# Proof that e is irrational

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In mathematics, the series expansion of the number *e*

can be used to prove that *e* is irrational.

Summary of the proof:

This will be a proof by contradiction. Initially *e* will be assumed to be rational. The proof is constructed to show that this assumption leads to a logical impossibility. This logical impossibility, or contradiction, implies that the underlying assumption is false, meaning that *e* must not be rational. Since any number that is not rational is by definition irrational, the proof is complete.

Proof:

Suppose *e* = *a*/*b*, for some positive integers *a* and *b*. Construct the number

We will first show that *x* is an integer, then show that *x* is less than 1 and positive. The contradiction will establish the irrationality of *e*.

- To see that
*x*is an integer, note that

- The last term in the final sum is (i.e. it can be interpreted as an empty product). Clearly, however, every term is an integer.

- To see that
*x*is a positive number less than 1, note that

and so . But:

- Here, the last sum is a geometric series.

If , then , which would imply is an integer. So and since there does not exist a positive integer less than 1, we have reached a contradiction, and so *e* must be irrational. Q.E.D.