This beautiful formula combines five fundamental numbers in one and is generally considered to be the most elegant result in mathematics. In fact, it was ranked first in a 1988 vote among the readers of "The Mathematical Intelligencer".

The formula is obtained by setting x= in the famous identity:
 

discovered by the Swiss mathematician Leonhard Euler (1707-1783). It is used extensively in science, for example to represent rotating objects:

Proving the identity is easy (its discovery was certainly not easy):

The Taylor series for cosx and sinx are:

therefore:

where we use the Taylor series of e^cx with c=i and the fact that that i²= -1, i³= -i, i⁴= +1, i⁵= i, i⁶= -1 etc.

Euler's identity allows to define the natural logarithm of -1 which is in fact multi-valued because e^ix= -1 for x=k, k odd.

Interestingly, we can use eⁱ= -1 to show that iⁱ is not an imaginary number: Taking the square root of both sides and then raising to the power i: e^i/2= i and e^-/2= iⁱ.

By the way, the theorem ranked second in the "competition" was also from Euler (sometimes called the Euler-Descartes theorem), namely the formula relating the numbers of vertices (V), facets (F) and edges (E) of a polyhedron: V+F=E+2.

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