Fig 4-1. Polar coordinates for complex numbers
Any point {x, y} on a Cartesian plane can be represented by polar coordinates {r, θ}. We can apply this transformation to complex numbers on the Argand plane. That is, we can write:
So that a complex number can be defined as:x = r·sin(θ) y = r·cos(θ)
Note that 2πn, where n is an integer, can be added to θ without changing the value of z. The length r is called the modulus of z, written |z|, wherez = r·[sin(θ) + cos(θ)i]
The angle θ, in radians, is called the argument of z, written arg(z), where|z| = √(x2 + y2)
The principle value of θ is always between 0 and 2π,θ = tan-1(y/x)
0 ≤ arg(z) < 2πSo, a complex number, z, has the polar coordinates
{|z|, tan-1(y/x)}
Fig 4-1. Polar coordinates for complex numbers
In figure 4-1, the complex number z1 = 3 + 2i can be represented as {r1, θ1} where
That is, z1 = {3.61, 0.322}. Similarly, the complex number z2 = -2.2 - 3.3i can be represented asr1 = √(9 + 4) = √13 = 3.61 θ1 = tan-1(2/3) = 0.322 [radians]
z2 = {3.966, 4.124}
z1·z2 = r1[cos(θ1) + sin(θ1)i]·r2[cos(θ2) + sin(θ2)i]
= r1·r2·[cos(θ1)·cos(θ2) - sin(θ1)·sin(θ2)
+ (cos(θ1)·sin(θ2) + sin(θ1)·cos(θ2)i]
Using trigonometric identities, we get:
So for multiplication,z = z1·z2 = r1·r2·[cos(θ1 + θ2) + sin(θ1 + θ2)i]
|z| = |z1|·|z2| arg(z) = arg(z1) + arg(z2)
So for division,z = z1/z2 = (r1/r2)[cos(θ1 - θ2) + sin(θ1 - θ2)i]
|z| = |z1|/|z2| arg(z) = arg(z1) - arg(z2)