Member-only story
Featured
Weird but True: i^i Is a Real Number
A step-by-step explanation of why the imaginary unit raised to an imaginary power is a real number
“Imaginary numbers are a fine and wonderful refuge of the divine spirit, almost an amphibian between being and non-being.”
— Gottfried Wilhelm Leibniz (1646–1716)
I n 1748, Leonhard Euler published his seminal work, Introductio in analysin infinitorum (Introduction to the Analysis of the Infinite). The book introduced ideas and techniques that would later become fundamental to the field of complex analysis. It included a relationship that leads to what is widely considered one of the most beautiful formulas in all of mathematics. This is where we will start.
Understanding Euler’s Identity
Euler’s identity is a cornerstone of complex analysis and has widespread application in fields from electrical engineering to quantum mechanics. It is often presented as
In a mere seven symbols, it unites the base of the natural logarithm e, the imaginary unit i, the constant π, 1, and 0.
It results from an even more remarkable relationship he discovered. Euler was studying the series expansion for eˣ:
