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?


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

5 Answers

Anonymous 0 Comments

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.

Anonymous 0 Comments

[Lindemann and Weierstrass created a mathematical proof that showed that e and pi are transcendental.](

Anonymous 0 Comments

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.

Anonymous 0 Comments

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!)

Anonymous 0 Comments

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:

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