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A Much Bigger Piece Of Pi
Punk_Rock_Johnny points to an AP story on Pi-obsessed Professor Yasumasa Kanada. A snippet from the story: "Kanada and a team of researchers set a new world record by calculating the value of pi to 1.24 trillion places, project team member Makoto Kudo said yesterday. The previous record, set by Kanada in 1999, was 206.158 billion places." Trillion!
"
How about we see this bad boy!? I'd sure like to paste it into my "info.txt" file for future referance. It could come in handy sometime.
No sig for you. YOU GET NO SIG!
We would have either found the end by now or discovered a pattern.
heh.
"Everything you know is wrong. (And stupid.)"
Moderation Totals: Wrong=2, Stupid=3, Total=5.
Because pi is there. And they still have found only the tiniest fraction of the total of decimals of pi...
Nyh!
The number Six!
Does the problem that pi can't be expressed in decimal notation extend to other base systems? For example, if you tried to write pi out in binary or hex would you encounter the same problem? Is there a special base system (other than base pi) which can describe pi in a finite number of digits?
Someone you trust is one of us.
Here's how it works. You'll need several boxes of toothpicks. Get a large piece of chart paper, and draw parallel lines on it, from one side to the other. The lines should be separated by a distance just slightly larger than the length of a toothpick.
From a height of about one metre, drop a measured number of toothpicks onto the chart paper, so that they all fall randomly somewhere on the paper. Count how many toothpicks are touching a line (or would be, if they weren't resting on another toothpick).
Repeat this process as many times as you can. Lots of people can do it at once. All that's important is that, each time you drop some toothpicks, you write down how many you dropped, and how many of those ended up touching a line. When you're done, find a total for each quantity.
You now have all the numbers you need to calculate Pi:
Now here's the formula you need to calculate Pi:
Fill them in the formula, and work out your own value of Pi!In the book version of Contact by Carl Sagan, but skipped in the Jodie Foster movie, was the notion that the aliens had discovered proof that the universe was created by a higher intelligence. A God or society of Gods far higher and more advanced than the aliens. The whole point of dragging Human-kind to that remote beach to talk with daddy was to tell Human-kind that it was time for them to look for God's signature on this universe.
As any artist, the creator signed the creation. Where? Deep into the insignificant but irrefutably valid digits of several of the fundamental mathematical constants such as pi and e.
The main character finds one of the signatures at the end of the book: if calculating digits of pi in base 11, after a few million or billion places, a 500x500 digit span is almost entirely zeros. If the span was rendered as a square of pixels, the non-zero digits drew a perfect circle inscribed in the square. A circle in a square. The key concept defining pi, in the digits of pi itself. The whole way the universe works is affected by that constant, so any such 'design' in it has, if you pardon the pun, a transcendental import.
Why base 11? It's left to the reader to decide, but I expect Sagan wrote it because it is considered one of the possible designs of the universe, one of the string theories is based on an 11-dimensional all-inclusive physics model. As the alien explains to the main character, it wouldn't be base 10, because what's the likelihood that the creator also happened to have ten fingers?
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Um, you have 1.24 trillion digits of pi. I think you can begin a statisticall analisys now.
If Mr. Edison had thought smarter he wouldn't sweat as much. --Nikola Tesla
You can calculate Pi by doing:
..) x 4
(1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 +
Obviously the more iterations you do, the closer you will be to the 'true' value of Pi.
42 really is the answer to life, the universe, and everything!!!
You could always just do it with Good ol' Calculus.
Trolls lurk everywhere. Mod them down.
Hmm..
size of the proton: ~ 1 fm = 10^-15 m
age of the universe: ~15 Gyr
speed of expansion ~ c = 3 x 10^8 m/s
gives:
proton/cosmic radius ~ 10^-42
So you need about 40 places for this. Of course, you might want to calculated it to the Plank scale, so maybe tack on a few more.. say 100 for safety. Yes, a trillion digits does seem a bit like overkill.
Dr. Math's Pi FAQ. Very informative.
Here's a program written in BrainF*ck to calculate pi: http://www.people.fas.harvard.edu/~jafowler/pi/pi. b
Here's the analysis of the program, and a link to what the Turing-inspired BrainF*ck programming language is about.
[
Pi is represented usually by a fraction or relatively simple equation, it's just the division that makes the number go on for ever.
