical sense, in that a program which gives you the next digit in pi does not need to be any longer than any of the multitude of algorithms for calculating pi (see liebnitz, ramanujan).
In fact, Bailey (the mathematician focused on in the article), does not even mention the word random. he is too busy trying to describe his work in which he is linking the BBP algorithm with a chaotic dynamics conjecture called "hypothesis a" which describes certain sequences (including those generated by the BBP algorithm).
random is just a convenient, if innacurate, way of saying that the probability of a given n-digit sequence appearing in pi (in base b) converges to 1/(b^n) the further out you look in the base-b expansion of pi.
stop saying how 'the digits of PI are not in fact random'. it's beating a dead horse, irrelevant, and worse than the constant drone of people correcting CmdrTaco's typos...
A: None. The Universe spins the bulb, and the Zen master merely stays out of the way.
So does this mean that those people that can recite 500 digits of pi are figuring them out as they go?
That would make it even easier to trip them up in the middle
Who is more foolish, the fool,
or the fool that follows him? (Obi-Wan Kenobi)
-Mark
-Mark
Dovie'andi se tovya sagain.
Besides, how else are you supposed to do to show you are a 1337 g33k if you can't rattle off a couple hundred digits of the most famous infinitely long constant.
D - M - C - A
If god had intended you to be naked, you would have been born that way.
Can someone tell me some down to earth, real reasons that anyone should care what the 12,345th digit of Pi is? I mean really, who cares?
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~ now you know
...the answer will be "42."
try to compress the "random" string of numbers; if you can compress a string of random numbers, it isn't
Not really. Since pi is some constant, and not generated by a random process, the most meaningful description of its compressibility is its Kolmogorov complexity, which refers to the shortest program capable of re-generating the original string. Unfortunately, Kolmogorov complexity is not computable in general.
Toronto-area transit rider? Rate your ride.
Depending of your algorithm (repetion, fractal regression,...) you will get VERY DIFFERENT RESULTS using the same original file.
+ PI having no end in itself, can you please send me the method you think you will use before actually compressing pi, and which involve calculating pi to it's end ? 8)
Please call me 5' before World's End, so I can come and check youir results 8)
It takes 40+ muscles to frown, but only four to extend your arm and bitchslap the motherfucker
Because a low probability is not a truth in itself.
I also have a LOW probability to win the Lotery 8|
(1/14 600 000, under French Lotery system)
It takes 40+ muscles to frown, but only four to extend your arm and bitchslap the motherfucker
if not, it could not be used as universal common point.
the famous "Golden Number" is more impressive, I think
It takes 40+ muscles to frown, but only four to extend your arm and bitchslap the motherfucker
At the man's homepage.
http://www.nersc.gov/~dhbailey/
Check out the piqp.c in the middle of the page.
The article says that "...the natural logarithm of 2, often written "log(2)"," but really, it would be written "ln(2)".
MAKE YOUR TIME
"In addition, a simple formula discovered makes it possible to calculate the Nth binary digit of Pi without computing any of the first N-1 digits, and do the computation with very little computing power."
h tm l
l
Someone developed a digit-extraction algorithm (w/o having any n-1 digits) back in 1995:
http://www.mathsoft.com/asolve/plouffe/plouffe.
A [now-defunct] distributed computing project that set records to compute the five trillionth, forty trillionth, and quadrillionth bit of pi:
http://www.cecm.sfu.ca/projects/pihex/index.htm
I remember Richard Crandall from Reed. He was a physics professor who was way into algorithms and number theory. At one time he held the record for highest prime number. He also did a bunch of stuff for Next early on; I think he was Steve Jobs professor while Steve was at Reed. In any event they know each other pretty well. Main thing I remember about his code was that he was really good at numerical algorithms, but he didnt like to put in spaces or newlines; his code was real hard to read. Somebody had said something about checking the floating point in the code, but I am sure there is no error, just obscure shortcuts being taken.
