Euler's identity
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
Therefore,
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]- ↑ 1.0 1.1 "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.
- ↑ 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.