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### Proving e is Irrational

```
Date: 11/19/97 at 21:42:47
From: Josh Decker
Subject: Proving e is irrational

For my Calculus II class I have to prove that e is irrational.  My
professor suggested using a proof by contradiction, but I don't
understand how to do it.  Your help would be greatly appreaciated.
```

```
Date: 11/20/97 at 15:21:17
From: Doctor Wilkinson
Subject: Re: Proving e is irrational

I'll give you a couple of hints.

You need to use the formula

e = 1 + 1/1! + 1/2! + 1/3! +...

Suppose e is rational, i.e. e = a/b, where a and b are positive
integers. Multiply both sides of the equation by b!.  This gives you

b!e = b! + b!/1! +... + b!/b! + other terms

Since e = a/b by assumption, be is an integer, so b!e is an integer.
So are the terms of the series up to b!/b!.  Now see if you can show
that the sum of the remaining terms is less than 1.  This will give
you a contradiction, since the difference of two integers cannot be a
number between 0 and 1.

-Doctor Wilkinson,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Exponents
High School Number Theory

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