Euler's identity

e ^ (πi) + 1 = 0
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E-to-the-i-pi.svg

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.[1]

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".[2]

Mathematical proof of Euler's Identity using Taylor SeriesEdit

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  , 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..

  •  

Then we know that

  •  

and

  •  

Therefore:

  •  
  •  

QED

ReferencesEdit

  1. Sandifer, C. Edward 2007. Euler's greatest hits. Mathematical Association of America, p. 4. ISBN 978-0-88385-563-8
  2. Crease, Robert P. (2004-10-06). "The greatest equations ever". IOP. Retrieved 2016-02-20.