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

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.

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, 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

- −1 (number)
- Euler's formulapl:Wzór Eulera#Tożsamość Eulera

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