Euler’s Identity, shown above, is often said to be the most beautiful equation in all of science and mathematics.

It links the three basic arithmetic operations:

- Addition, in the +1 term.
- Multiplication, in the
*i*π term. - Exponentiation, in the e
^{iπ}term.

It also links five of the most important mathematical constants:

- e, the base of the natural logarithms.
*i*, the imaginary number (√−1) on which complex numbers are based.- π, the ratio of a circle’s circumference to its diameter.
- 1, the multiplicative identity, the basic unit of counting.
- 0, the additive identity that leaves a number unchanged.

Understanding how *e*^{iπ} + 1 can equal zero is more difficult.

To begin to understand how to evaluate Euler’s Identity we must first understand about series expansions. The series expansion of a function can be used to calculate the value of this function to an arbitrary degree of precision.

The series expansion of *e*^{x} is given by:

If only the first term is used then* e* = 1.00, if two terms are used *e* = 2.00, if three terms are used *e* = 2.50, if four terms are used *e* = 2.66, and so on, until an infinite number of terms are used and the exact value of *e* is given. After only ten terms the value found for *e* is accurate to better than one part in a million.

The series expansion of *e*^{iπ} is therefore given by:

The even powered terms (e.g. *i*^{2}π^{2}, *i*^{4}π^{4}) become negative because *i*^{2}=−1 and the odd powered terms (e.g. *i*^{3}π^{3}, *i*^{5}π^{5}) gain a negative multiple of *i*.

There are two other important expansions, the expansion of sin(*x*) and cos(*x*), two basic trigonometric functions.

If we collect the terms in the expansions of sin(*x*) and cos(*x*) and compare them with the expansion of *e ^{ix}* we find that:

Inserting π in place of *x* in that expression yields:

Because cos(π) = −1 and sin(π) = 0 we find that indeed, e^{iπ} = −1 and therefore e^{iπ} + 1 = 0.