Euler's identity

File:E-to-the-i-pi.svg

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

<math>e^{i\pi} + 1 = 0</math>

It features the following mathematical constants:

• <math>\pi</math>, pi

  <math>\pi \approx 3.14159</math>

• <math>e</math>, Euler's Number

  <math>e \approx 2.71828</math>

• <math>i</math>, imaginary unit

  <math>i = \sqrt{-1}</math>

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 <math>e ^{x}</math> can be written as the Taylor series

<math>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!}</math>

As well, the sine function can be written as

<math>\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}} </math>

and cosine as

<math>\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}} </math>

Here, we see a pattern take form. <math>e^{x} </math> 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 <math>e^{ix} = \cos(x) + i \sin(x)</math>.

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

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 <math>\cos(x) + i \sin(x)</math>, whose Taylor series is the Taylor series of cosine, plus i times the Taylor series of sine, which can be shown as:

<math>( 1 - {x^{2} \over 2!} + {x^{4} \over 4!} \cdots) + (ix - {ix^{3} \over 3!} + {ix^{5} \over 5!}\cdots)</math>

if we add these together, we have

<math>1 + ix - {x^{2} \over 2!} - {ix^{3} \over 3!} + {x^{4} \over 4!} + {ix^{5} \over 5!} \cdots</math>

Therefore,

<math>e^{ix} = \cos(x) + i \sin(x)</math>

Now, if we replace x with <math>\pi</math>, we have:

<math>e^{i\pi} = \cos(\pi) + i \sin(\pi)</math>

Since we know that <math>\cos(\pi) = -1</math> and <math>\sin(\pi) = 0</math>, we have:

• <math>e^{i\pi} = -1</math>
• <math>e^{i\pi} + 1 = 0</math>

which is the statement of Euler's identity.

Related pages

• −1 (number)
• Euler's formula