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Pre-Calculus Complex numbers (advanced)

Operations on complex numbers in polar form

20 practice questions 0 video lessons Theory + worked examples

Operations on Complex Numbers in Polar Form

Common Core Pre-Calculus • Standard N-CN.5 • Complex Numbers

Operations on Complex Numbers in Polar Form is a topic in Complex Numbers in the Common Core State Standards. It is aligned to Standard N-CN.5, which requires students to represent the multiplication of complex numbers geometrically in polar form.

In polar form, complex numbers multiply by multiplying moduli and adding arguments, and divide by dividing moduli and subtracting arguments.

Common Core Pre-Calculus › Complex Numbers › Operations on Complex Numbers in Polar Form  —  Standard N-CN.5

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Theory

Polar form turns multiplication and division into simple arithmetic on the modulus and argument:

\[z_1z_2=r_1r_2\,\text{cis}(\theta_1+\theta_2),\qquad \dfrac{z_1}{z_2}=\dfrac{r_1}{r_2}\,\text{cis}(\theta_1-\theta_2).\]

Geometrically, multiplying by a complex number scales by its modulus and rotates by its argument.

This is why polar form matters. The messy FOIL of rectangular multiplication becomes “multiply the sizes, add the angles.”
Product Product Product z₁z₂ = r₁r₂ cis(θ₁+θ₂) multiply moduli, add arguments
Product: multiply moduli, add arguments.
Quotient Quotient Quotient z₁z₂ = r₁r₂ cis(θ₁−θ₂) divide moduli, subtract arguments
Quotient: divide moduli, subtract arguments.

The product and quotient rules:

\[z_1z_2=r_1r_2\,\text{cis}(\theta_1+\theta_2),\qquad \dfrac{z_1}{z_2}=\dfrac{r_1}{r_2}\,\text{cis}(\theta_1-\theta_2)\]
multiply moduli and add arguments for a product; divide moduli and subtract arguments for a quotient
Keep arguments in range by adding or subtracting \(360^\circ\) (or \(2\pi\)) when they run past a full turn.

How to multiply or divide in polar form

  1. Write both numbers in polar form \(r\,\text{cis}\,\theta\).
  2. Multiply: multiply moduli, add arguments.
  3. Divide: divide moduli, subtract arguments.
  4. Reduce the argument to a standard range if needed.
Example 1 — Product in polar form
Multiply \(z_1=2\,\text{cis}\,40^\circ\) and \(z_2=3\,\text{cis}\,70^\circ\).
Solution

Multiply the moduli and add the arguments.

\(z_1 z_2\)\(=\)\((2\cdot 3)\,\text{cis}\,(40^\circ+70^\circ)\)
\(=\)\(6\,\text{cis}\,110^\circ\)
product is 6 cis 110 degrees
Example 2 — Quotient in polar form
Divide \(z_1=10\,\text{cis}\,100^\circ\) by \(z_2=2\,\text{cis}\,30^\circ\).
Solution

Divide the moduli and subtract the arguments.

\(\dfrac{z_1}{z_2}\)\(=\)\(\dfrac{10}{2}\,\text{cis}\,(100^\circ-30^\circ)\)
\(=\)\(5\,\text{cis}\,70^\circ\)
quotient is 5 cis 70 degrees
Example 3 — Product needing a wrap-around
Multiply \(z_1=4\,\text{cis}\,200^\circ\) and \(z_2=2\,\text{cis}\,250^\circ\).
Solution

Add arguments; if the sum exceeds \(360^\circ\), subtract a full turn.

\(z_1 z_2\)\(=\)\(8\,\text{cis}\,(200^\circ+250^\circ)\)
\(=\)\(8\,\text{cis}\,450^\circ=8\,\text{cis}\,90^\circ\)
product is 8 cis 90 degrees after subtracting 360
Example 4 — Why polar form helps
Interpret multiplying by \(z_2=\text{cis}\,90^\circ\) geometrically.
Solution

Its modulus is 1 and argument \(90^\circ\), so multiplying by it rotates a number \(90^\circ\) without changing its size.

\(r\cdot 1\)\(=\)\(r,\quad \theta+90^\circ\)
multiplying by cis 90 rotates a number 90 degrees

Common pitfalls

Multiply moduli but add arguments. Don't add the moduli or multiply the angles.
Reduce large angles. Subtract \(360^\circ\) (or \(2\pi\)) to keep the argument standard.
Convert to polar first. These rules need both numbers in polar form.

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.

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