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Properties of the complex conjugate

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Conjugate first, or work first?

You have two numbers and need the conjugate of their product. Do you multiply first and then flip the sign of i? Or flip each one first and then multiply? Your book proves it does not matter.

The same is true for sums and quotients. Taking the conjugate twice brings you back to where you started.

The four properties (pp. 11–13)

1. The conjugate of z1 + z2 is z̄1 + z̄2.

2. The conjugate of z1z2 is z̄1 z̄2.

3. The conjugate of z1 ÷ z2 is z̄1 ÷ z̄2, when z2 ≠ 0.

4. The conjugate of z̄ is z.

Do you know? (p. 17)

Re(z) = (z + z̄) ÷ 2 and Im(z) = (z − z̄) ÷ 2i.

Conjugating and doing the arithmetic can be done in either order. Conjugating twice gives back z.

Step 1 / 7

Notes, short questions and MCQs

Read the full notes: key terms, model answers and MCQs with answers

Conjugate first, or work first?

You have two numbers and need the conjugate of their product. Do you multiply first and then flip the sign of i? Or flip each one first and then multiply? Your book proves it does not matter.

The same is true for sums and quotients. Taking the conjugate twice brings you back to where you started.

The four properties (pp. 11–13)

1. The conjugate of z1 + z2 is z̄1 + z̄2.

2. The conjugate of z1z2 is z̄1 z̄2.

3. The conjugate of z1 ÷ z2 is z̄1 ÷ z̄2, when z2 ≠ 0.

4. The conjugate of z̄ is z.

Do you know? (p. 17)

Re(z) = (z + z̄) ÷ 2 and Im(z) = (z − z̄) ÷ 2i.

Conjugating and doing the arithmetic can be done in either order. Conjugating twice gives back z.

Key terms

Conjugate of a sum
Property 1: add, then conjugate, or the other way.
Conjugate of a product
Property 2: multiply, then conjugate, or the other way.
Conjugate of a quotient
Property 3, with z2 ≠ 0.
Double conjugate
Property 4: flipping the sign twice undoes it.

Short questions with model answers

  1. Q1. Example 11 (p. 11): z1 = 4 + 3i, z2 = 5 + 2i. Show conj(z1 + z2) = z̄1 + z̄2.

    • z1 + z2 = 9 + 5i, so conj(z1 + z2) = 9 − 5i
    • z̄1 + z̄2 = (4 − 3i) + (5 − 2i) = 9 − 5i

    Both sides are 9 − 5i.

  2. Q2. Example 12 (p. 12): z1 = 3 + 4i, z2 = 2 + 3i. Show conj(z1z2) = z̄1 z̄2.

    • z1z2 = 6 + 9i + 8i + 12i2 = −6 + 17i
    • conj(z1z2) = −6 − 17i
    • z̄1 z̄2 = (3 − 4i)(2 − 3i) = 6 − 17i − 12 = −6 − 17i

    Both sides are −6 − 17i.

Common mistakes

  • ✗ conj(−6 + 17i) = 6 − 17i.

    ✓ Only the i part changes sign: −6 − 17i.

  • ✗ conj(z1z2) = z̄1 z2.

    ✓ Both factors are conjugated: z̄1 z̄2.

  • ✗ z + z̄ = 8 + 8i for z = 4 + 4i.

    ✓ The i parts cancel: z + z̄ = 2Re(z) = 8.

MCQs

  1. 1. If z = 4 + 4i, then z + z̄ =

    1. (a) 8
    2. (b) 8 + 8i
    3. (c) 8i
    4. (d) 0
    Show answer

    (a) Review Q1(ix): (4 + 4i) + (4 − 4i) = 8.

  2. 2. The conjugate of the conjugate of 7 + 3i is:

    1. (a) 7 − 3i
    2. (b) −7 + 3i
    3. (c) 7 + 3i
    4. (d) −7 − 3i
    Show answer

    (c) Property 4, Example 14 (p. 13).

  3. 3. conj(z1 ÷ z2) equals:

    1. (a) z̄1 ÷ z2
    2. (b) z1 ÷ z̄2
    3. (c) z̄2 ÷ z̄1
    4. (d) z̄1 ÷ z̄2
    Show answer

    (d) Property 3 (p. 12), with z2 ≠ 0.

  4. 4. For z1 = 5 + 4i and z2 = 3 + 2i, conj(z1 ÷ z2) is (Example 13):

    1. (a) (23 + 2i) ÷ 13
    2. (b) (23 − 2i) ÷ 13
    3. (c) (7 − 2i) ÷ 5
    4. (d) (23 − 2i) ÷ 5
    Show answer

    (b) z1 ÷ z2 = (23 + 2i) ÷ 13, so its conjugate is (23 − 2i) ÷ 13.

  5. 5. Which formula gives Im(z)? (p. 17)

    1. (a) (z + z̄) ÷ 2
    2. (b) (z − z̄) ÷ 2
    3. (c) (z − z̄) ÷ 2i
    4. (d) z z̄
    Show answer

    (c) z − z̄ = 2iy, so dividing by 2i leaves y.

Quick revision

  • The conjugate of a sum, product or quotient is the sum, product or quotient of the conjugates.
  • Conjugating twice gives back z.
  • z + z̄ = 2Re(z) is real; z − z̄ = 2i Im(z) is pure imaginary.

Before you start

What is Class 10 Maths Properties of the complex conjugate?
Properties of the complex conjugate is the open Class 10 Maths lesson for Matric Part 2, Punjab board. It is a step-by-step lesson, not a PDF of the book.
Which boards is this Class 10 Maths lesson for?
The same Punjab textbook: Lahore, Rawalpindi, Gujranwala, Faisalabad, Multan, Sargodha, Bahawalpur, Dera Ghazi Khan, Sahiwal.
Can I study this in Urdu?
Yes. English, Urdu or Roman Urdu. Science terms stay as the exam writes them.
Is this a notes PDF?
No. The book stays with you. This page is the lesson and the check.

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