# I want an exact value of Pi

Use 1, 2, 2, 2 (Each number used exactly once) to form π.
I want an exact value of π, not any approximation.

Available operators:

+ - * / ( ) ^ !
You can use any one of them for any times.

Hint:

Think about a special feature of the "!".

• MathJax is very slow here. Oct 27, 2019 at 13:03
• Yes, I wouldn't recommend using MathJax unless it's necessary for things like fractions - for some people, it takes a while to load. (Here, you could easily just use the π character rather than switching to MathJax.)
– Deusovi
Oct 27, 2019 at 15:06
• I want an exact value of π, not any approximation - I wholeheartedly endorse this message.
– HTM
Oct 27, 2019 at 20:35
• Instead of the factorial do you mean the gamma function? It seems that the only answer provided erroneously uses the factorial rather than the gamma function (the factorial is only defined for positive integers). Oct 27, 2019 at 22:43
• Well, not exact, but I remember it to 19 digits: 3.141592653589793238 Oct 27, 2019 at 23:52

It is

$$(2 \times (\frac{1}{2})!)^2$$

This video tells why.

• By my calculations, this adds up to 4 times domain error squared.
– Bass
Oct 27, 2019 at 14:10
• Factorial is only defined for positive integers. This answer clearly means to use the gamma function (one of - but not the only - way to extend factorials to the complex numbers), but that was not one of the allowed functions in the question... Oct 27, 2019 at 21:30
• @BlueRaja-DannyPflughoeft But we all know it's only possible if factorial is interpreted as gamma function - which is the only point of this question... Oct 28, 2019 at 4:02
• @BlueRaja-DannyPflughoeft The gamma function is the standard way to extend factorials to the complex numbers. If you write $\alpha!$ for $\alpha\not\in\mathbb{N}$ in mathematics, this is assumed to mean $\Gamma(\alpha+1)$. See for example Wolfram Alpha and my experience as a mathematician in a field where gamma functions are used all over the place. Oct 28, 2019 at 6:15
• @Bass I disagree. Saying "the gamma function" explicitly would make the puzzle trivial and pointless. The exclamation point is not strictly integer-only in mathematics, and using it in this way makes for a very clever puzzle - kudos to the OP. If you don't believe me, all I can do is point to the evidence in my previous comment. I work with factorials and gamma functions every day, and use $\alpha!$ completely interchangeably with $\Gamma(\alpha+1)$. Oct 28, 2019 at 21:30