Operations on complex numbers in polar form
Theory
Polar form turns multiplication and division into simple arithmetic on the modulus and argument:
Geometrically, multiplying by a complex number scales by its modulus and rotates by its argument.
The product and quotient rules:
How to multiply or divide in polar form
- Write both numbers in polar form \(r\,\text{cis}\,\theta\).
- Multiply: multiply moduli, add arguments.
- Divide: divide moduli, subtract arguments.
- Reduce the argument to a standard range if needed.
Multiply the moduli and add the arguments.
| \(z_1 z_2\) | \(=\) | \((2\cdot 3)\,\text{cis}\,(40^\circ+70^\circ)\) |
| \(=\) | \(6\,\text{cis}\,110^\circ\) |
Divide the moduli and subtract the arguments.
| \(\dfrac{z_1}{z_2}\) | \(=\) | \(\dfrac{10}{2}\,\text{cis}\,(100^\circ-30^\circ)\) |
| \(=\) | \(5\,\text{cis}\,70^\circ\) |
Add arguments; if the sum exceeds \(360^\circ\), subtract a full turn.
| \(z_1 z_2\) | \(=\) | \(8\,\text{cis}\,(200^\circ+250^\circ)\) |
| \(=\) | \(8\,\text{cis}\,450^\circ=8\,\text{cis}\,90^\circ\) |
Its modulus is 1 and argument \(90^\circ\), so multiplying by it rotates a number \(90^\circ\) without changing its size.
| \(r\cdot 1\) | \(=\) | \(r,\quad \theta+90^\circ\) |
Common pitfalls
Frequently asked questions
How do you multiply complex numbers in polar form?
Multiply the moduli and add the arguments: \(z_1z_2=r_1r_2\,\text{cis}(\theta_1+\theta_2)\).
How do you divide complex numbers in polar form?
Divide the moduli and subtract the arguments: \(\dfrac{z_1}{z_2}=\dfrac{r_1}{r_2}\,\text{cis}(\theta_1-\theta_2)\).
What does multiplying by a complex number do geometrically?
It scales by the modulus and rotates by the argument. Multiplying by \(\text{cis}\,90^\circ\) is a pure \(90^\circ\) rotation.
Why is polar form better for multiplication?
Because it replaces FOIL with simple operations on sizes and angles, and it reveals the scaling-and-rotation meaning.