One of the more famous identities that Euler discovered was the identity relating the five great constants of mathematics:

- Euler's number,
- pi,
- the imaginary unit,
- the real unit number, 1
- zero, 0

These five constants of nature, Euler discovered, could neatly be tied together in a single, simple equation:

Note that the identity also uses three fundamental operations of arithmetic (as extended to complex numbers):

- addition,
- multiplication, and
- exponentiation.

The proof of Euler's identity is trivial if one uses the more generalized Euler's formula.

## Algebra

Algebraic manipulations of this simple identity can demonstrate each of the five constants in terms of the other four:

These above equivalences are greatly applicable as substitutions in more complex mathematics. For example, evaluating the logarithms of negative values:

Thus, the natural logarithm of a negative real value, , is a complex number: