eulers formula

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 this famous equation:

eiπ + 1 = 0

It seems absolutely magical that such a neat equation combines:

• e (Euler's Number)
• i (the unit imaginary number)
• π (the famous number pi that turns up in many interesting areas)
• 0 and 1 (also wonderful numbers!)

But if you want to take an interesting trip through mathematics, then read on to find out why it is true.

Euler's Formula

It actually comes from Euler's Formula:

eix = cos x + i sin x

When we calculate it 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

eiπ + 1 = 0

So eiπ + 1 = 0 is just a special case of a much more useful formula that Euler discovered.

Discovery

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

An imaginary number, when squared gives a negative result

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

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

(You can use the Sigma Calculator to play with this.)

And he put i into it:

And because i2 = -1, it simplifies to:

Now gather the terms with i and put them at the end:

And here is the miracle ...

the first group is the Taylor Series for cos

the second group is the Taylor Series for sin

 

He must have been so happy when he discovered this!

The result is:

Example: when x = 3

eix = cos x + i sin x

e3i = cos 3 + i sin 3

e3i = −0.990 + 0.141 i   (to 3 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):

Here we show the number −0.990 + 0.141 i

Which is the same as e3i

A Circle!

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

eix produces a circle of radius 1

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