4.4 The Modulus and the Conjugate of a Complex Number

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Modulus and Conjugate of a Complex Number | Class 11 Maths JEE & KCET

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Learn Class 11 Maths NCERT concept – Modulus and Conjugate of a Complex Number. Understand definitions, properties, geometric meaning, solved examples & JEE/KCET exam applications.

Introduction

In the study of complex numbers, two concepts play a central role: modulus and conjugate. They help us measure the size (magnitude) and provide symmetry across the real axis in the complex plane. These tools simplify algebraic manipulation, proofs of identities, and geometry of complex numbers. For JEE, KCET, and COMEDK exams, mastery of these concepts is crucial since many questions involve quick evaluation of modulus and conjugates.

1. The Modulus of a Complex Number

Definition

If z = a + ib, where a and b are real, then the modulus of z is:

|z| = √(a² + b²).

• It represents the distance of the point (a, b) from the origin in the Argand plane.
• Modulus is always non-negative real.

Properties of modulus

1. |z| ≥ 0, and |z| = 0 ⇔ z = 0.
2. |z₁ z₂| = |z₁| · |z₂|.
3. |z₁ / z₂| = |z₁| / |z₂| (z₂ ≠ 0).
4. |z|² = z z̅.
5. |z + w| ≤ |z| + |w| (Triangle Inequality).
6. |z − w| ≥ ||z| − |w||.

Examples

1. If z = 3 − 4i, then |z| = √(9 + 16) = 5.
2. If z = 7i, then |z| = √(0² + 7²) = 7.
3. If z = 2 + i, then |z|² = (2 + i)(2 − i) = 5.

2. The Conjugate of a Complex Number

Definition

If z = a + ib, then its conjugate is:

z̅ = a − ib.

• The conjugate of a complex number is its mirror image about the real axis in the Argand plane.

Properties of conjugate

1. (z̅)̅ = z.
2. z + z̅ = 2 Re(z).
3. z − z̅ = 2i Im(z).
4. z z̅ = |z|².
5. (z + w)̅ = z̅ + w̅.
6. (z w)̅ = z̅ w̅.
7. (z / w)̅ = z̅ / w̅ (w ≠ 0).
8. If z is real, then z̅ = z.

Examples

1. If z = 5 − 2i, then z̅ = 5 + 2i.
2. If z = 4i, then z̅ = −4i.
3. (3 + 2i)(3 − 2i) = 9 + 4 = 13 = |z|².

3. Relationship between Modulus and Conjugate

• The modulus can always be expressed in terms of conjugate:
  |z|² = z z̅.
• Division is simplified by multiplying numerator and denominator with the conjugate of the denominator:
  Example: (3 + 4i)/(2 − i) = (3 + 4i)(2 + i) / [(2 − i)(2 + i)] = (6 + 3i + 8i + 4i²) / (4 + 1) = (2 + 11i) / 5.

4. Geometric Interpretation

• The modulus |z| gives the length of the vector representing z.
• The conjugate z̅ gives the reflection across the x-axis.

Example:
z = 3 + 4i → represented by point (3, 4).

• |z| = 5 (distance from origin).
• z̅ = 3 − 4i (point reflected below x-axis).

5. Important Results

1. |z|² = z z̅ (always real).
2. Re(z) = (z + z̅)/2, Im(z) = (z − z̅)/(2i).
3. If |z| = 1, then z̅ = 1/z.
4. For any two complex numbers z, w:
   • |z + w|² = |z|² + |w|² + 2 Re(z w̅).
   • |z − w|² = |z|² + |w|² − 2 Re(z w̅).

Practice Examples

1. If z = 7 − 24i, find |z| and z̅.
   → |z| = √(49 + 576) = 25, z̅ = 7 + 24i.
2. If |z| = 13 and Re(z) = 5, find Im(z).
   → |z|² = a² + b² → 169 = 25 + b² → b² = 144 → b = ±12. So z = 5 ± 12i.
3. Prove that |z| = |z̅|.
   → |z̅| = √(a² + (−b)²) = √(a² + b²) = |z|.

Marks Allotment in Exams

• NCERT/Board exams:
  • Short 2–3 mark questions: calculating |z|, finding z̅, proving identities like z z̅ = |z|².
  • Long 4–5 mark questions: application in quadratic equations with complex roots.
• JEE Main / KCET / COMEDK:
  • 1 mark MCQs: cycle of i, direct modulus evaluation.
  • 2–3 mark questions: simplifying (z₁/z₂), verifying |z + w|² = |z|² + |w|² + 2 Re(z w̅).
  • Important weightage: ~2–3 questions typically in JEE/board level.

