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Comment by StefanKarpinski | Hacker News Reader

There is no mystery here. If you pick a random real number and find the best quality approximation of it — as calculated by log_b 1/|x-a/b| — then the median quality of the best such approximations within a tolerance of 1/1024 is around 3.19. So with respect to the measure this article is talking about, pi is completely normal and 355/113 is a completely unexceptional best approximation with a denominator of at most nine binary digits (about 3 decimal digits). 355/113 isn't even the best quality approximation of pi by the given criterion — that would be 22/7, which, as mentioned in the article, has a quality measure of 3.429288337281781.

Julia code to figure this out can be found here:

https://gist.github.com/3170899

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These kinds of articles about pi, or numbers, where they try to summon up some mystery and wonder around it, come across to me as not very genuine. I get the impression it gets upvoted because people want to feel like they're smarter, or more geeky and hip for having looked at it.

One of the most "mind boggling" areas of math is number theory

Math dealing with real/complex numbers has been vastly explored, and there are very powerful tools, and computers are very good at these kind of problems (Calculus, etc)

With number theory, progress is much harder, things work in a completely different logic (think for example that 5+3 can be 0 for example)

One of the places where these mysterious sides of math comes together is this: http://en.wikipedia.org/wiki/Riemann_zeta_function

>With number theory, progress is much harder, things work in a completely different logic (think for example that 5+3 can be 0 for example)

I wasn't aware that the modulo operation had been raised to a field of mathematics.

Modulo operations is part of "number theory 101"

And ok, addition is nice and fun. Then you go to multiplication

Then you end up with Galois Fields.

Never underestimate the amount of discussion that goes into things like 1+1=2

> ...for example that 5+3 can be 0 for example

When is that example true? It seems like you made that up.

mod 8?

Hardly the 'mystery of number theory' the poster was talking about. There was no 'mod 8' syntax expressed in the example.

Yes, it is http://en.wikipedia.org/wiki/Modular_arithmetic

But you don't need to write "mod 8" in a congruence class

Oh, you want "the mystery of number theory" you can start with http://en.wikipedia.org/wiki/Fermat%27s_little_theorem

It was solved, and he (Fermat) was likely wrong, given the resulting proof.

(5 + 3) % 8 = 0 is not the same as 5 + 3 = 0

I'm not looking for mystery, I'm wondering why people think it exists, and are faking its existence by being imprecise.

I don't know, I've met folks for whom math is so magical they do nothing but talk about it all the time. I can recognize that passion in other pursuits as well, from knitting to physics.

I find a close correlation between people who play with numbers to people who like to solve puzzles.

Or perhaps math are fun and people upvote a post they liked.

I wonder if mathematicians recite beautiful proofs as fourplay...

"I wonder if mathematicians recite beautiful proofs as fourplay ... "

I see what you did there.

Probably not, since most (?) beautiful proofs are symbolic and not very recitable.

Maybe there is a little mystery.

Enumerate all the fractions with at most 3-digit denominator (0, 1, 1/2, 1/3, 2/3, 1/4, ..., 998/999) between 0 and 1. Now what is the probability that amongst them we find an approximation to a random real that is at least as good as 355/113 is to pi?

Seems to be about 15%, so not much mystery here. But we allow denominators as large as 999 because we use the decimal system. In hexadecimal it would be at most 2-digit denominators, up to ff = 255.

In this case, the probability is about 1%. Much more mystery. So it comes down to, how large a set of denominators we think 113 is drawn from, and whether living in a 1 in 100 universe is enough to be mysterious in this case.

Sampled by this python code (takes minutes to run): https://gist.github.com/3172978

So just a note, you've got a great deal of repetition in your search. 0.5, for instance, turns up once for every even denominator. That's going to skew your percentages a bit.

Yes, the constructed list contains a lot of duplicates (removing duplicates with the python set operations shortens it by 40% - would be a worthwhile optimization for the runtime, too), but my percentages are not based on that.

But: I sampled a random real, and looked at whether any fraction in the list is close enough to this real. And repeated the sampling 1000 times. This is how the percentages are calculated, and the duplicates in the list don't affect this.

And in base-128 it would be only denominators up to 127, which is an even more amazing 0.2% ;)

Why not go till fff in base 16? And also atleast double digit in base 127? It can turn out in both cases that the probability is way higher.

...and an overflow in your numerator.

Seriously: this kind of thinking is more numerology than number theory.

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