The Most Beautiful Equation in the World

According to some, Euler’s identity is one of the most beautiful mathematical expressions in the world. The identity is
1 + ei? = 0 (although electrical engineers like myself are more familiar with seeing it written as 1 + ej? = 0).

Euler’s Identity, Technical Museum Berlin*

On a recent visit to the Mathema Exhibition at the German Technical Museum, Berlin, I took this photo of some art that represents this equation.

Lots of people like to take a break by sitting in the zero.

Why so beautiful? Because it neatly ties together five of the most mysterious numbers in Mathematics: 0, 1, e, ? and the square root of –1.

What does it mean? Euler’s formula is essentially a statement of the transformation from cartesian coordinates to polar coordinates for imaginary numbers: that is numbers that have a real part represented as a point on the x-axis, and an imaginary part that is represented by a point on the y-axis. It takes the general form
eix* = cos x +* i* sin x.

For the special case when x = ?: sin ? = 0 and cos ? = -1, thus ei? = -1 or ei? + 1 = 0.