How can the ratio of those two numbers be irrational?

[jnorman]

if the diameter of a circle is a finite value, and the circumference of a circle is a finite value, how can the ratio of those two numbers be irrational?

a new team, yee and kondo, just determined the value of pi to 5 trillion digits. (wow)

it takes only 39 digits of pi to make a circle the size of the observable universe accurate to within one hydrogen atom.

i assume something like maybe 50 digits (?) to be accurate to within the Planck length. does anything past that point actually make any sense?

 

[Petek]

I can't come up with a citation, but I understand that new supercomputers can be tested for flaws having them compute pi to billions of places. A mismatch in, for example, the trillionth place would indicate a problem.

 

[jnorman]

thanks petek - good point. btw, the yee and kondo team used a single desktop PC (3.33 ghz 2 x xeon x5680 processor with 12 physical cores, 24 hyperthreaded, 96 gb of ddr3 ram, and 3 hdd's) to obtain the 5 trillion digit record - it took 90 days, and the result - a compressed output of decimal and hexadecimal digits - took up about 3.8TB. fun...

 

[Mute]

First, the diameter of a circle and its circumference being "finite means only that they are not infinite. It does not imply they are integers. Because pi is (provably) irrational, you cannot have both the diameter and the circumference be integers.

Second, be careful not to mistake physics with math. Whether or not distances below the Planck scale are meaningful has nothing to do with the irrationality of pi. All it means is that the idealized circles in mathematics do not exist physically because there is a smallest length scale. Pi is entirely independent of the physical world, however, so the non-existence of a perfect circle in the physical universe implies nothing about pi.

 

[CRGreathouse]

i assume something like maybe 50 digits (?) to be accurate to within the Planck length. does anything past that point actually make any sense?

Good intuition. The diameter of the observable universe is about 5.3 × 10^61 Planck lengths, so nothing beyond the 62nd decimal place or so has meaning for physical circles in the universe. But we care about pi for reasons other than measuring big circles.

 

[jnorman]

thanks all - good answers.

 

[Sakha]

How do they actually compute pi with such many digits?

 

[adriank]

There are a number of algorithms; a few are listed at http://en.wikipedia.org/wiki/Category:Pi_algorithms.

 

[jnorman]

sakha - generally, they use a differential between two polygons - one inside the circle and one outside the circle - and try to approach an infinite number of sides to each polygon. as the number of sides on each polygon approach infinity, they get can closer and closer to a representation of the value of pi. but i am sure a real mathemetician can explain that much better than me...

 

[adriank]

That's a very slow way to compute pi.

 

[hotvette]

Besides an algorithm, a way to compute and store numbers of many digits is necessary. Not sure how the big guys do it, but there are many 'extended' precision schemes available. The ones I've seen are text based (i.e. the 'numbers' are stored as text and special routines are needed to add, multiply, and divide text based numbers).

 

[CRGreathouse]

That's a very slow way to compute pi.

Pfft, that's nothing. This is a slow way to compute pi.

 

[Sakha]

There are a number of algorithms; a few are listed at http://en.wikipedia.org/wiki/Category:Pi_algorithms.

There are some really creative ways of doing it while there are others that I don't have a clue what they mean.

Pfft, that's nothing. This is a slow way to compute pi.

This is definitely a very creative way of doing it! I liked very much the little JavaScript app that shows it working. After around 10000 darts it stabilized at 3.14~~, very interesting.


I just think its a big waste of supercomputer use to calculate the billionth digit of pi, it's almost completely useless.

 

[disregardthat]

The last one there is not very reliable though, the accuracy of the estimate will depend on the accuracy of determining whether the arrow hits inside the circle or not. I suspect the probability distribution is not evenly distributed over the square (for real values), so its not really computing pi even if it was 100% accurate.

As to the use of calculating large decimal expansions of pi, its more a prestigious thing than a useful thing, like the olympics.

 

[CRGreathouse]

I just think its a big waste of supercomputer use to calculate the billionth digit of pi, it's almost completely useless.

