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 *e*^{z}. 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 i^{i}, 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*. <https://www.math.hmc.edu/funfacts>.

**References:**

Paul Nahin, *An Imaginary Tale*.

**Fun Fact suggested by: **

Ed Poncin