Proving a number is irrational is mostly done by **contradiction**. So first suppose *e* is rational: e = p/q where p, q are coprime integers.

We know that q≥2 as *e* is not an integer (in fact, it’s in between 2 and 3). Then

Note that, as *q!e* and *n* are natural numbers, we must have that *x* is a natural number.

However,

And so we can bound *x* in the following way

This is a contradiction since *q!e* must be a natural number, but it is a sum of an integer *n* plus a non-integer *x*. Hence, e is irrational.

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