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!
or
No subject limitations
Search only in selected subjects
    Algebra
    Calculus or Analysis
    Combinatorics
    Geometry
    Number Theory
    Probability
    Topology
    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:

i to the i is a Real Number

If you are familiar with complex numbers, the "imaginary" number i has the property that the square of i is -1. It is a rather curious fact that i raised to the i-th power is actually a real number!

In fact, its value is approximately 0.20788.

Presentation Suggestions:
This makes a great exercise after learning the basics about complex numbers.

The Math Behind the Fact:
From Euler's formula, we know that exp(i*x) = cos(x) + i*sin(x), where "exp(z)" is the exponential function ez. Then


exp(i*Pi/2) = cos(Pi/2) + i*sin(Pi/2) = i.

Raising both sides to i-th power, we see that the right side is the desired quantity ii, while the left side becomes exp(i*i*Pi/2), or exp(-Pi/2), which is approximately .20788.

(Actually, this is one of many possible values for i to the i, because, for instance, exp(5i*Pi/2)=i. In complex analysis, one learns that exponentiation with respect to i is a multi-valued function.)

How to Cite this Page:
Su, Francis E., et al. "i to the i is a Real Number." Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

Share

References:
Keywords:    i^i, i to the i-th power, i raised to the i, imaginary power
Subjects:    calculus, analysis
Level:    Medium
Fun Fact suggested by:   Ed Poncin
Suggestions? Use this form.
4.33
 
current
rating 		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!