The Most Beautiful Equation in the World
By Dr Chris P. Jobling
According to some, Euler’s identity is one of the most beautiful mathematical expressions in the world. The identity is
1 + e__i? = 0 (although electrical engineers like myself are more familiar with seeing it written as 1 + e__j? = 0).
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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
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eix* = cos x +* i* sin x.
For the special case when x = ?: sin ? = 0 and cos ? = -1, thus ei? = -1 or ei? + 1 = 0.