MathematicsChapter 1: Complex Numbers
Conjugate and Modulus
The conjugate of z = x + i y is z-bar = x - i y (flip the sign of the imaginary part).
Learning objectives
- State the main ideas and notation used in conjugate and modulus.
- Apply the rules of conjugate and modulus in a clear, step-by-step solution.
- Check signs, conditions, and the final answer for mathematical accuracy.
Definition
The conjugate of z = x + i y is z-bar = x - i y (flip the sign of the imaginary part). The modulus (magnitude, absolute value) is |z| = sqrt(x^2 + y^2), the distance from 0 on the Argand plane. Always |z| >= 0. Meaning: conjugate reflects in the real axis; modulus is a length, so it is a real number.
Mathematical form
z = x + i y is z-bar
bar = 3+4i (3-4i)(3+4i)
Where / symbols
- CONJUGATE AND MODULUS
- The conjugate of z = x + i y is z-bar = x - i y (flip the sign of the imaginary part). The modulus (magnitude, absolute value) is |z| = sqrt(x^2 + y^2), the distance from 0 on the Argand plane. Always |z| >= 0. Meaning: conjugate reflects in the real axis; modulus is a length, so it is a real number.
- 1. CONJUGATE RULES
- z + z-bar = 2x (real). z - z-bar = 2 i y. (z-bar)-bar = z. Product conjugate: (zw)-bar = z-bar w-bar. z * z-bar = |z|^2.
- 2. MODULUS RULES
- |z-bar| = |z|. |zw| = |z| |w|. |z/w| = |z|/|w| if w != 0. Triangle: |z+w| <= |z|+|w|. |z| = 0 iff z = 0.
Relationship
- |z| = sqrt(x2+y2)
Overview
Flip, then length:
Conjugate Rules
z + z-bar = 2x (real). z - z-bar = 2 i y. (z-bar)-bar = z. Product conjugate: (zw)-bar = z-bar w-bar. z * z-bar = |z|^2.
Examples (3-2i)-bar = 3+2i · z z-bar = |z|^2 · sum of conjugates
Modulus Rules
|z-bar| = |z|. |zw| = |z| |w|. |zw| = |z|/|w| if w != 0. Triangle: |z+w| <= |z|+|w|. |z| = 0 iff z = 0.
Examples |3+4i| = 5 · |i| = 1 · |-2| = 2
WHAT — CONJUGATE AND MODULUS
The conjugate of z = x + i y is z-bar = x - i y (flip the sign of the imaginary part). The modulus (magnitude, absolute value) is |z| = sqrt(x^2 + y^2), the distance from 0 on the Argand plane. Always |z| >= 0. Meaning: conjugate reflects in the real axis; modulus is a length, so it is a real number.
RULE — 1. CONJUGATE RULES
z + z-bar = 2x (real). z - z-bar = 2 i y. (z-bar)-bar = z. Product conjugate: (zw)-bar = z-bar w-bar. z * z-bar = |z|^2.
METHOD — 2. MODULUS RULES
|z-bar| = |z|. |zw| = |z| |w|. |zw| = |z|/|w| if w != 0. Triangle: |z+w| <= |z|+|w|. |z| = 0 iff z = 0.
WORKED EXAMPLE
Worked: |3 - 4i| and conjugate. |3-4i| = sqrt(9+16) = 5 (3-4i)-bar = 3+4i (3-4i)(3+4i) = 9 + 16 = 25 = 5^2
How to answer a conjugate and modulus 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
- |3 - 4i| and conjugate.
- |3-4i| = sqrt(9+16) = 5
- (3-4i)-bar = 3+4i
- (3-4i)(3+4i) = 9 + 16 = 25 = 52
In simple words
|z| is never negative. Do not write |3-4i| = 3-4i.
Key points
- z-bar flips Im
- |z| = sqrt(x2+y2)
- flip Im only
- length
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Conjugate and Modulus
Sticky Notes · Quick Revision
Core Idea
- The conjugate of z = x + i y is z-bar = x - i y (flip the sign of the imaginary part).
- The modulus (magnitude, absolute value) is |z| = sqrt(x^2 + y^2), the distance from 0 on the Argand plane.
- Always |z| >= 0.
1. CONJUGATE RULES
- z + z-bar = 2x (real).
- z - z-bar = 2 i y.
- (z-bar)-bar = z.
- Product conjugate: (zw)-bar = z-bar w-bar.
2. MODULUS RULES
- |z-bar| = |z|.
- |zw| = |z| |w|.
- |zw| = |z|/|w| if w != 0.
- Triangle: |z+w| <= |z|+|w|.
Worked Example
- Worked: |3 - 4i| and conjugate.
- |3-4i| = sqrt(9+16) = 5
- (3-4i)-bar = 3+4i
- (3-4i)(3+4i) = 9 + 16 = 25 = 52
Common Misconception
- |z| is never negative. Do not write |3-4i| = 3-4i.
FBISE Exam Check
- State the rule for conjugate and modulus before substituting values.
- Show the main algebra steps, keep exact values where possible, and check every condition.
60-second revision
- The conjugate of z = x + i y is z-bar = x - i y (flip the sign of the imaginary part).
- The modulus (magnitude, absolute value) is |z| = sqrt(x^2 + y^2), the distance from 0 on the Argand plane.
- Always |z| >= 0.
- |z| is never negative. Do not write |3-4i| = 3-4i.
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