Euler's identity

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):

• 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:

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

$e=\sqrt[\pi i]{-1}$

$1=-e^{\pi i}$

$i=\frac{\ln(-1)}{\pi}$

$\pi=-i\cdot\ln(-1)$

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:

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