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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.
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.