The Euler Identity

Originally posted August 24, 2016

Today’s Eponymy in August is the Euler identity.

It’s often referred to as “The Most Beautiful Equation in Mathematics” and for good reason. It is simple, it contains the four fundamental numbers that make most mathematics possible, (1, 0, e, and $\pi$) and “$i$” from complex numbers, and the combination of them in this way isn’t intuitive but is fairly easy to derive.
The Euler identity follows from Euler’s formula, which is a reconciliation of trigonmetric math and logarithmic/exponential math in the complex plane (where complex numbers are sums of a real part containing a real number, and an imaginary part containing a multiple of the square root of -1). Complex numbers can be visually represented in 2d graphs, plotting the real part onto the x axis and the imaginary part onto the y axis. This means that they can be reasoned about using trigonometry (sines, cosines, the Pythagorean equation, etc.). Euler showed that the exponentiation $e^{i{\cdot}x}$ was equal to $cos x + i sin x$, and since the radian system of measure of angles is based on $\pi$ (and unitless scalar values in this system), plugging $\pi$ in as the value of x yielded $cos\pi + i\cdot sin\pi$, which is $(-1) + i \cdot (0)$. (The identity can also be written $e^{i \cdot \pi} = -1$, but that’s less nice looking).

Leonhard Euler was an 18th-century Swiss-Russian polymath, and one of the most brilliant mathematicians in history. Euler first coined the idea of a function in mathematics (see the post on Currying for more on that topic), named the numbers e and i, first used the capital Greek sigma to denote summation, was the first to use exponentiation and logarithms, and first used such a notation to describe infinitely added power series. He also wrote several papers on optics that helped to popularize Huygens’ wave theory of light, and provided numerous other gifts to applied mathematics, science, and engineering. Euler was for a time head of mathematics at the Academy in St. Petersburg, and also held a position at Berlin Academy for 25 years until Frederick the Great got tired of having someone so unsophisticated (in all areas but math) in his court. After this, Euler returned to St. Petersburg and the Academy, and remained there for the rest of his years.