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?

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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

[Lindemann and Weierstrass created a mathematical proof that showed that e and pi are transcendental.](https://en.wikipedia.org/wiki/Lindemann%E2%80%93Weierstrass_theorem)

The exact proof for either is about a full page of math.
But the general gist is:
If pi/e wasn’t transcendental, then there would be this way of representing pi/e in a not-transcendental way, and that it would have to match this other way of representing pi/e in a not-transcendental way.
And making them match is impossible.

At least for pi there are people that figure out the new digits using super computers. There was recently 9 trillion digits new digits compute. So 31.4 trillion digits total. https://cloud.google.com/blog/products/compute/calculating-31-4-trillion-digits-of-archimedes-constant-on-google-cloud

Although it goes beyond ELI5, since no one has actually provided a proof yet, this video does the best job explaining it that I’ve seen:
https://youtu.be/WyoH_vgiqXM

He covers both the transcendence of e and Pi in a relatively straightforward way.

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!)