Euler's identity

Explanations

Imaginary exponents

Euler's identity asserts that ${\displaystyle e^{i\pi }}$ is equal to −1. The expression ${\displaystyle e^{i\pi }}$ is a special case of the expression ${\displaystyle e^z}$, where z is any complex number. In general, ${\displaystyle e^z}$ is defined for complex z by extending one of the definitions of the exponential function from real exponents to complex exponents. For example, one common definition is:

$${\displaystyle e^z=\underset{n\to \mathrm{\infty }}{lim}{\left(1+\frac{z}{n}\right)}^n.}$$

Euler's identity therefore states that the limit, as n approaches infinity, of ${\displaystyle (1+\frac{i\pi }{n}{)}^n}$ is equal to −1. This limit is illustrated in the animation to the right.

Euler's identity is a special case of Euler's formula, which states that for any real number x,

$${\displaystyle e^{ix}=\cos x+i\sin x}$$

where the inputs of the trigonometric functions sine and cosine are given in radians.

In particular, when x = π,

$${\displaystyle e^{i\pi }=\cos \pi +i\sin \pi .}$$

Since

$${\displaystyle \cos \pi =-1}$$

and

$${\displaystyle \sin \pi =0,}$$

it follows that

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

which yields Euler's identity:

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

Geometric interpretation

Any complex number ${\displaystyle z=x+iy}$ can be represented by the point ${\displaystyle (x,y)}$ on the complex plane. This point can also be represented in polar coordinates as ${\displaystyle (r,\theta )}$, where r is the absolute value of z (distance from the origin), and ${\displaystyle \theta }$ is the argument of z (angle counterclockwise from the positive x-axis). By the definitions of sine and cosine, this point has cartesian coordinates of ${\displaystyle (r\cos \theta ,r\sin \theta )}$, implying that ${\displaystyle z=r(\cos \theta +i\sin \theta )}$. According to Euler's formula, this is equivalent to saying ${\displaystyle z=r{e}^{i\theta }}$.

Euler's identity says that ${\displaystyle -1=e^{i\pi }}$. Since ${\displaystyle e^{i\pi }}$ is ${\displaystyle r{e}^{i\theta }}$ for r = 1 and ${\displaystyle \theta =\pi }$, this can be interpreted as a fact about the number −1 on the complex plane: its distance from the origin is 1, and its angle from the positive x-axis is ${\displaystyle \pi }$ radians.

Additionally, when any complex number z is multiplied by ${\displaystyle e^{i\theta }}$, it has the effect of rotating ${\displaystyle z}$ counterclockwise by an angle of ${\displaystyle \theta }$ on the complex plane. Since multiplication by −1 reflects a point across the origin, Euler's identity can be interpreted as saying that rotating any point ${\displaystyle \pi }$ radians around the origin has the same effect as reflecting the point across the origin. Similarly, setting ${\displaystyle \theta }$ equal to ${\displaystyle 2\pi }$ yields the related equation ${\displaystyle e^{2\pi i}=1}$, which can be interpreted as saying that rotating any point by one turn around the origin returns it to its original position.

Generalizations

Euler's identity is also a special case of the more general identity that the nth roots of unity, for n > 1, add up to 0:

$${\displaystyle \sum _{k=0}^{n-1}{e}^{2\pi i\frac{k}{n}}=0.}$$

Euler's identity is the case where n = 2.

A similar identity also applies to quaternion exponential: let {i, j, k} be the basis quaternions; then, $${\displaystyle e^{\frac{1}{\sqrt{3}}(i\pm j\pm k)\pi }+1=0.}$$

More generally, let q be a quaternion with a zero real part and a norm equal to 1; that is, ${\displaystyle q=ai+bj+ck,}$ with ${\displaystyle a^2+b^2+c^2=1.}$ Then one has $${\displaystyle e^{q\pi }+1=0.}$$

The same formula applies to octonions, with a zero real part and a norm equal to 1. These formulas are a direct generalization of Euler's identity, since ${\displaystyle i}$ and ${\displaystyle -i}$ are the only complex numbers with a zero real part and a norm (absolute value) equal to 1.

History

Euler's identity is a direct result of Euler's formula, first published in his monumental 1748 work of mathematical analysis, Introductio in analysin infinitorum, but it is questionable whether the particular concept of linking five fundamental constants in a compact form can be attributed to Euler himself, as he may never have expressed it.

Robin Wilson writes:

We've seen how [Euler's identity] can easily be deduced from results of Johann Bernoulli and Roger Cotes, but that neither of them seem to have done so. Even Euler does not seem to have written it down explicitly—and certainly it doesn't appear in any of his publications—though he must surely have realized that it follows immediately from his identity [i.e. Euler's formula], e^ix = cos x + i sin x. Moreover, it seems to be unknown who first stated the result explicitly