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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.
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Conjugate and Modulus
What is it?
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.
In simple words
- 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.
Important terms
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|. |zw| = |z|/|w| if w != 0. Triangle: |z+w| <= |z|+|w|. |z| = 0 iff z = 0.
What you should learn
- 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.
Examples
- (3-2i)-bar = 3+2i
- z z-bar = |z|2
- sum of conjugates
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.
MODULUS RULES
|z-bar| = |z|. |zw| = |z| |w|. |zw| = |z|/|w| if w != 0. Triangle: |z+w| <= |z|+|w|. |z| = 0 iff z = 0.
Recommended for Class 11
Full school-level explanation
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.
Flip, then length:
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
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.
Remember
- flip Im only
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.
Remember
- length
Examples
- |3+4i| = 5
- |i| = 1
- |-2| = 2
How to answer a conjugate and modulus 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
- z-bar flips Im
- |z| = sqrt(x2+y2)
- 5 and 3+4i
- modulus is real >= 0
- flip Im only
- length
- z-bar flips Im
- |z| = sqrt(x2+y2)
How the ideas connect
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
Deeper understanding
Go beyond memorising Conjugate and Modulus. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
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
Common mistakes
Exam focus
Core Idea
1. CONJUGATE RULES
2. MODULUS RULES
Worked Example
Exam focus
Quick revision
- z-bar flips Im
- |z| = sqrt(x2+y2)
- 5 and 3+4i
- modulus is real >= 0
- 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 (
- 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| is never negative. Do not write |3-4i| = 3-4i.
For long answers on Conjugate and Modulus: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- z-bar flips Im
- |z| = sqrt(x2+y2)
- 5 and 3+4i
- modulus is real >= 0
Important formulas
- z-bar flips Im
- |z| = sqrt(x2+y2)
- 5 and 3+4i
Common exam traps
- |z| is never negative. Do not write |3-4i| = 3-4i.
- 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
|z| is never negative. Do not write |3-4i| = 3-4i.
- 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
|z| is never negative. Do not write |3-4i| = 3-4i.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
z-bar flips Im
|z| = sqrt(x2+y2)
5 and 3+4i
modulus is real >= 0
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 dis
|z| = sqrt(x2+y2)
- 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 dis
- Formula
- |z| = sqrt(x2+y2)
- Example
- (3-2i)-bar = 3+2i
- Critical warning
- |z| is never negative. Do not write |3-4i| = 3-4i.
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Exam Practice
Define
Q1. Define Conjugate and Modulus.
Explain
Q2. Explain: CONJUGATE RULES.
Example
Q3. Give one clear example of Conjugate and Modulus and say why it fits.
Mistake
Q4. State one common mistake students make about Conjugate and Modulus.
Exam note
Q5. Write a short exam note on Conjugate and Modulus: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.