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:
But if you want to take an interesting trip through mathematics, then read on to find out why it is true.
It actually comes from Euler's Formula:
eix = cos x + i sin x
When we calculate that for x = π we get:
So eiπ + 1 = 0 is just a special case of a much more useful formula that Euler discovered.
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 in it:
And because i2 = -1, it simplifies to:
Now, separating the terms without and with i gets:
And here is the miracle ...
So we get:
Example: when x = 3
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 even 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
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)
Example: the number 3 + 4i
To turn into reix form we do a Cartesian to Polar conversion:
- r = √(32 + 42) = √(9+16) = √25 = 5
- x = tan-1 ( 4 / 3 ) = 0.927 (to 3 decimals)
So 3 + 4i can also be 5e0.927 i
There are many cases (such as multiplication) where it is easier to use reix than a+bi
Lastly, here is the point created by eiπ (where our discussion began):
eiπ = −1