What irrational number has the simplest calculation in terms of computation?

Question

I came across https://github.com/philipl/pifs which is a fancy way of storing data. And a thought struck my mind, is it so that Pi is the simplest irrational number to calculate?

So the Question is. What irrational number is simplest to calculate eg. contains fewer/simpler operators?

Cost of operators are from low -> high (+-,*,/)

It seems that you want to talk about Kolmogorov complexity of computable numbers. — lhf, Aug 21 '13 at 09:55

score 6 · Answer 1 · answered Aug 21 '13 at 09:10

6

Firstly, I wouldn't call $\sqrt {-1}$ an irrational number.

To answer your question, recall that any rational number has either a finite or a repeating decimal expantion. So when you say 'easiest to calculate,' I might give you an irrational number exactly so that there is no calculation. For example, consider $0.1234567890112233...$, where it has each digit, then two copies of each digit, then 3 copies, etc. The fractional part (i.e. the part after the decimal) never repeats, so it's irrational. And there's no real 'calculation' involved.

This is all to say that your question doesn't admit a rigorous answer.

answered Aug 21 '13 at 09:10

davidlowryduda

91,687

it has to be reproducable by calculation – Thomas Andreè Wang Aug 21 '13 at 09:26
1

@Thomas: I don't understand. My number is easily reproducable by a couple of additions. – davidlowryduda Aug 21 '13 at 09:29
you do understand my question? is there a simpler formula that could be used to reproduce any file? – Thomas Andreè Wang Aug 21 '13 at 09:30
1

@Thomas: No, I do not understand your question. And I don't understand what you mean by "file." – davidlowryduda Aug 21 '13 at 09:33
there is a link in the question, go there and read the page it is fairly short. I'm wondering is there a number which is faster to compute with the same properties as Pi in terms of randomness of digits? – Thomas Andreè Wang Aug 21 '13 at 09:36
1

So the OP is asking for another normal number which is "easier" to calculate than $\pi$. – Fixed Point Aug 21 '13 at 09:38
pretty much, what i thought i asked for. – Thomas Andreè Wang Aug 21 '13 at 09:39
The confusion is because you said "irrational" number when you meant "normal". – Fixed Point Aug 21 '13 at 09:44
i thought Pi and all numbers without an end was irrational. – Thomas Andreè Wang Aug 21 '13 at 09:45
"Without end" doesn't mean irrational. It must be non-terminating AND non-repeating. And yes, non-terminating and non-repeating is irrational but irrational doesn't mean normal. You are looking for normal. See my answer for more detail. – Fixed Point Aug 21 '13 at 09:48
i did. and it teaches me, but a answer to the question (albeit faulty worded) is to say formula x for the number e is simpler to calculate. or maybe formula z for the number g is the simplest known calculation. – Thomas Andreè Wang Aug 21 '13 at 09:51

Fixed Point · Accepted Answer · 2013-08-21T09:59:11.647

First, we already know that "almost all" of the real numbers are normal. Second, despite the first fact, it is quite difficult to prove a given number as normal or non-normal. Third, a number may be normal in one base but not in another. We don't know if $\pi$ is normal in base 16 and we don't know if $\pi$ is normal in base 10. Learn more about normal numbers here.

Further, irrational doesn't automatically imply normal. Liouville numbers are irrational but not normal because for example they contain way more zeros than any other digit. A normal must have a uniform distribution of all the digits in its expansion.

Lastly, what about something like Champernowne constant which can be obtained by just adding one and concatenation. And this number is proven to be normal. And the proof is easy. Every number (and hence every file in base 10) is in there because every number is in there. And not just that but we also know exactly where a given number is. You are probably looking for a normal number in base 2 or base 16, not in base 10.

Do you have a reference for $\pi$ being normal in base 16? I thought that (as well as other bases) was still open. — hmakholm left over Monica, Aug 21 '13 at 09:51
For some reason I thought I read somewhere that the BBP expansion proves $\pi$'s normality in base 16 but I can't immediately find where. I will retract that statement immediately. — Fixed Point, Aug 21 '13 at 09:57

What irrational number has the simplest calculation in terms of computation?

2 Answers2