One of the most famous formulas in mathematics, indeed in all of science is commonly written in two different ways:

eπi = −1 or eπi + 1 = 0.

Moreover, it is variously known as the Euler identity (the name we will use in this column), the Euler formula or the Euler equation. Whatever its name or form, it consistently appears at or near the top of lists of people's “favorite” results. It finished first in a 1988 survey by David Wells for Mathematical Intelligencer of “most beautiful theorems.” It finished second in a 2004 survey by the editors of Physics World to select the “greatest equations” and it was third in a 2007 survey of participants in an MAA Short Course of “Euler's greatest theorems.”

Whether people call it a formula, an equation or an identity, and regardless of which form they use, almost everyone credits the result to Euler. But it is not entirely clear why people give him credit for this result, because he never wrote it down in anything remotely like this form, because he wasn't the first one to know the fact behind the formula, and because he himself credited that fact to hismentor, Johann Bernoulli. In this column we will look at the origins of the Euler identity, see what Euler contributed, and consider whether it is correctly named.

Phase 1: 1702 to 1729

There are two formulas that are closely related to the Euler identity. The first we will call the “Euler formula”:

eiθ = cosθ + i sin θ

The Euler identity is an easy consequence of the Euler formula, taking θ = π. The second closely related formula is DeMoivre's formula:

(cos θ + i sin θ)n = cos nθ + i sin θ.

This, too, is an easy consequence of the Euler formula, since

(cos θ + i sin θ)n = (eiθ)n = einθ = cos nθ + i sin nθ.