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Revision as of 05:39, 17 November 2005 by 67.87.73.86(talk)(added executive summary, one intermediate step in proof, slight wording changes)
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 contradition, 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. The contradiction will establish the irrationality of e.
To see that x is an integer, note that
Here, the last term in the final sum is to be interpreted as an empty product.
To see that x is a positive number less than 1, note that