e is irrational

From the Taylor seriesMathworldPlanetmath for ex we know the following equation:

e=k=01k!. (1)

Now let us assume that e is rational. This would there are two natural numbersMathworldPlanetmath a and b, such that:


This yields:


Now we can write e using (1):


This can also be written:


The first sum is obviously a natural number, and thus


must also be . Now we see:


Since 1b1 we conclude:


We have also seen that this is an integer, but there is no integer between 0 and 1. So there cannot exist two natural numbers a and b such that e=ab, so e is irrational.

Title e is irrational
Canonical name EIsIrrational
Date of creation 2013-03-22 12:33:02
Last modified on 2013-03-22 12:33:02
Owner mathwizard (128)
Last modified by mathwizard (128)
Numerical id 13
Author mathwizard (128)
Entry type Theorem
Classification msc 11J82
Classification msc 11J72
Related topic ErIsIrrationalForRinmathbbQsetminus0
Related topic NaturalLogBase