The last two records weren't even done with supercomputers!

 

[CRGreathouse]

The last one there is not very reliable though, the accuracy of the estimate will depend on the accuracy of determining whether the arrow hits inside the circle or not. I suspect the probability distribution is not evenly distributed over the square (for real values), so its not really computing pi even if it was 100% accurate.

There's error in all methods, they key is controlling the error. (*Realistic* methods don't use interval arithmetic or proven error bounds, they just use a few thousand guard digits and know that this suffices.) So this method really isn't that different in spirit from the actual methods. But its performance... ah, the performance. I expect that getting n digits of precision would take at least 100^n steps. Surely appropriate analysis could show that there's less chance of error due to getting a bad distribution of inputs (assuming the numbers are truly random!) than of a cosmic ray interfering with the calculation if you used, say, 200^(n+10) steps. So instead of taking ~30 microseconds per digit to get a trillion digits, it would take 10^2301029995658 years per digit.

 

[disregardthat]

There's error in all methods, they key is controlling the error. (*Realistic* methods don't use interval arithmetic or proven error bounds, they just use a few thousand guard digits and know that this suffices.) So this method really isn't that different in spirit from the actual methods. But its performance... ah, the performance. I expect that getting n digits of precision would take at least 100^n steps. Surely appropriate analysis could show that there's less chance of error due to getting a bad distribution of inputs (assuming the numbers are truly random!) than of a cosmic ray interfering with the calculation if you used, say, 200^(n+10) steps. So instead of taking ~30 microseconds per digit to get a trillion digits, it would take 10^2301029995658 years per digit.

I'd say its very different form other methods by that the random distribution is not over all real numbers. That fact makes it incapable of determining pi to any given accuracy. I don't consider it an error in the method if the computation is actually converging to pi. Its bad sign when your error relies on physical limitations.

 

[CRGreathouse]

I'd say its very different form other methods by that the random distribution is not over all real numbers. That fact makes it incapable of determining pi to any given accuracy. I don't consider it an error in the method if the computation is actually converging to pi. Its bad sign when your error relies on physical limitations.

What do you mean by "not over all real numbers" and "physical limitations"?

 

[disregardthat]

What do you mean by "not over all real numbers" and "physical limitations"?

The "random distribution" used in the calculation may rely on physical limitations (such as data from the surroundings), and not all real numbers can be generated by a any given random distribution, as far as I know. One would need a completely random distribution in order to actually calculate pi.

 

[CRGreathouse]

One would need a completely random distribution in order to actually calculate pi.

Well... maybe. There's a nice result (no citation at the moment, but I'll give it tomorrow if you care) that randomness can be 'amplified': that given a source generating slightly-random bits (say, the likelihood of a 1 is known to be between 60% and 98%), the bits can be combined to create bits of arbitrary randomness (a 1 has a 49.999% to 50.001% likelihood). Of course this may consume many bits from the original source to generate one output bit.

Combine this with the knowledge of your target probability and you can determine how much randomness is needed. For safety's sake, set your bounds narrower than that and calculate away.

 

[disregardthat]

Well... maybe. There's a nice result (no citation at the moment, but I'll give it tomorrow if you care) that randomness can be 'amplified': that given a source generating slightly-random bits (say, the likelihood of a 1 is known to be between 60% and 98%), the bits can be combined to create bits of arbitrary randomness (a 1 has a 49.999% to 50.001% likelihood). Of course this may consume many bits from the original source to generate one output bit.

That's interesting, if its not too much trouble I'd like to see a reference.

 

[CRGreathouse]

That's interesting, if its not too much trouble I'd like to see a reference.

I have the paper in my hands now. It is:
Miklos Santha, "Generating quasi-random sequences from slightly-random sources"

A Google search produces
M. Santha, U.V. Vazirani, "Generating Quasi-Random Sequences From Slightly-Random Sources," focs, pp.434-440, 25th Annual Symposium on Foundations of Computer Science (FOCS 1984), 1984.
which may be a subset, a superset, or neither.