You signed in with another tab or window. Reload to refresh your session.You signed out in another tab or window. Reload to refresh your session.You switched accounts on another tab or window. Reload to refresh your session.Dismiss alert
In mathematics, **Euler’s identity** (also known as _Euler’s
4
+
equation_) is the equality
5
+
$$ e^(i pi) + 1 = 0 $$
6
+
where
7
+
8
+
- $e$ is Euler's number, the base of natural logarithms,
9
+
- $i$ is the imaginary unit, which by definition satisfies $i^2 = -1$,
10
+
and
11
+
- $pi$ is pi, the ratio of the circumference of a circle to its
12
+
diameter.
13
+
14
+
Euler’s identity is named after the Swiss mathematician Leonhard
15
+
Euler. It is a special case of Euler’s formula $e^(ix) = cos x + i sin
16
+
x$ when evaluated for $x=pi$. Euler's identity is considered an
17
+
exemplar of mathematical beauty, as it shows a profound connection
18
+
between the most fundamental numbers in mathematics. In addition, it
19
+
is directly used in a proof that $pi$ is transcendental, which implies
20
+
the impossibility of squaring the circle.
21
+
22
+
## Imaginary exponents
23
+
24
+
Euler’s identity asserts that $e^(i pi)$ is equal to $-1$. The
25
+
expression $e^(i pi)$ is a special case of the expression $e^z$, where
26
+
$z$ is any complex number. In general, $e^z$ is defined for complex $z$
27
+
by extending one of the definitions of the exponential function from
28
+
real exponents to complex exponents. For example, one common
29
+
definition is:
30
+
31
+
$$
32
+
e^z = lim_(n->oo) (1 + z/n)^n.
33
+
$$
34
+
35
+
Euler’s identity therefore states that the limit, as $n$ approaches
36
+
infinity, of $(1 + (i pi)/n)^n$ is equal to $-1$.
37
+
38
+
Euler’s identity is a special case of Euler’s formula, which states
39
+
that for any real number $x$,
40
+
41
+
$$ e^(ix) = cos x + i sin x $$
42
+
43
+
where the inputs of the trigonometric functions sine and cosine are
44
+
given in radians.
45
+
46
+
47
+
48
+
In particular, when $x = pi$,
49
+
50
+
$$
51
+
e^(i pi) = cos pi + i sin pi.
52
+
$$
53
+
54
+
Since
55
+
$$ cos pi = -1 $$
56
+
and
57
+
$$ sin pi = 0 $$
58
+
59
+
it follows that
60
+
$$ e^(i pi) = -1 + 0i, $$
61
+
62
+
which yields Euler's identity:
63
+
64
+
$$ e^(i pi) + 1 = 0. $$
65
+
66
+
## Generalizations
67
+
68
+
Euler’s identity is also a special case of the more general identity
69
+
that the $n$th roots of unity, for $n > 1$, add up to $0$:
70
+
71
+
$$
72
+
sum_(k=0)^(n-1) e^(2pi i k/n) = 0 .
73
+
$$
74
+
75
+
Euler’s identity is the case where $n = 2$.
76
+
77
+
A similar identity also applies to quaternion exponential: let ${i, j,
78
+
k}$ be the basis quaternions; then,
79
+
80
+
$$
81
+
e^(1/sqrt(3) (i +- j +- k) pi) + 1 = 0 .
82
+
$$
83
+
84
+
More generally, let q be a quaternion with a zero real part and a norm
85
+
equal to $1$; that is, $q = ai + bj + ck$, with $a^2 + b^2 + c^2 =
86
+
1$. Then one has
87
+
88
+
$$
89
+
e^(q pi) + 1 = 0 .
90
+
$$
91
+
92
+
The same formula applies to octonions, with a zero real part and a
93
+
norm equal to $1$. These formulas are a direct generalization of
94
+
Euler’s identity, since $i$ and $-i$ are the only complex numbers with
95
+
a zero real part and a norm (absolute value) equal to $1$.
0 commit comments