Complex numbers, Imaginary axis, Imaginary unit, Algebraic form, Polar form, Absolute value, Argument,Trigonometric form, Sine, Cosine, Addition, Multiplication, Complex conjugate, Division, nth power.
There is a big problem with the algebraic form of complex numbers. That is, it is almost impossible to raise them to powers.
Let's try to compute the value of
Well, this is it.
But this could only be some sick joke...
There must be a simple way to raise complex numbers to powers.
This is the usual algebraic form of complex numbers,
and now we replace it with a polar form (trigonometric form).
The main idea of thispolar form (trigonometric form) is that it describes complex numbers using two new attributes: one is the absolute value and the other is the angle.
We denote the absolute value by r,
and the angle... well, the angle by theta. Here it is:
The polar form (trigonometric form) makes it surprisingly simple to multiply complex numbers,
and to divide them.
And now, let’s get back to the issue of powers.
We want to compute the value of .
Here comes the polar form (trigonometric form):
And now we start computing the powers.
The nth power is computed by raising r to the nth power, and multiple the angle by n:
and that, if we feel like it, can be transformed back into algebraic form.
Now let's try to compute this:
Let's see first the polar form (trigonometric form).
But there is a snag.
This equation here:
has another solution.
As to which one we need, we could decide by flipping a coin,
but it is better to draw a figure.
It seems we need the negative solution.
And now we are ready for the multiplication.