Multiplication and Division in Polar Form
In polar form, multiplying complex numbers multiplies their moduli and adds their arguments, and dividing divides the moduli and subtracts the arguments. In brief: if z1 = r1(cos θ1 + i sin θ1) and z2 = r2(cos θ2 + i sin θ2), then z1z2 = r1r2[cos(θ1 + θ2) + i sin(θ1 + θ2)] and z1/z2 = (r1/r2)[cos(θ1 − θ2) + i sin(θ1 − θ2)].
Multiplication of complex numbers is a rotation combined with a scaling. This is why complex numbers describe rotations in the plane and oscillations in physics. The formula follows from the sum formulas for sine and cosine. This guide proves the rule, shows its geometric meaning, solves examples with multiplication, division and reciprocals, and gives a table that compares algebraic and polar methods.
How do you multiply in polar form?
Let z1 = r1(cos θ1 + i sin θ1) and z2 = r2(cos θ2 + i sin θ2). Multiply:
z1z2 = r1r2[(cos θ1 cos θ2 − sin θ1 sin θ2) + i(sin θ1 cos θ2 + cos θ1 sin θ2)].
By the sum formulas, the brackets equal cos(θ1 + θ2) and sin(θ1 + θ2). So
z1z2 = r1r2[cos(θ1 + θ2) + i sin(θ1 + θ2)].
The modulus of a product is the product of the moduli, |z1z2| = |z1||z2|, and the argument of a product is the sum of the arguments (modulo 2π). The result may need an adjustment of the angle to the principal range.
How do you divide in polar form?
z1/z2 = (r1/r2)[cos(θ1 − θ2) + i sin(θ1 − θ2)], for z2 ≠ 0. The reciprocal is 1/z = (1/r)[cos(−θ) + i sin(−θ)] = (1/r)(cos θ − i sin θ), which is also z̄/|z|2.
What is the geometric meaning?
Multiplication by z2 = r2(cos θ2 + i sin θ2) is a spiral similarity: it rotates every point by θ2 about the origin and scales it by r2. For |z2| = 1, it is a pure rotation. Multiplication by i is a rotation by 90°. Division is the inverse transformation: a rotation by −θ2 and a scaling by 1/r2.
Worked examples
Example 1: multiplication
z1 = 2(cos 30° + i sin 30°) and z2 = 3(cos 45° + i sin 45°). Then z1z2 = 6(cos 75° + i sin 75°) ≈ 6(0.2588 + 0.9659i) = 1.553 + 5.796i.
Example 2: angle adjustment
z1 = 4(cos 200° + i sin 200°) and z2 = 0.5(cos 250° + i sin 250°). The product has the modulus 2 and the argument 450°, which equals 90°. So z1z2 = 2(cos 90° + i sin 90°) = 2i.
Example 3: division
z1 = 8(cos 150° + i sin 150°) and z2 = 2(cos 60° + i sin 60°). Then z1/z2 = 4(cos 90° + i sin 90°) = 4i.
Example 4: from algebraic form
(1 + i)(√3 + i): 1 + i = √2(cos 45° + i sin 45°) and √3 + i = 2(cos 30° + i sin 30°). The product is 2√2(cos 75° + i sin 75°). Algebraically: (1 + i)(√3 + i) = √3 + i + √3 i + i2 = (√3 − 1) + (√3 + 1)i ≈ 0.732 + 2.732i. Check: 2√2 · cos 75° = 2.828 · 0.2588 = 0.732 ✓ and 2.828 · 0.9659 = 2.732 ✓. This computation also gives the exact value cos 75° = (√3 − 1)/(2√2) = (√6 − √2)/4.
Example 5: a quotient in polar form
(−1 + i)/(1 + √3 i): −1 + i = √2(cos 135° + i sin 135°) and 1 + √3 i = 2(cos 60° + i sin 60°). The quotient is (√2/2)(cos 75° + i sin 75°) ≈ 0.7071(0.2588 + 0.9659i) = 0.183 + 0.683i.
Example 6: rotating a point
Rotate the point 3 + i by 60° counterclockwise about the origin. Multiply by cos 60° + i sin 60° = 1/2 + (√3/2)i: (3 + i)(1/2 + (√3/2)i) = 3/2 + (3√3/2)i + i/2 + (√3/2)i2 = (3 − √3)/2 + (3√3 + 1)/2 · i ≈ 0.634 + 3.098i. The modulus is unchanged: √(0.402 + 9.598) = √10 ✓, matching |3 + i| = √10.
Example 7: a regular hexagon
The vertices of a regular hexagon centered at the origin with a vertex at 2 are 2, 2ω, 2ω2, …, with ω = cos 60° + i sin 60°. The vertex 2ω = 1 + √3 i, 2ω2 = −1 + √3 i, 2ω3 = −2.
Comparison of the methods
| Operation | Algebraic form | Polar form | Preferred when |
|---|---|---|---|
| z1 + z2 | add parts | convert first | algebraic |
| z1z2 | (ac − bd) + (ad + bc)i | r1r2, θ1 + θ2 | polar for many factors |
| z1/z2 | multiply by the conjugate | r1/r2, θ1 − θ2 | polar |
| 1/z | z̄/|z|2 | 1/r, −θ | either |
| zn | repeated multiplication | rn, nθ | polar |
Where does the rule come from?
Abraham de Moivre discovered related formulas in the early eighteenth century, and Leonhard Euler gave the complete picture of the multiplication rule using the exponential function: eiθ1 eiθ2 = ei(θ1 + θ2). The geometric picture of multiplication as rotation and scaling was part of the work of Wessel, Argand and Gauss.
Where is it used today?
In AC circuit analysis, the impedance of series elements is a sum and the combined voltage is the product of current and impedance, a multiplication in polar form: a current of 2 A at 30° through an impedance of 5 ohms at 40° gives a voltage of 10 V at 70°. In computer graphics, complex multiplication rotates 2D objects. In signal processing, a filter multiplies each frequency component by a complex number: it scales its amplitude and shifts its phase.
Which mistakes are common?
Multiplying the arguments
The arguments are added when multiplying, not multiplied. The moduli are multiplied.
Adding moduli in a product
|z1z2| = |z1||z2|, not |z1| + |z2|.
Using polar form for addition
There is no simple polar rule for sums. Convert to algebraic form first.
Not reducing the angle
An argument such as 450° is valid, but the principal value is 90°. State the result clearly.