Euler's identity

Euler's identity, sometimes called Euler's equation, is this equation:

$e^{i π} + 1 = 0$

It features the following mathematical constants:

$π$ , pi
$π \approx 3.14159$
$e$ , Euler's Number
$e \approx 2.71828$
$i$ , imaginary unit
$i = \sqrt{- 1}$

It also features three of the basic mathematical operations: addition, multiplication and exponentiation.

Euler’s identity is the result of using Euler’s formula with θ equal to pi.

Euler's identity is named after the Swiss mathematician Leonhard Euler. It is not clear that he invented it himself.

Respondents to a Physics World poll called the identity "the most profound mathematical statement ever written", "uncanny and sublime", "filled with cosmic beauty" and "mind-blowing".

Mathematical proof of Euler's Identity using Taylor Series

Many equations can be written as a series of terms added together. This is called a Taylor series.

The exponential function $e^{x}$ can be written as the Taylor series

$e^{x} = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \frac{x^{4}}{4!} \dots = \sum_{k = 0}^{\infty} \frac{x^{n}}{n!}$

As well, the sine function can be written as

$\sin x = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \frac{x^{7}}{7!} \dots = \sum_{k = 0}^{\infty} \frac{(- 1)^{n}}{(2 n + 1)!} x^{2 n + 1}$

and cosine as

$\cos x = 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \frac{x^{6}}{6!} \dots = \sum_{k = 0}^{\infty} \frac{(- 1)^{n}}{(2 n)!} x^{2 n}$

Here, we see a pattern take form. $e^{x}$ seems to be a sum of sine and cosine's Taylor series, except with all of the signs changed to positive. The identity we are actually proving is $e^{i x} = \cos (x) + i \sin (x)$ .

So, on the left side is $e^{i x}$ , whose Taylor series is $1 + i x - \frac{x^{2}}{2!} - \frac{i x^{3}}{3!} + \frac{x^{4}}{4!} + \frac{i x^{5}}{5!} \dots$

We can see a pattern here, that every second term is i times sine's terms, and that the other terms are cosine's terms.

On the right side is $\cos (x) + i \sin (x)$ , whose Taylor series is the Taylor series of cosine, plus i times the Taylor series of sine, which can be shown as:

$(1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} \dots) + (i x - \frac{i x^{3}}{3!} + \frac{i x^{5}}{5!} \dots)$

if we add these together, we have

$1 + i x - \frac{x^{2}}{2!} - \frac{i x^{3}}{3!} + \frac{x^{4}}{4!} + \frac{i x^{5}}{5!} \dots$

Therefore,

$e^{i x} = \cos (x) + i \sin (x)$

Now, if we replace x with $π$ , we have:

$e^{i π} = \cos (π) + i \sin (π)$

Since we know that $\cos (π) = - 1$ and $\sin (π) = 0$ , we have:

$e^{i π} = - 1$
$e^{i π} + 1 = 0$

which is the statement of Euler's identity.

Related pages

References

1 2 "Euler's Formula: A Complete Guide — Euler's Identity". Math Vault. 2020-09-30. Retrieved 2020-10-02.
↑ Weisstein, Eric W. "Euler Formula". mathworld.wolfram.com. Retrieved 2020-10-02.
↑ Hogenboom, Melissa. "The most beautiful equation is... Euler's identity". www.bbc.com. Retrieved 2020-10-02.
↑ Weisstein, Eric W. "Euler Formula". mathworld.wolfram.com. Retrieved 2026-06-15.
↑ Sandifer, C. Edward 2007. Euler's greatest hits. Mathematical Association of America, p. 4. ISBN 978-0-88385-563-8
↑ Crease, Robert P. (2004-10-06). "The greatest equations ever". IOP. Retrieved 2016-02-20.

[:0-1] 1 2 "Euler's Formula: A Complete Guide — Euler's Identity". Math Vault. 2020-09-30. Retrieved 2020-10-02.

[2] Weisstein, Eric W. "Euler Formula". mathworld.wolfram.com. Retrieved 2020-10-02.

[3] Hogenboom, Melissa. "The most beautiful equation is... Euler's identity". www.bbc.com. Retrieved 2020-10-02.

[4] Weisstein, Eric W. "Euler Formula". mathworld.wolfram.com. Retrieved 2026-06-15.

[5] Sandifer, C. Edward 2007. Euler's greatest hits. Mathematical Association of America, p. 4. ISBN 978-0-88385-563-8

[6] Crease, Robert P. (2004-10-06). "The greatest equations ever". IOP. Retrieved 2016-02-20.