Euler's Formula for Complex Numbers

(There is another "Euler's Formula" about Geometry,
this page is about the one used in Complex Numbers)

First, you may have seen the famous "Euler's Identity":

eiπ + 1 = 0

It seems absolutely magical that such a neat equation combines:

And also has the basic operations of add, multiply, and an exponent too!

But if you want to take an interesting trip through mathematics, you will discover how it comes about.

Interested? Read on!

Discovery

It was around 1740, and mathematicians were interested in imaginary numbers.

An imaginary number, when squared gives a negative result

imaginary squared is negative

This is normally impossible (try squaring any number, remembering that multiplying negatives gives a positive), but just imagine that you can do it, call it i for imaginary, and see where it carries you:

i2 = −1

Leonhard Euler

 

Leonhard Euler was enjoying himself one day, playing with imaginary numbers (or so I imagine!), and he took this Taylor Series which was already known:

ex = 1 + x + x22! + x33! + x44! + x55! + ...

And he put i into it:

eix = 1 + ix + (ix)22! + (ix)33! + (ix)44! + (ix)55! + ...

 

And because i2 = −1, it simplifies to:

eix = 1 + ix − x22!ix33! + x44! + ix55! − ...

 

Now group all the i terms at the end:

eix = ( 1 − x22! + x44! − ... )  +  i( x − x33! + x55! − ... )

 

And here is the miracle ... the two groups are quite neatly the Taylor Series for cos and sin

cos x = 1 − x22! + x44! − ...
sin x = x − x33! + x55! − ...

And so it simplifies to:

eix = cos x + i sin x

He must have been so happy when he discovered this!

And it is now called Euler's Formula.

 

Let's give it a try:

Example: when x = 1.1

eix = cos x + i sin x
e1.1i = cos 1.1 + i sin 1.1
e1.1i = 0.45 + 0.89 i   (to 2 decimals)

Note: we are using radians, not degrees.

The answer is a combination of a Real and an Imaginary Number, which together is called a Complex Number.

We can plot such a number on the complex plane (the real numbers go left-right, and the imaginary numbers go up-down):

graph real imaginary 0.45 + 0.89i
Here we show the number 0.45 + 0.89 i

Which is the same as e1.1i

Let's plot some more!

graph real imaginary many e^ix values

A Circle!

Yes, putting Euler's Formula on that graph produces a circle:

e^ix = cos(x) + i sin(x) on circle
e
ix produces a circle of radius 1

 

Include a radius of r and we can turn any point (such as 3 + 4i) into reix form (by finding the correct value of x and r)

Example: the number 3 + 4i

To turn 3 + 4i into reix form we do a Cartesian to Polar conversion:

 

So 3 + 4i can also be 5e0.927 i

3+4i = 5 at 0.927

It is Another Form

It is basically another way of having a complex number.

This turns out to very useful, as there are many cases (such as multiplication) where it is easier to use the reix form rather than the a+bi form.

Plotting eiπ

Lastly, when we calculate Euler's Formula for x = π we get:

eiπ = cos π + i sin π
eiπ = −1 + i × 0   (because cos π = −1 and sin π = 0)
eiπ = −1

And here is the point created by eiπ (where our discussion began):

e^ipi = -1 + i on circle

And eiπ = −1 can be rearranged into:

eiπ + 1 = 0

The famous Euler's Identity.