All Rational Approximations of Pi Are Useless • Hacker News Board

Re: All Rational Approximations of Pi Are Useless

#51

Post by marshray » Mon, Oct 01, 2012, 9:37 PM UTC

beefman wrote: post #14

guard-of-terra wrote: post #5

Earlier quoted context omitted.

No it isn't. It isn't "a shorthand". It is a notation. Of those two notations, no one is "a shorthand" of another.

I think the point is that McLoone is using two different notations and measuring an artifact of this. There's nothing special about base-10 denominators.

Perhaps we could say he's comparing Shannon entropy per symbol for the two systems.

It seems that what's special about rationals with power-of-10 (with power > 1) denominators is that we have a readily available shorthand, uh notation, for them.

Notation can be very significant. The transition from Roman numberals to positional notation with 0 took a thousand or so years. But boy did it ever make long division easier!

Re: All Rational Approximations of Pi Are Useless

#52

Post by ColinWright » Mon, Oct 01, 2012, 9:38 PM UTC

tailrecursion wrote: post #50

22/7 is a better approximation of pi than 3.14 because 22/7 is closer to pi than 3.14. The hypothesis of the article is wrong. I believe Colin [EDIT: I meant Jon McLoone, the author of the article. Sorry Colin!] is counting only full digit matches. It is this metric that is useless.

Are you confusing the person who actually wrote the article with me, who simply submitted it here?

If so, perhaps you can explain why people do that so often. I see it quite frequently, and am baffled by it.

Re: All Rational Approximations of Pi Are Useless

#53

Post by DanBC » Mon, Oct 01, 2012, 9:40 PM UTC

JackC wrote: post #46

derekp7 wrote: post #44

Earlier quoted context omitted.

This also makes sense from data compression theory. If the digits of an irrational / transcendental number share some of the properties of a random string, then you shouldn't be able to compress it. And finding a fractional representation with fewer total digits is a form of data compression.

Nitpick: data compression theory says you can't have a general-purpose algorithm that on average compresses random strings. The best any algorithm can do is make some strings shorter and some strings longer, which is why compression is only useful on strings with known properties. But given a particular random finite string (such as N digits of pi) you can very likely (certainly?) find an algorithm that compresses it…

Here's a mildly amusing challenge and response to that challenge:

(http://www.patrickcraig.co.uk/other/compression.htm)

This page has some nice compression gimmicks (a file that compresses well with one algorithm but hardly at all with another; a file that uncompresses to itself)

(http://www.maximumcompression.com/compression_fun.php)

The large text compression benchmark has some nice finely tuned compression software and statistics.

(http://mattmahoney.net/dc/text.html)

And the Hutter prize is interesting. (Get the 100 Mb file of enwiki8 and the de-compressor to under 16 Mb)

(http://prize.hutter1.net/)

Re: All Rational Approximations of Pi Are Useless

#54

Post by Xcelerate » Mon, Oct 01, 2012, 9:41 PM UTC

I stumbled across this interesting paper about the subject at some point: http://cogprints.org/3667/1/APRI-PH-2004-12b.pdf I know a lot of you will gloss over this, but I think you will be surprised at how interesting it is if you read it.

Basically, it tries to see if mathematical equations have meaning by determining how well they "compress" the results. For instance, the author says the equation eπ−π=19.9990999... is compressible (the equation generates more bits of π than it takes up itself), and thus it is likely that there is some mathematical reason for this -- it's not just a coincident. On the other hand, 314100 gives an approximation to π but does not compress its representation, so there is nothing intriguing about this formula.

Re: All Rational Approximations of Pi Are Useless

#55

Post by scott_s » Mon, Oct 01, 2012, 9:43 PM UTC

kdavis wrote: post #41

scott_s wrote: post #34

Earlier quoted context omitted.

The natural log - ln - is not rational, as it is the logarithm with base e . That is, ln(x) answers the question, to what power would we have to raise e in order for it to equal x ?

Two points: 1. As soon as you allow square roots, you allow irrational numbers. (This sqrt(2).) 2. "The natural log - ln - is not rational" is a different statement than "e is irrational". A rational function is one that can be written as the ratio of two polynomials.

