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In mathematics, Euler's identity (also known as Euler's equation) is the equality
e
i
π
+
1
=
0
{\displaystyle e^{i\pi }+1=0}
where
e
{\displaystyle e}
is Euler's number, the base of natural logarithms,
i
{\displaystyle i}
is the imaginary unit, which by definition satisfies
i
2
=
−
1
{\displaystyle i^{2}=-1}
, and
π
{\displaystyle \pi }
is pi, the ratio of the circumference of a circle to its diameter.
Euler's identity is named after the Swiss mathematician Leonhard Euler. It is a special case of Euler's formula
e
i
x
=
cos
x
+
i
sin
x
{\displaystyle e^{ix}=\cos x+i\sin x}
when evaluated for
x
=
π
{\displaystyle x=\pi }
. Euler's identity is considered an exemplar of mathematical beauty, as it shows a profound connection between the most fundamental numbers in mathematics.
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