So you want to know what pi equals???
Sometimes I think mathematics has a built-in sense of humor. This is a good approximation to pi:
22/7 = 3.142857142857142857....
but hilarious part is that the difference
22/7 - π = 0.00126448926...
is given by the elegant integral shown here!
Why is this true? I don't know any good way to answer that. I'm sure with work I could do the integral and see that it is true, and that would be one answer. But the question "why should there be such a cute formula for the difference between pi and everybody's favorite approximation to pi?" would remain.
Who first discovered this formula? I don't know that either. Wolfram MathWorld says:This integral was known by Kurt Mahler in the mid-1960s and appears in an exam at the University of Sydney in November 1960.
So, maybe Mahler discovered it, or maybe not.
Kurt Mahler did other cool things. One of the cute things he proved was that like pi, the Chapernowne constant
is a transcendental number. In other words: it's not the root of any polynomial with integer coefficients!
But he also did more important things.
For example, he proved Mahler's inequality. The geometric mean of the sum of two lists of n positive numbers is greater than or equal to the sum of their geometric means!
That's pretty easy. Mahler's theorem is harder, and I'll throw it in here just for people who need some stronger stuff for their daily dose of math. Mahler's theorem says that any continuous function from the p-adic integers to the p-adic numbers can be expressed in terms of difference operators using the same formula that works for polynomial functions from the integers to the real numbers.
I won't write down the formula, but Newton probably knew it, and you should too - you can see it here:https://en.wikipedia.org/wiki/Mahler%27s_theorem
So, p-adic integers are in some ways better than ordinary integers!
You can see more shocking formulas for pi here:http://mathworld.wolfram.com/PiFormulas.html#pi