# Euler's identity

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

• Euler's number, $e\approx 2.718282\ldots$
• pi, $\pi\approx 3.141592\ldots$
• the imaginary unit, $i=\sqrt{-1}$
• the real unit number, 1
• zero, 0

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

$e^{\pi i}+1=0$

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

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:

$0=e^{\pi i}+1$
$e=\sqrt[\pi i]{-1}$
$1=-e^{\pi i}$
$i=\frac{\ln(-1)}{\pi}$
$\pi=-i\cdot\ln(-1)$

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

$\ln(-5)=\ln(-1\cdot5)=\ln(-1)+\ln(5)=\pi i+\ln(5)$

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

$\ln(x)=\ln(|x|)+\pi i$