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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!
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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.
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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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.
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
Why waste all that space? You can write any number from 0 to 255 in 1 byte, any number from 0 to 65355 in 2 bytes, and 0 to 4.2billion in 4 bytes.
Anyone care to do the math about how much space you need (uncompressed)? (I cant cause I'm dumb)
P.S. Does Pi compress very well?
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
Actually, to store a number this large it will have to be stored in IEEE 754 format.
In IEEE 754 a string of either 32 bits is divided into a sign bit, 8 or more bits for the exponent, and 23 or more bits for the mantissa.
You'll probably need a little more than 4119190837661 bits to tell you the truth.
Then again, who cares.
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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.
With a few extra bits, you can make it easy to extract any digit of the number.
... ...
:)
Split the 1.24TB number into triplets, and store each triplet in 10 bits:
(leaving off the initial 3):
141 592 653
encode into 30 bits:
0010001101 1001010000 1010001101
Pack these together (and on and on), and you use only ~517 GB, but you can index to any digit by a simple division, shift, and decimal conversion.
(assuming you decided to put together a RAID array for the storage in the first place
- The Sigless Wonder