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.
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.
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.