How do we know numbers like “e” & “pi” are transcendental? If its true, how do people calculate all those digits of them, down to the millions of digits?

Question

Thank you in advance, im dumbfounded how we are able to calculate these numbers as if they are truely transcendental, there would be no way to express them as a true fraction.

In: Mathematics

Answer 1

So, calculating Pi and e is really easy (at least for the first few digits).

These numbers are special because they pop up in different formulas. For example:

Some expression = pi^2

Then we can obviously take the square root on both sides and compute the thing on the left side.

Here’s two suggestions for you:

e = 1 + 1/1 + 1/2 + 1/6 + 1/24 + 1/120 + 1/720 + …

(divide by ever increasing factorials. 720 = 6!)

If you do this until 6!, you already get 2.7180 which is not too bad.

pi = sqrt(6 * (1 + 1/4 + 1/9 + 1/16 + 1/25 + …))

(divide by ever increasing square numbers, then multiply by 6, and finally take the square root.)

If you do this up until 30^2, you get 3.110 which is not very close. But as you go further down, you get more and more accuracy.

Edit: There might be a misunderstanding of what “transcendental” means: Transcendental doesn’t mean “can’t be expressed as a formula”. It just means: There is no *polynomial with coefficients that are whole numbers* whose root is the transcendental number. I.e., the number doesn’t pop out at the end of a calculation using a *finite* number of multiplications, additions, exponentiations using whole numbers.

Nobody claimed anything about infinite sum, as these are not polynomials. In fact, e can be explicitly written as an infinite sum (and it’s the usual definition given!)