Regarding point 1, you had many things wrong. I picked what was the most obvious to me at the moment - in order for it not to apply, there only needs to be one thing wrong with it. And my point with the natural logarithm is that once you introduce it, you have introduced an irrational number. Overall, I'm not sure what your point has been.

Re: All Rational Approximations of Pi Are Useless

#56

Post by recursive » Mon, Oct 01, 2012, 9:43 PM UTC

Jach wrote: post #27

jfoutz wrote: post #21

Earlier quoted context omitted.

Rational approximations are good for finding reasonable answers without a calculator. My contrived example is the area of a circle, r=7. i can do 22 * 7 in my head and get 154, which is pretty darn close to 153.86. 7 is awkward (for me anyway) but it does offer opportunities to cancel.

How fast can you do 22 * 7 in your head? I get a lot of mileage out of simply approximating pi=3, and optionally adding in a factor of 10% later. For your example, I know 7^2=49 immediately, which I round to 50, 50 * 3=150 which is fairly close to 153.938. (Use more digits of pi!) I can quickly improve my estimate by adding 4.9. (Edit: It's also fairly trivial to get even better estimates quickly from here, but at th…

> optionally adding in a factor of 10% later

You're better off not adding anything. If you could add 5% instead, then you'd be much better still.

Re: All Rational Approximations of Pi Are Useless

#57

Post by ColinWright » Mon, Oct 01, 2012, 9:51 PM UTC

scott_s wrote: post #25

kdavis wrote: post #17

This seems slightly silly if one is allowed to use, as Ed Pegg suggested, log and square root. If log and square root are allowed, then the obvious solution is log(-1)/(sqrt(-1)*log(e)) which is accurate to an infinite number of digits.

e is not rational.

I'm confused - he's addressing the point made in the article by Ed Pegg that things are interseting if you allow log and sqrt. Surely that means he's allowed to use log and sqrt, and in particular, to use them in rational expressions.

So I don't really understand what your point is.

Re: All Rational Approximations of Pi Are Useless

#58

Post by kdavis » Mon, Oct 01, 2012, 9:53 PM UTC

mhartl wrote: post #36

kdavis wrote: post #30

Earlier quoted context omitted.

ln(-1)/sqrt(-1)

Maybe you're just trolling, but rational numbers are numbers that can be expressed as p / q , where p and q are integers . Neither i e nor i is an integer, so your proposed quotient has no bearing on the (ir)rationality of e .

I'll assume you are not trolling. My point is that

log(-1)/(sqrt(-1)*log(e)) = ln(-1)/sqrt(-1)

I am just writing the same equation using a log with another base so I don't have e in the equation explicitly.

Re: All Rational Approximations of Pi Are Useless

#59

Post by scott_s » Mon, Oct 01, 2012, 9:57 PM UTC

ColinWright wrote: post #57

scott_s wrote: post #25

Earlier quoted context omitted.

e is not rational.

I'm confused - he's addressing the point made in the article by Ed Pegg that things are interseting if you allow log and sqrt . Surely that means he's allowed to use log and sqrt , and in particular, to use them in rational expressions. So I don't really understand what your point is.

The point of the exercise is to find a short approximation of pi, no? If you allow the use of e, then you can define pi. What he wrote is not an approximation, it is pi. Surely at that point you've defeated the point of the exercise.

Re: All Rational Approximations of Pi Are Useless

#60

Post by tailrecursion » Mon, Oct 01, 2012, 10:22 PM UTC

ColinWright wrote: post #52

tailrecursion wrote: post #50

22/7 is a better approximation of pi than 3.14 because 22/7 is closer to pi than 3.14. The hypothesis of the article is wrong. I believe Colin [EDIT: I meant Jon McLoone, the author of the article. Sorry Colin!] is counting only full digit matches. It is this metric that is useless.

Are you confusing the person who actually wrote the article with me, who simply submitted it here? If so, perhaps you can explain why people do that so often. I see it quite frequently, and am baffled by it.

Yes that was my mistake. Sorry about that. It must happen because of the byline above the box where one types in the top-level comment. Thanks for mentioning that it's a frequent error, that makes me feel better.