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Properties of complex numbers

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Same rules, bigger family

Real numbers follow rules you trust without thinking: 2 + 3 = 3 + 2, and x × 1 = x. Do complex numbers still obey them after we invented i? Your book checks each rule, for addition and for multiplication.

They do. The sum and product of complex numbers are complex (closure). Order and grouping do not matter (commutative, associative). Each number has an identity and an inverse.

Identities and inverses (pp. 7–8)

Additive identity 0 = 0 + 0i: z + 0 = 0 + z = z.

Multiplicative identity 1 = 1 + 0i: z × 1 = 1 × z = z.

Additive inverse −z: z + (−z) = 0. Multiplicative inverse z−1 (z ≠ 0): z × z−1 = 1.

Distributive law (p. 8)

z1(z2 + z3) = z1z2 + z1z3 and (z1 + z2)z3 = z1z3 + z2z3.

Complex numbers keep the real-number rules: closure, commutative, associative, identities 0 and 1, inverses −z and z−1, and the distributive law.

Step 1 / 7

Notes, short questions and MCQs

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

Same rules, bigger family

Real numbers follow rules you trust without thinking: 2 + 3 = 3 + 2, and x × 1 = x. Do complex numbers still obey them after we invented i? Your book checks each rule, for addition and for multiplication.

They do. The sum and product of complex numbers are complex (closure). Order and grouping do not matter (commutative, associative). Each number has an identity and an inverse.

Identities and inverses (pp. 7–8)

Additive identity 0 = 0 + 0i: z + 0 = 0 + z = z.

Multiplicative identity 1 = 1 + 0i: z × 1 = 1 × z = z.

Additive inverse −z: z + (−z) = 0. Multiplicative inverse z−1 (z ≠ 0): z × z−1 = 1.

Distributive law (p. 8)

z1(z2 + z3) = z1z2 + z1z3 and (z1 + z2)z3 = z1z3 + z2z3.

Complex numbers keep the real-number rules: closure, commutative, associative, identities 0 and 1, inverses −z and z−1, and the distributive law.

Key terms

Additive identity
Adding it leaves z unchanged.
Multiplicative identity
Multiplying by it leaves z unchanged.
Additive inverse (−z)
Change the sign of both parts.
Multiplicative inverse (z−1)
1 á z, written as x + iy; z must not be 0.

Short questions with model answers

  1. Q1. Example 9 (p. 8): find the additive inverse of z = 7 − 10i.

    • −z = −7 + 10i
    • z + (−z) = 7 − 10i − 7 + 10i = 0

    −7 + 10i

  2. Q2. Example 10 (p. 8): find the multiplicative inverse of 4 − 3i.

    • z−1 = 1 á (4 − 3i) = (4 + 3i) á ((4 − 3i)(4 + 3i))
    • = (4 + 3i) á (16 + 9) = (4 + 3i) á 25

    4/25 + (3/25)i

Common mistakes

  • ✗ The additive inverse of 5 − 2i is −5 − 2i.

    ✓ Both signs change: −5 + 2i.

  • ✗ The multiplicative inverse of 1 + i is 1 − i.

    ✓ (1 + i)(1 − i) = 2, not 1. The inverse is (1 − i) ÷ 2.

  • ✗ Every complex number has a multiplicative inverse.

    ✓ Only non-zero ones. Nothing times 0 gives 1.

MCQs

  1. 1. Additive inverse of 5 − 2i is:

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

    (d) Review Q1(v) (p. 20): (5 − 2i) + (−5 + 2i) = 0.

  2. 2. Multiplicative inverse of z = 1 + i is:

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

    (c) Review Q1(vi): (1 − i) ÷ ((1 + i)(1 − i)) = (1 − i) ÷ 2.

  3. 3. Is 0 a complex number? (Review Q2(i))

    1. (a) Yes: 0 = 0 + 0i
    2. (b) No: it has no i
    3. (c) Only if i = 0
    4. (d) No: it is only real
    Show answer

    (a) Every real number is x + 0i, and 0 is the additive identity (p. 7).

  4. 4. z1z2 = z2z1 is the:

    1. (a) Associative property
    2. (b) Commutative property of multiplication
    3. (c) Distributive property
    4. (d) Closure property
    Show answer

    (b) Order does not matter (p. 7).

  5. 5. Which number has no multiplicative inverse?

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

    (d) The inverse needs z ≠ 0 (p. 8).

Quick revision

  • Closure, commutative and associative laws hold for both addition and multiplication.
  • Identities: 0 + 0i for addition, 1 + 0i for multiplication.
  • Inverses: −z flips both signs; z−1 = 1 á z needs the conjugate and z ≠ 0.

Before you start

What is Class 10 Maths Properties of complex numbers?
Properties of complex numbers 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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