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e is irrational

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

Screen Shot 2017-06-14 at 10.40.11 AM.png

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

However,

Screen Shot 2017-06-14 at 10.41.02 AM.png

And so we can bound x in the following way

Screen Shot 2017-06-14 at 10.41.06 AMThis 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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