Complex Numbers 2

Last post Complex Numbers 1, we saw that:

Positive numbers like 3 stretches vectors.
Pure positive i numbers like 2i rotate by 90^o and stretches.
Negative numbers like -3 rotate by 180^o and stretches.
Pure negative i numbers like -2i rotate a vector by 270^o and stretches.

What you guess composite numbers like 1+i or 2+3i do?

Returning with our vector v we can phrase the question as what is (1+i)*v ? It is (1+i)*v = v + iv, by distribution.

The vector iv forms 90^o with v, so v + iv forms a square! and the sum is the diagonal of that square. That is v + iv has 45^o, with v and is a little longer by a stretching factor of sqrt(2)=1,41 approximately.

And (1+i)² should rotate by 45^o + 45^o = 90^o and stretch by sqrt(2) * sqrt(2) = 2.

So it should be 2i!! Checking algebraically
(1+i)² = (1+i)*(1+i) = 1 + 2i + i² = 1 + 2i -1 = 2i.

Now what is (3+4i)*v ? It is the diagonal of the rectangle with sides 3v and 4iv, so it is a rotation by the angle of the diagonal with v and a stretching by 5 of v. So what is (3+4i)² ? Before doing the multiplication we now that the angle will be the double of 3+4i and the stretching will be of 25!

Take a complex number z₁ that rotates by 30^o and stretches by 7 and another z₂ that rotates by 120^o and stretches by 4 then z₁ * z₂ rotates by 30^o +120^o = 150^o and stretches by 4*7= 28.

In the last paragraph I didn't say what those numbers were, with a little trigonometry you find that they are:

7(cos(30^o) + i sin(30^o))
4(cos(120^o) + i sin(120^o))
28(cos(150^o) + i sin(150^o))

And that is the rule of sum the angles and multiply the stretching factors.

This way to write complex numbers is called the polar form it is more useful when we want to multiply or divide complex numbers, and it also makes powering much more easier.

Like if we want to calculate

(2(cos(30^o) + i sin(30^o)))¹⁵

We get 2¹⁵ of stretching and an angle of 15*30^o = 450^o = 90^o since 360^o is a complete turn.

More on the next post.