A Proof that e is Irrational

Introduction

In this article, I’ll try and show that e, sometimes called Euler’s number, is an irrational number 2.718281828459045… and so on. Euler’s number is a fantastic number, and it plays a role in just about every aspect of physics, maths, and statistics. There are many ways of calculating the value of e, but none of them ever give a totally exact answer, because e is irrational and its digits go on forever without repeating.

Beautiful Property

This exponential function, , is the only one that if we differentiate it, we get the same function .

The next differential of it we get

and it goes on an on and on, and that’s the beauty of this particular function that all its derivatives and slopes match the actual function itself. It means it carries on forever, because the slope just keeps the same no matter how far it goes, whereas other functions will die off if we differentiate them.

A Proof that e is Irrational

In this section, we want to prove that e can’t be represented as the ratio of two integers, that’s what an irrational number is.

We assume that e can be represented as the ratio of two integers, and write with p and q integers. By cancelling, we may assume that p and q are not both even (if they are, we can simply keep cancelling powers of 2 until one of them is not). If our original assumption (that e is rational) is not correct, then we know that e is irrational.

One thing to mention is that e has a nice infinite series expansion, it’s

where in mathematics, the factorial of a positive integer n, denoted by n!, is the product of all positive integers less than or equal to n.

To simplify, we call function 1 and function 2. And we’re going to multiply this function 2 by q factorial. Then we will get function 3

From our assumption, we know that , that means

So we know that must be an integer. And also we know that is an integer. Then the rest of function 3 (components after ) should be an integer right?

Let’s derive the rest of the function 3 after ,

and we can get

which is clearly

According to Gauss, if , then . In this case, , so we have . Finally, rearranging this then we can get .

Let’s come back to the remainder R, the rest, we said it must be an integer.

Hence we know R is bounded between 0 and and q is bigger than 1, so this is a fraction less than 1. It means R can not be an integer. It contradicts! Therefore e can not be represented as a rational number, so it has to be irrational.

Conclusion

This is the most well-known proof by Joseph Fourier using contradiction. Hope you like it!

References

1. https://www2.math.upenn.edu/~kazdan/202F13/hw/e-irrat.pdf
2. https://math.stackexchange.com/questions/713467/e-is-irrational