No, it was neither Texas nor Kansas (Kansas did do the 'evolution' thing, though).
There was, a year or so back, a hoax on the Internet about Alabama doing the same thing, but it was a hoax. The only factual evidence that anything of the sort ever happened was in Indiania, 1897, and it never even got close to passing: the Senators considered it to be a complete joke.
The number PI never has a repeating decimal in the decimal system(their are actual mathematical system where pi does not have a repeating decimal but that's another story.) The number PI is not random. It is just a constant. A number that can easily be calculated in decimal with reasonable accuracy. Because we can calculate it, it is not random. If a number has decimals that are just their without reason, it is random. This is not the case.
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Just because a bunch of people believe or do something stupid, doesn't make it any less stupid.
Do you think that enough of us emailed them, Opera would consider revising that error message to This site is slashdotted. Please try again in a couple of days?
A word can paint a thousand pictures
Darnit I would really like to see this article, but the dang link is broke, can someone please send me the correct link...thanks, or repost or something...
-rt-
501 Not Implemented
1. The Borwein-formula for pi has certain properties that let you compute the n-th digit behind the comma in far less time than by the conventional method of computing all the other n-1 digits. But the computing-time still raises with n.
) -1/(8k+6))
2. This method works for base 16, but not for base 10. As far as I know, no formula for base 10 has been found, yet.
3. Borwein's formula:
pi = sum[k=0..inf]((1/16^k)*(4/(8k+1)-2/(8k+4)-1/(8k+5
4. How to compute the n-th digit fast (in base 16):
Multiply the formula with 16^(n-1). Now the n-th digit is directly behind the comma. You get:
16^(n-1)*pi=sum[k=0..inf]( (16^(n-1-k))/(8k+1) - (2*(16^(n-1-k)/....
And here's the trick: We are only interested in the digits *behind* the comma, so we can calculate each nominator (16^(n-1-k)) modulo its denominator, losing the part before the comma and therefore dealing with shorter numbers which is faster. E.g. for the first term: ( 16^(n-1-k) mod (8k+1) ) / (8k+1). [hint: n mod m is the remainder of n/m]
Once again: if you have a term a/b with a>b then there's a term (b*c+d)=a with d<b [d=a mod b]. Then (b*c+d)/b=a/b and d/b is the part behind the comma and c is the part before the comma (because b/b=1).
Note that each term of the infinite sum has a *positive* value (compared to other formulas, where the terms are alternating positiv and negative). Otherwise this method couldn't work, because you couldn't be sure that the part before the comma is really not relevant.
Of course, the infinite series can be compressed into two ASCII symbols representing the english letter 'p' followed by the letter 'i', or by the english phrase 'the exact value of the ratio of the circumference of a circle to its diameter'. These are technically both accurate compressions of the transcendental value of pi, and of course require that the metadescription of the compression scheme is external to the compressed data, but that is also true of most encoding schemes. The digits of pi are definitely not random, the are determinate and calculable; they are however present in a random probability distribution, or at least presumed to be, since no one can calculate all of the digits of a transcendental number in finite time. The digits of pi are calculable, and there has been inexorable progress in coming up with better algorithms for calculating pi.
Here's some more stuff i found lookin around for crazy people with too much time on their hands: http://www.ac.wwu.edu/~mnaylor/pi.html
for the goatse fear in all of us
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practically an AC
Where do they find these guys? He memorizes pi, i play snake on my cellphone. eh
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practically an AC
Try this link instead: http://www.lbl.gov/Science-Articles/Archive/pi-ran dom.html
How many cycles does this calculation take? A software co. could right some software and run it through a cluster to find where in pi the program was. The size of the software could be 10 bits or 10 gigs, but you would only need to download simple equation and plug it into your PiWare program. But then what would we all do with our bandwidth?
Math is like sex. People who get it are popular in class, people who don't are not.