... but these are just approximations; 22/7 is a good enough approximation a lot of the time, but that's just an approximation too)
Nope. If pi was rational (a fraction), it wouldn't go on for ever without repeating. (reference)
In fact pi is irrational, i.e. there are no integers p, q such that pi = p / q. (proof)
You can approximate pi as a fraction, which is what projects like this do. (pi is approximately equal to 31/10, or 314/100, or 31416/1000, or
Kanada and a team of researchers
MPAA forces have today invaded Canada, when asked their reasons they replied:
"While we were looking through through the binary version of Pi, and one of our special forces noticed that hidden in from digit 12,166,133,883 onwards was a c source to DeCSS. Obviously these terrorists must be stopped!"
When pointing out that it was Kanada, the researcher, and not Canada the country, the Canadian government sued for trademark violation.
The case is not expected to hold up, as it is doubtful canada will be able to proove it has the computing power to calculate Pi beyond 4 decimal places - and no confusion can occur.
Damn, if I could only compute e^(i*pi) with a trillion digits precision...
Buy the complete six DVD set!
You'll need to insert all six one after the other next time you #include <math.h>
Once more unto the breach, dear friends, once more, Or close the wall up with our American dead!
3.14159265358979323846264338327950288...... ah, this is gonna take a while :-/
A google only has 100 zeros, thus 100 places.
:-P)
:-)
10^trillion is 1 followed by 1 trillions 0's... Assuming we are following the american system that would be equivalent to.
10^(10^12)
Okay... now.. let's get some interesting facts with this.
The absolutely smallest length measurable by quantum theory is the planck length which is approx 10^-34 m. Needless to say, if we have a diameter of an incredibly small perfect circle, we'll know it's circumference beyond what is possible by quantum theory (but since there are no perfect circles, and quantum theory adds probability, this doesn't mean anything really useful.
Now, since we know the smallest measurable... lets look at what the estimates for the size of the universe are. Recent estimates put it as 10 billion light years in radius source
Which works out to about... (assuming american notation on billion)
10^9 * 300,000,000 m/s* 365*24*3600 ~= 10^25 m
Okay... now if we were to measure the circumference to as accurate as allowed by quantum theory we'd have.
pi*2.10^25 ~= 6.28*10^26 10^27 with an accuracy of about 34 decimals...
So... to get perfect accuracy as allowed by quantum theory we would have at most 35 decimal places afterwards... therefore, we'd need pi with an accuracy of
~10^63...
We have pi with an accuracy of 10^(10^12) which is
63 : 10^12 ~= 1: 1.59x10^11
Way more accuracy then we really need.
That's absolutely insane, but it is fun math.
Just some food for thought.
~ kjrose
is EXACTLY 3.
Sorry about that. I just wanted to get your attention. Glayvin!
I can't believe you report this and don't even include the value of Pi he calculated in the article!
I guess I'll have to wait for one of the page widening trolls to post it.
Nope. Any number with a recurring pattern is automatically rational, and pi is not.
This isn't too hard to see. For example, if pi repeated after 1.5 trillion digits, we could write its value (where [1.5tril] represents all those repeating digits:
Then multiply this number by 10^(1.5 trillion).
such that the repeating part starting with .14159 still follows the decimal point.
Then subtract the top equation from the bottom one, so the repeating part gets subtracted away.
Then just divide both sides by (10^(1.5 trillion) - 1) and you've written pi as a ratio of two integers.
Hrm.. Well, as one of my Computer Science teachers once told me (in a discrete math class).. Mathemeticians do things because it interests them. The fact that it often has no practical application is why they are often cold, bitter and broke. :)
Now I finally have the measurements needed to make my cookies PERFECTLY round.
Why?
Well, if you read the article, you would know why. Mapping out a very large number like that is useful for testing the accuracy of supercomputers. Also, the research process spins off lots of discoveries. Someone who mapped out pi to 1.24 trillion decimal places probably learned a couple neat tricks along the way.
You could at least give credit where due ;)
Here's one of the nicer sites I've seen that has a java applet to simulate this.
"Not knowing when the dawn will come, I open every door." - Emily Dickinson
Comment removed based on user account deletion
"Google" - search engine.
"Googol" - 10^100.
Imagine this program screaming along calculating a few more trillion places when all of a sudden it stops. Pi is NOT infinite after all.
Imagine the hiliarity that would ensue (oops, wrong web site...)
They don't need to do so, as one can prove that Pi is irrational in advance.
put a disclaimer on these posts moderators ;-)
I don't need a 20 page proof to tell me the moderators are irrational.
Only in Indiana
You, Sir, despite your low member number, would get an F- for information theory at the university I was tought and now teach.
There is nothing that compresses to one bit. There is such thing as a most efficient way of encoding any message. Counted in bits. and no, not just one bit. One bit would just contain enough information to say "Pi" or "Not Pi". "Not Pi" would according to my intuition not be an acceptable answer, you also have to say "What kind of 'Not Pi'". And that takes bits. You forgot that your algorithm is supposed to possibly generate all possible messages, or else it's "not fair".
