Eye to Eye: Imaginary Exponentiation

The term “imaginary number” is an unfortunate one. It makes these numbers seem strange and separate from more familiar “real” numbers, when in fact there is very little difference. I prefer the term complex numbers that encompasses the closed set of all real and imaginary numbers with the usual arithmetic operators. Recall that the imaginary numbers are numbers that are less then zero when squared, with the imaginary constant i representing the square root of -1:

i 2 = -1

One can add, subtract, multiply and divide with it just like other numbers. One can not only square it to get -1, but also take its square root, which turns out to be another complex number.

i  = 2/2 + i 2/2

But what about raising i to the ith power?

Surely, that must be some sort of weird “very imaginary” number, right? But in fact, it is just a real number, approximately 0.2078796…

The same mechanism that allows us to take the square root of i can be used to explain why ii is real. Just as real numbers can be visualized on the familiar number line, complex numbers can be represented by a plane where the horizontal axis represents real numbers and the vertical axis represents imaginary numbers.

Any complex number x + yi can also be expressed with an angle and a radius: rcosθ+risinθ. Using the angular representation on the plane, we can then visualize any exponentiation operation (take the square, the square root, etc.) as a rotation around the origin.

Squaring a number means doubling the angle. Taking the square root means cutting the angle in half. The imaginary constant i has a radius of 1 and an angle of 90 degrees (or π/2 radians). Doubling it to 180 degrees rotates to the position of -1 on the complex plane. SImilarly, taking the square root of i reduces the angle to 45 degrees, moving it into the position of 2/2 + i 2/2.

But how does one rotate an angle by an imaginary amount? To accomplish this, we turn to one of my favorite formulas in all of mathematics, Euler’s identity:

e = cosθ+isinθ

This identity unites trigonometry and exponentiation using the complex plane and rotations. It is more than just a curiosity and has practical applications including signal processing that we use for synthesizers and audio effects. However, it does allow us to also calculate the value of ii:

ii = cos(πi/2) + isin(πi/2) = eiπi/2 = e-π/20.20787957635076193…

It is odd how rotating an imaginary number by an imaginary factor yields a real number.