8 min read
Division and the Inverse
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.
Choose Your Level
Division and the Inverse
What is it?
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.
In simple words
- 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.
- If the denominator is 0, the quotient is not defined. Check |w| first.
Important terms
DIVISION AND THE INVERSE: 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.
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 * (1z) = 1. If z = x+iy, 1z = x-iyx2+y2. Useful in linear systems with complex coefficients.
What you should learn
- 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.
Examples
- 1+i1-i = i
- Real and Im separately
- No leftover i below
DIVIDE
Write the fraction. Times conjugate of denominator on top and bottom. Split into real and imaginary parts. Check |w| != 0.
INVERSE
z * (1z) = 1. If z = x+iy, 1z = x-iyx2+y2. Useful in linear systems with complex coefficients.
Recommended for Class 11
Full school-level explanation
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.
Clear the bottom, then the inverse:
WHAT — DIVISION AND THE INVERSE
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.
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
DIVIDE
Write the fraction. Times conjugate of denominator on top and bottom. Split into real and imaginary parts. Check |w| != 0.
Remember
- conjugate the bottom
Examples
- 1+i1-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.
Remember
- z != 0
Examples
- 1i = -i
- 11+i = 1-i2
- |z|^2 in the bottom
How to answer a division and the inverse question
How it works
- 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.
Remember
- multiply by conjugate of the bottom
- 1z = z-bar / |z|2
- 0.1 + 0.7 i
- w != 0
- conjugate the bottom
- z != 0
- multiply by conjugate of the bottom
- 1z = z-bar / |z|2
How the ideas connect
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
Deeper understanding
Go beyond memorising Division and the Inverse. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
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
Common mistakes
Exam focus
Core Idea
1. DIVIDE
2. INVERSE
Worked Example
Exam focus
Quick revision
- multiply by conjugate of the bottom
- 1z = z-bar / |z|2
- 0.1 + 0.7 i
- w != 0
- DIVISION AND THE INVERSE: To divide, multiply top and bottom by the conjugate of the bottom: (zw) = z w-bar / |w|2. The mult
- 1. DIVIDE: Write the fraction. Times conjugate of denominator on top and bottom. Split into real and imaginary
If the denominator is 0, the quotient is not defined. Check |w| first.
For long answers on Division and the Inverse: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- multiply by conjugate of the bottom
- 1z = z-bar / |z|2
- 0.1 + 0.7 i
- w != 0
Important formulas
- multiply by conjugate of the bottom
- 1z = z-bar / |z|2
- 0.1 + 0.7 i
Common exam traps
- If the denominator is 0, the quotient is not defined. Check |w| first.
- Do not use a remembered formula before checking that its conditions fit the question.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
If the denominator is 0, the quotient is not defined. Check |w| first.
- Mistake
Do not use a remembered formula before checking that its conditions fit the question.
- Mistake
Do not hide sign changes or restrictions; write them on the line where they are used.
- Mistake
If the denominator is 0, the quotient is not defined. Check |w| first.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
multiply by conjugate of the bottom
1z = z-bar / |z|2
0.1 + 0.7 i
w != 0
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. Th
1z = z-bar / |z|2
- 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. Th
- Formula
- 1z = z-bar / |z|2
- Example
- 1+i1-i = i
- Critical warning
- If the denominator is 0, the quotient is not defined. Check |w| first.
Did you understand this topic?
Your answer is saved with your learning record. It does not mark the topic as mastered.
Ready to test yourself?
You've completed the learning section.
Exam Practice
Define
Q1. Define Division and the Inverse.
Explain
Q2. Explain: DIVIDE.
Example
Q3. Give one clear example of Division and the Inverse and say why it fits.
Mistake
Q4. State one common mistake students make about Division and the Inverse.
Exam note
Q5. Write a short exam note on Division and the Inverse: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.