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 *i*^{i}; that is, 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, *i*^{i} is a real number!

Euler proved that for any angle expressed in radians,

If we take radians (that is, ), then Euler’s formula yields

Therefore,

which is clearly real! Moreover, like π, *i*^{i} 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, *i*^{i} 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 for any integer *k*. Thus, using the reasoning above, .)

Happy π Day Eve! ( Day? Day?)

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