Euler's identity

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

It features the following mathematical constants:

  • , pi
  • , Euler's Number
  • , imaginary unit

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[change | change source]

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

The exponential function can be written as the Taylor series

As well, the sine function can be written as

and cosine as

Here, we see a pattern take form. 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 .

So, on the left side is , whose Taylor series is

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 , whose Taylor series is the Taylor series of cosine, plus i times the Taylor series of sine, which can be shown as:

if we add these together, we have


Now, if we replace x with , we have:

Since we know that and , we have:

which is the statement of Euler's identity.

Related pages[change | change source]

References[change | change source]

  1. 1.0 1.1 "Euler's Formula: A Complete Guide — Euler's Identity". Math Vault. 2020-09-30. Retrieved 2020-10-02.
  2. Weisstein, Eric W. "Euler Formula". Retrieved 2020-10-02.
  3. Hogenboom, Melissa. "The most beautiful equation is... Euler's identity". 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.