Pi would not compress at all, given it's an infinitely long number. (To be precise, it's length would be reduced from inf to inf/(alphabet entropy) which is still inf, although a "smaller" inf). If you are content with a finite number of digits, its length would be reduced by about a little more than three bits per decimal (because log2(10)=3.???) with any decent entropy encoder. You could try to reduce this further by taking two decimal digits at once, but unfortunately it would not work, as not only are Pi's digits uniformly distributed from 0 to 9, pairs of digits are also distributed uniformly from 0-99, so you would remain with 6.???? bits (log2(100)) per decimal digits pair.
Another approach you might take, if you want infinite precision (silly on a finite machine), or more generally random precision, is to write a code in a predetermined programming language, in this case a series developement, or whatever the number thorists use nowadays to calculate pi, and decide that the "decompression algorithm" is a compiler (that is perfectly legal, as any finite message can be passed that way, eg "#include <iostream> int main(){cout << "The message";}").
My idea is that the c compression algorithm would be beat by a perl compression. Maybe try in BrainFuck, it might beat perl, but BF sucks at multiplications.
Anyway, the most optimal compression for pi is probably saying "Pi" by itself. Any decent geek knows at least one way to calculate that/ find it on project gutenberg/whatever. But don't ever think that you could compress it to two bytes or less : you gotta be sure that I will not understand "the string of decimal digits a.k.a. Pi, do write it in numbers when decompressing", not just "mu turned over", "Pi the string" or "Private investigator". This certainty takes bytes.
Another example is : "you cannot encode '3 4 8 15 3.141592653 78 54' as '3 4 8 15 pi 78 54', because that would increase the number of symbols in the alphabet, and all the other symbols would have to contain more bits as a result, so the compressed message length would suffer- hope there are a lot of 'pi' in the compresed message".
I must leave now, gotta go bowling with friends. Start your flames, I can see blatant holes in my reasonments. Hope you get the point. Mailing a link to the message to my signal theory professor (formally one of my bosses), so I will suffer if I told bullshit.
while it's true (I think) that any fininte sequence of digits will eventually appear in a non-repeating, infinite sequence, I think the point in the book was that the odds of our being able to find it, given the tiny tiny tiny tiny tiny tiny portion of the number space we're able to search with our extremely finite computing power, would be evidence that it was placed there if we ever did manage to find it.
Put another way, it would have to be hanging in easy reach for us to be able to find such an insanely improbable thing as (say) a 500x500 block of pre-arranged digits. In base 11, that would be 11^(25,000), a number too hideous to contemplate, and think of the size of the space you'd need to search before such a number would be found just based on probability. So if we found such a thing, we either beat bazillion^bazillion-to-one odds, or we found something that was left there for us. Interesting.
Actually, base converting Pi in to Base 11 is actually pretty damn EASY.
Here is the number
3.1415926535897932384626433832795
3 in Base 11 = 3
Now the rest is simple.
Multiply 0.1415926535897932384626433832795 by 11
Take the number past to the left of the decimal point and use this as your first digit of Base 11 Pi. In this case it is = 1.
Subtract that number and multiply by 11 again.
The number you get is 6.
Now if you Wash - Rinse - Repeat you'll arrive at the number in Base 11 (3.16150702865A485235215...)
Pretty simple? You can do this quickly with other bases without hitting negative powers of the base number. You can also convert a number in another base quickly using the technique from my earlier post in this Slashdot chat. The trick is to convert your target base number into the base that you're converting from. It works for all decimals just like RADIX works for all integers. Do a find for "matrix29" on this page and you'll hit my previous post right off.
You can also convert to non-integer bases (ergo Base 7.886) but the method is a tad more awkward.
"Face it, a nation that maintains a 72% approval rating on George W. Bush is a nation with a very loose grip on reality.
aparently you've never eaten Thanksgiving dinner at my place. Give it an hour or two, and you're bound to see it make a reappearance...
Dude, they measure it to 1.24 Trillion, not 10^(Trillion).Someone had pointed that out, but...
If you think about it, you could not have fitted the entire observable universe with enough paper to record (even if you write in very very very very small fonts) the number of decimals if you know PI to 10^(Trillion).
In fact the entire observable universe had about 10^120 atoms. So you are out of luck very soon. (You can imagine packing more atoms, but then the universe will become too dense and collapse on herself so fast you won't have time to expand to her current volume).
Mode (3) smart-aleck mode. Press * to return to main menu.