Euler's identity

Euler's identity, sometimes called Euler's equation, is this equation:^[1][2]

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

It features the following mathematical constants:

• ${\displaystyle \pi }$, pi

  ${\displaystyle \pi \approx 3.14159}$

• ${\displaystyle e}$, Euler's Number

  ${\displaystyle e\approx 2.71828}$

• ${\displaystyle i}$, imaginary unit

  ${\displaystyle i={\sqrt {-1}}}$

It also features three of the basic mathematical operations: addition, multiplication and exponentiation.^[1][3]

Euler's identity is named after the Swiss mathematician Leonard Euler. It is not clear that he invented it himself.^[4]

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".^[5]

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 ${\displaystyle e^{x}}$ can be written as the Taylor series

${\displaystyle e^{x}=1+x+{x^{2} \over {2!}}+{x^{3} \over {3!}}+{x^{4} \over {4!}}\cdots =\sum _{k=0}^{\infty }{x^{n} \over n!}}$

As well, the sine function can be written as

${\displaystyle \sin {x}=x-{x^{3} \over 3!}+{x^{5} \over 5!}-{x^{7} \over 7!}\cdots =\sum _{k=0}^{\infty }{(-1)^{n} \over (2n+1)!}{x^{2n+1}}}$

and cosine as

${\displaystyle \cos {x}=1-{x^{2} \over 2!}+{x^{4} \over 4!}-{x^{6} \over 6!}\cdots =\sum _{k=0}^{\infty }{(-1)^{n} \over (2n)!}{x^{2n}}}$

Here, we see a pattern take form. ${\displaystyle e^{x}}$ 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 ${\displaystyle e^{ix}=\cos(x)+i\sin(x)}$.

So, on the left side is ${\displaystyle e^{ix}}$, whose Taylor series is ${\displaystyle 1+ix-{x^{2} \over 2!}-{ix^{3} \over 3!}+{x^{4} \over 4!}+{ix^{5} \over 5!}\cdots }$

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

${\displaystyle (1-{x^{2} \over 2!}+{x^{4} \over 4!}\cdots )+(ix-{ix^{3} \over 3!}+{ix^{5} \over 5!}\cdots )}$

if we add these together, we have

${\displaystyle 1+ix-{x^{2} \over 2!}-{ix^{3} \over 3!}+{x^{4} \over 4!}+{ix^{5} \over 5!}\cdots }$

Therefore,

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

Now, if we replace x with ${\displaystyle \pi }$, we have:

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

Since we know that ${\displaystyle \cos(\pi )=-1}$ and ${\displaystyle \sin(\pi )=0}$, we have:

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

which is the statement of Euler's identity.

Related pages

• −1 (number)
• Euler's formula

References

1. "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. Sandifer, C. Edward 2007. Euler's greatest hits. Mathematical Association of America, p. 4. ISBN 978-0-88385-563-8
5. Crease, Robert P. (2004-10-06). "The greatest equations ever". IOP. Retrieved 2016-02-20.