euler’s formula

Preface

Whenever I do a post like this, I hope I don’t make a silly mistake; but then again, posts of this nature contain content copied and pasted from other sources.

I first encountered the following equation in high school. The class was further mathematics, and only had 3 people. I believe it was for the A-levels module FM1 (sadly, this was ages ago, FM stands for further mathematics). Our teacher was teaching the Taylor series and he proved the equation that way. Later, in this post, I will just prove the equation via differentiation.

I don’t remember encountering this equation ever again, not even at university (though I didn’t study mathematics, sorry for being so cryptic, maybe later I will tell you what I studied at uni). Actually, maybe we touched it at university, but it didn’t stick in my mind that much.

Recently, I was researching mathematical proofs (because this is what I do in my life now), and came across this neat equation again.

 

Main point: Euler’s formula

e^{i\theta } = cos\theta + isin\theta discovered by Swiss mathematician Euler, born in 1707.

 

Beauty

It is beautiful for many reasons. When you set \theta = \pi, you get

e^{i\pi}+1=0.

This equation contains:

  • the basic arithmetic operations (each only occuring once):
    • addition,
    • multiplication, and
    • exponentiation
  • the five fundamental mathematical constants:
    • 0 (additive identity),
    • 1 (multiplicative identity),
    • π (3.141592653),
    • e (2.718281828),
    • i (square root of negative 1, basic unit of imaginary part of complex numbers)

 

Proof

By differentiation.

  • define function f(\theta)=e^{-i\theta}(cos\theta+isin\theta)
  • with product rule, it is easy to get the function’s derivative f'(\theta)=0
  • this means the function is constant in \theta
  • evaluate f(0)=1
  • this function is 1 everywhere
  • rearrange this to get Euler’s identity

QED.

 

Picture

euler's identity

Euler’s equation uses polar coordinates (distance, angle). e^{i\theta } = cos\theta + isin\theta separates the complex growth into its components: real growth (distance from origin) and imaginary growth (anti-clockwise rotation).

The equation in that picture is explained by starting at (1, 0). Operating (1, 0) with e^{i\pi}. This explains why e^{i\pi} is -1.

But there is more! Let’s not stop there… since this is periodic (goes around in a circle), this also applies to anything that is +2\pi.

It also shows why e^{\pi/2} = i.

Then we can show that i^i = e^{-\pi/2} = 0.20787... a real number!

Other usefulness of this equation is it replaces a lot of dreadful trig formulas you had to memorise in high school like double angle formulas – yuk! Just substitute in 2x in the theta and do some expansions, comparing imaginary parts with imaginary parts etc… and you will get what you want!

 

Salt

You might want to check other sources to gain a better intuitive understanding of the equation – it covers a lot of mathematical ground so it is worthwhile doing that. I warn you now that I do not have a complete understanding myself so take everything above with a grain of salt.

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