The Most Imaginary Number… is Real!

On the eve of “π Day,” a hush has fallen on the mathematical internet. Everyone is gearing up, getting ready to celebrate the beauty of the transcendental number π. Indeed, π is an amazing number. I have a book coming out in October about the geometric problems of antiquity (Tales of Impossibility: The 2000-Year Quest to Solve the Mathematical Problems of Antiquity, Princeton University Press), and one of the problems—squaring the circle—is all about the nature of π.

However, because it is not yet π Day, I thought I’d introduce you to another number that appears in the book—one that is not as well known but that has a very surprising property. The number is ii; that is, \sqrt{-1}^{\sqrt{-1}}. At a glance, this looks like the most imaginary number possible—an imaginary number raised to an imaginary power. But in fact, as Leonhard Euler pointed out to Christian Goldbach in a 1746 letter, ii is a real number!

Euler proved that for any angle \theta, expressed in radians,

e^{i\theta}=\cos(\theta)+i \sin(\theta).

If we take \theta=\pi/2 radians (that is, \theta=90^{\circ}), then Euler’s formula yields

e^{i\pi/2}=\cos(\pi/2)+i \sin(\pi/2)=0+i\cdot 1=i.


i^{i}=(e^{i\pi/2})^{i}=e^{i\cdot i\pi/2}=e^{-\pi/2}=0.2078795763\ldots,

which is clearly real! Moreover, like π, ii is a transcendental number (and hence an irrational number). This fact was proved in 1929 by the 23-year-old Russian Aleksander Gelfond.

(Actually, as Euler pointed out, ii does not have a single value; rather, it takes on infinitely many real values. The angle i makes with the real axis can be expressed as 2\pi k+\pi/2 for any integer k. Thus, using the reasoning above, i^{i}=e^{-2\pi k-\pi/2}.)

Happy π Day Eve! (\pi-1 Day? \pi-0.01 Day?)