Math Fun Facts!
hosted by the Harvey Mudd College Math Department created, authored and ©1999-2010 by Francis Su
Subscribe to our RSS feed   or follow us on Twitter.
Get a random Fun Fact!
No subject limitations
Search only in selected subjects
    Calculus or Analysis
    Number Theory
    Other subjects
  Select Difficulty  
Enter keywords 

  The Math Fun Facts App!
  List All : List Recent : List Popular
  About Math Fun Facts / How to Use
  Contributors / Fun Facts Home
© 1999-2010 by Francis Edward Su
All rights reserved.

From the Fun Fact files, here is a Fun Fact at the Medium level:

e is irrational

If e were rational, then e = n/m for some integers m, n. So then 1/e = m/n. But the series expansion for 1/e is

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

Call the sum of the first n terms of this alternating series S(n). How good is this approximation to 1/e? Well, the error is bounded by the next term of the alternating series:

0 < | 1/e - S(n) | = | m/n - S(n)| < 1/(n+1)!

But multiplying through by n!, you will see that

0 < | integer - integer | < 1/(n+1) < 1.
But there is no integer strictly between 0 and 1, so this is a contradiction; e must be irrational.

Presentation Suggestions:
Use the series expansion for 1/e as a fun fact on a previous day.

The Math Behind the Fact:
Anytime you have an alternating series in which the terms decrease, then each partial sum is not farther from the limit than the next term in the series!

On an unrelated note, it is much harder to show that Pi is irrational.

How to Cite this Page:
Su, Francis E., et al. "e is irrational." Math Fun Facts. <>.

Keywords:    alternating series, proof by contradiction
Subjects:    calculus, analysis, number theory
Level:    Medium
Fun Fact suggested by:   Arthur Benjamin
Suggestions? Use this form.
Click to rate this Fun Fact...
    *   Awesome! I totally dig it!
    *   Fun enough to tell a friend!
    *   Mildly interesting
    *   Not really noteworthy
and see the most popular Facts!
New: get the MathFeed iPhone App!

Brings you news and views on math:
showcasing its power, beauty, and humanity

Want another Math Fun Fact?

For more fun, tour the Mathematics Department at Harvey Mudd College!