The Equation That Ties Five Constants Together

Set x to π in Euler's formula and the equation collapses to e^(iπ) + 1 = 0. Five constants — e, i, π, 1, 0 — each from a different province of mathematics, sitting in a single line with nothing else.

e shows up in calculus, where d/dx of e^x is e^x. π shows up in geometry, the ratio of a circle's circumference to its diameter. i is the imaginary unit, the square root of -1, invented in the 16th century by Italians who needed it for cubics. 1 and 0 are the multiplicative and additive identities of arithmetic. Before Euler, nobody had a reason to expect those five to sit at the same table.

The path runs through the formula Euler proved in 1748: e^(ix) = cos x + i sin x. Plug in x = π. cos π = -1, sin π = 0, so e^(iπ) = -1. Add 1 to both sides. The identity is what falls out.

What the formula is really saying is that exponentiating a pure imaginary number rotates a unit vector around the origin, by an angle equal to the imaginary part. Travel a distance π around the unit circle starting at 1 and you land at -1. The identity is the snapshot at exactly half a turn.

A 1990 Mathematical Intelligencer reader poll voted Euler's identity the most beautiful theorem in mathematics. Richard Feynman, in his Lectures on Physics, called the underlying formula "our jewel" and "the most remarkable formula in mathematics." The beauty isn't in what it lets you compute — most physicists use Euler's formula directly, not the special case. It's in the fact that the special case exists at all.