Key Properties of Modulus

1. |z| ≥ 0; |z| = 0 ⇔ z = 0
2. |z₁ z₂| = |z₁| · |z₂|
3. |z₁ / z₂| = |z₁| / |z₂| (z₂ ≠ 0)
4. |z|² = z z̅
5. |z + w| ≤ |z| + |w| (Triangle inequality)
6. |z − w| ≥ ||z| − |w||

Key Properties of Conjugate

1. (z̅)̅ = z
2. z + z̅ = 2 Re(z)
3. z − z̅ = 2i Im(z)
4. z z̅ = |z|²
5. (z + w)̅ = z̅ + w̅
6. (z w)̅ = z̅ w̅
7. (z / w)̅ = z̅ / w̅ (w ≠ 0)
8. If z is real → z̅ = z

Relationship Between Modulus and Conjugate

• |z|² = z z̅
• If |z| = 1 → z̅ = 1/z

Formula Box (for fast recall)

🔹 Geometric Meaning

• |z| = Distance of point (a, b) from origin.
• z̅ = Reflection of z across the real axis.

🔹 Exam Importance

• Boards (2–5 marks): Compute modulus, conjugates, prove |z|² = z z̅, apply in quadratic roots.
• JEE Main / KCET / COMEDK (1–3 marks): Simplifications using modulus & conjugates, triangle inequality, quick MCQs.

Practice Problems on Modulus & Conjugate

MCQ 1:

If z = 3 – 4i, then |z| is:
A) 3
B) 4
C) 5
D) √7

Solution:
|z| = √(3² + (–4)²) = √(9 + 16) = √25 = 5.
Answer: C (5)

MCQ 2:

If z = 5 + 12i, find z z̅.
A) 13
B) 25
C) 144
D) 169

Solution:
z̅ = 5 – 12i
z z̅ = (5 + 12i)(5 – 12i) = 25 + 144 = 169
Answer: D (169)

MCQ 3:

If z = a + ib, then Re(z) = ?
A) (z + z̅)/2
B) (z – z̅)/2
C) (z – z̅)/2i
D) None of these

Solution:
Re(z) = (z + z̅)/2
Answer: A

MCQ 4:

If |z| = 1, then which is true?
A) z̅ = z
B) z̅ = –z
C) z̅ = 1/z
D) z̅ = z²

Solution:
For |z| = 1 → z z̅ = 1 → z̅ = 1/z
Answer: C

MCQ 5 (JEE Style):

If z = cos θ + i sin θ, then |z| equals:
A) cos θ
B) sin θ
C) 1
D) θ

Solution:
|z| = √(cos²θ + sin²θ) = 1
Answer: C (1)

Solved Examples

Example 1:

Find the modulus and conjugate of z = –7 + 24i.
Solution:
|z| = √((–7)² + (24)²) = √(49 + 576) = √625 = 25.
z̅ = –7 – 24i.

Example 2:

Prove that |z|² = z z̅.
Solution:
Let z = a + ib.
z̅ = a – ib.
z z̅ = (a + ib)(a – ib) = a² + b² = |z|².

Example 3 (Triangle Inequality):

If z₁ = 3 + 4i and z₂ = 1 – 2i, show that |z₁ + z₂| ≤ |z₁| + |z₂|.
Solution:
z₁ + z₂ = (3+1) + (4–2)i = 4 + 2i.
|z₁ + z₂| = √(4² + 2²) = √20 = 2√5.
|z₁| = √(3² + 4²) = 5, |z₂| = √(1² + (–2)²) = √5.
|z₁| + |z₂| = 5 + √5 ≈ 7.24, while |z₁ + z₂| = 2√5 ≈ 4.47.
Hence, inequality holds.

Example 4 (KCET Type):

If z = 2 + i√3, find |z| and the argument of z.
Solution:
|z| = √(2² + (√3)²) = √(4 + 3) = √7.
arg(z) = tan⁻¹(√3/2).

Example 5 (Application in Quadratic):

Solve x² + 4x + 13 = 0.
Solution:
Discriminant = (4)² – 4(1)(13) = 16 – 52 = –36 < 0.
Roots = (–4 ± √(–36))/2 = (–4 ± 6i)/2 = –2 ± 3i.
Here modulus = √((-2)² + (3)²) = √13.

 Final Summary

The modulus and conjugate are two fundamental tools in complex number algebra.

• Modulus (|z|): Represents the distance of a complex number from the origin in the Argand plane. For z = a + ib, |z| = √(a² + b²).
• Conjugate (z̅): Represents the reflection of a complex number across the real axis. For z = a + ib, z̅ = a − ib.

Key Properties:

• |z|² = z × z̅
• |z₁z₂| = |z₁||z₂|
• (z + w)̅ = z̅ + w̅, (z w)̅ = z̅ w̅
• |z + w| ≤ |z| + |w| (triangle inequality)
• If |z| = 1 → z̅ = 1/z

Geometric Meaning:

• |z| is the distance from origin to (a, b).
• z̅ is the mirror image of z across the x-axis.

Exam Perspective:

• Board Exams (2–5 marks): Basic modulus/conjugate computation & proofs like |z|² = z z̅.
• JEE / KCET / COMEDK (1–3 marks): MCQs involving triangle inequality, simplifications, and properties.