MathematicsChapter 1: Complex Numbers

Division and the Inverse

To divide, multiply top and bottom by the conjugate of the bottom: (z/w) = z w-bar / |w|^2.

Learning objectives

State the main ideas and notation used in division and the inverse.
Apply the rules of division and the inverse in a clear, step-by-step solution.
Check signs, conditions, and the final answer for mathematical accuracy.

Definition

To divide, multiply top and bottom by the conjugate of the bottom: (zw) = z w-bar / |w|². The multiplicative inverse is z^-1 = z-bar / |z|² when z != 0. Then Re(1z) = x / |z|² and Im(1z) = -y / |z|^2. Meaning: you never 'cancel i'; you clear i from the denominator.

Where / symbols

DIVISION AND THE INVERSE: To divide, multiply top and bottom by the conjugate of the bottom: (z/w) = z w-bar / |w|^2. The multiplicative inverse is z^{-1} = z-bar / |z|^2 when z != 0. Then Re(1/z) = x / |z|^2 and Im(1/z) = -y / |z|^2. Meaning: you never 'cancel i'; you clear i from the denominator.
1. DIVIDE: Write the fraction. Times conjugate of denominator on top and bottom. Split into real and imaginary parts. Check |w| != 0.
2. INVERSE: z * (1/z) = 1. If z = x+iy, 1/z = (x-iy)/(x^2+y^2). Useful in linear systems with complex coefficients.

Relationship

1z = z-bar / |z|²

Overview

Clear the bottom, then the inverse:

Divide

Write the fraction. Times conjugate of denominator on top and bottom. Split into real and imaginary parts. Check |w| != 0.

Examples (1+i)/(1-i) = i · Real and Im separately · No leftover i below

Inverse

z * (1z) = 1. If z = x+iy, 1z = x-iyx²+y². Useful in linear systems with complex coefficients.

Examples 1/i = -i · 1/(1+i) = (1-i)/2 · |z|^2 in the bottom

WHAT — DIVISION AND THE INVERSE

RULE — 1. DIVIDE

Write the fraction. Times conjugate of denominator on top and bottom. Split into real and imaginary parts. Check |w| != 0.

METHOD — 2. INVERSE

z * (1z) = 1. If z = x+iy, 1z = x-iyx²+y². Useful in linear systems with complex coefficients.

WORKED EXAMPLE

Worked: 1 + 2i3 - i. times 3+i3+i top: (1+2i)(3+i) = 3 + i + 6i + 2 i² = 1 + 7i bottom: 9+1 = 10 = 110 + (710) i

How to answer a division and the inverse question

Identify the given quantities, notation, and required result.
Choose the matching rule and write it before substitution.
Work one algebraic step at a time, keeping brackets and signs clear.
Check restrictions, simplify the result, and verify it where possible.

Worked example

1 + 2i3 - i.
times 3+i3+i
top: (1+2i)(3+i) = 3 + i + 6i + 2 i² = 1 + 7i
bottom: 9+1 = 10
= 110 + (710) i

In simple words

If the denominator is 0, the quotient is not defined. Check |w| first.

Key points

multiply by conjugate of the bottom
1z = z-bar / |z|²
conjugate the bottom
z != 0

Exam preparation

Exam Focus

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Review the key facts, formulas and checks before you move on.