*Kiddle Encyclopedia.*

# Euler's identity facts for kids

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

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

Euler's identity is named after the Swiss mathematician Leonard 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 can be written as the Taylor series

As well, Sine 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 , who's 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 , who's 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..

Then we know that

and

Therefore:

QEDpl:Wzór Eulera#Tożsamość Eulera