π in the title is a character I haven't seen before, [Mathematical Italic Small Pi](https://unicode-table.com/en/1D70B/).
A formula for the nth digit of π and π^n
61β70 of 143 posts
Re: A formula for the nth digit of π and π^n
#62Earlier quoted context omitted.
The article title says "powers of π", so presumably an arbitrary digit.
Iβm not sure thatβs right. The abstract and text say the nth digit of \pi^n, and the worked example on page 2 uses the same n=1000 to find the thousandth digit of \pi^{1000}.
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Re: A formula for the nth digit of π and π^n
#63Finally, we can find the last digit of pi
Now we just have to figure out what the last integer is
Re: A formula for the nth digit of π and π^n
#64Earlier quoted context omitted.
Exactly, the only possible "last digit" it could have (in keeping with the silly premise that it has one), is zero.
Why is zero different than any other number?
It acts as a terminator or terminal, see https://news.ycombinator.com/item?id=34391483.
Re: A formula for the nth digit of π and π^n
#65It is all curious, but it seems that procedure that calculates the n-th digit using some other functions that require O(n) calculations (i.e. Bernoulli numbers) is not that exciting, as it's just a speedup comparing to a naive calculation (maybe a big one but still).
Although because pi is a transcendental number maybe it is impossible to have an algorithm to return the n-th digit in O(1) operations? Does anyone know ?
Upd: thinking more about it, just to read all the digits of number n, one needs log(n) operations, so I guess anything that has performance of O(log(n)) can be considered easy (i.e. the digits of rational numbers can be computed in O(logn).
Re: A formula for the nth digit of π and π^n
#66But... pi is in the formula. If you already know it, then why bother?
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Re: A formula for the nth digit of π and π^n
#67Re: A formula for the nth digit of π and π^n
#68Wanted to compliment this paper for being well written. I'm not a practicing mathematician but I was able to easily follow along here and that was a cool feeling.
Could someone who is a practicing mathematician speak to the practical application of this? From what I understand from reading this seems like an interesting curiosity but the Chudnovsky formula it refers to seems to be better at doing the same thing for any practical purpose.
Re: A formula for the nth digit of π and π^n
#69Earlier quoted context omitted.
This is one of my favorite software projects of all time. I remember seeing this several years ago and it has driven my imagination crazy since then.
But why?
For me, it stimulates a mental shift that could be good exercise elsewhere: instead of storing the string, you find the string somewhere and store its index.
It also stimulate the imagination: what other transcendental numbers might this work with? How long do you have to search in the digits to find your string? What can you say about the size of the index (how far you searched) vs your string length? Etc. It's patterns all the way down.
Re: A formula for the nth digit of π and π^n
#70It is all curious, but it seems that procedure that calculates the n-th digit using some other functions that require O(n) calculations (i.e. Bernoulli numbers) is not that exciting, as it's just a speedup comparing to a naive calculation (maybe a big one but still). Although because pi is a transcendental number maybe it is impossible to have an algorithm to return the n-th digit in O(1) operations? Does anyone knowβ¦
The normal method for computing digits of pi is not O(n), so this could still be interesting. I think it's probably better to compare this with the BBP formula [1]. The commonly used algorithm for pi I think still takes O(n log^3 n) time [2]
[1] https://en.wikipedia.org/wiki/Bailey%E2%80%93Borwein%E2%80%9...