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Properties of real numbers: the rules behind every calculation

Your guide: Sir AhmedSays brackets have saved more marks than any calculator.

Why 2 + 5 = 5 + 2 is a rule

Your book says all addition, subtraction, multiplication and division of real numbers rest on a few properties. Each one has a name you must quote in exams.

For addition and multiplication: closure, commutative (order does not matter), associative (grouping does not matter), identity (0 for addition, 1 for multiplication) and inverse.

The distributive property links them: a(b + c) = ab + ac is left distributive and (a + b)c = ac + bc is right distributive. Zero has no multiplicative inverse.

Order properties handle inequalities. Trichotomy: a = b, a > b or a < b. Multiplying by a negative c flips the sign: a > b gives ac < bc.

The book's table examples (p. 7)

Commutative: 2 + 5 = 5 + 2; 2 × 3 = 3 × 2

Associative: 2 + (3 + 5) = (2 + 3) + 5

Identity: 5 + 0 = 5; 5 × 1 = 5

Inverse: 6 + (−6) = 0; 7 × 1/7 = 1

Equality and order (p. 8)

Reflexive a = a; symmetric a = b ⇒ b = a

Transitive: a > b and b > c ⇒ a > c

Additive: a > b ⇒ a + c > b + c

Multiplicative: a > b ⇒ ac > bc if c > 0, ac < bc if c < 0

Order of numbers: commutative. Brackets: associative. Bracket times a sum: distributive. Inequality: an order property.

Step 1 / 7

Notes, short questions and MCQs

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

Why 2 + 5 = 5 + 2 is a rule

Your book says all addition, subtraction, multiplication and division of real numbers rest on a few properties. Each one has a name you must quote in exams.

For addition and multiplication: closure, commutative (order does not matter), associative (grouping does not matter), identity (0 for addition, 1 for multiplication) and inverse.

The distributive property links them: a(b + c) = ab + ac is left distributive and (a + b)c = ac + bc is right distributive. Zero has no multiplicative inverse.

Order properties handle inequalities. Trichotomy: a = b, a > b or a < b. Multiplying by a negative c flips the sign: a > b gives ac < bc.

The book's table examples (p. 7)

Commutative: 2 + 5 = 5 + 2; 2 × 3 = 3 × 2

Associative: 2 + (3 + 5) = (2 + 3) + 5

Identity: 5 + 0 = 5; 5 × 1 = 5

Inverse: 6 + (−6) = 0; 7 × 1/7 = 1

Equality and order (p. 8)

Reflexive a = a; symmetric a = b ⇒ b = a

Transitive: a > b and b > c ⇒ a > c

Additive: a > b ⇒ a + c > b + c

Multiplicative: a > b ⇒ ac > bc if c > 0, ac < bc if c < 0

Order of numbers: commutative. Brackets: associative. Bracket times a sum: distributive. Inequality: an order property.

Key terms

Commutative
Swapping the order changes nothing.
Associative
Regrouping changes nothing.
Distributive
Multiplying through a bracket.
Trichotomy
a = b, a > b or a < b.

Short questions with model answers

  1. Q1. Example 5 (i): a = 2/3, b = 3/2, c = 5/3. Verify a(b + c) = ab + ac.

    • LHS = 2/3 (3/2 + 5/3) = 2/3 (9 + 10)/6 = 2/3 × 19/6 = 19/9
    • RHS = (2/3)(3/2) + (2/3)(5/3) = 1 + 10/9 = 19/9

    LHS = RHS = 19/9, so a(b + c) = ab + ac is verified.

  2. Q2. Example 6 (iv): which property gives −5 < −4 ⇒ 20 > 16?

    • (−5) × (−4) > (−4) × (−4)
    • 20 > 16

    Order property w.r.t. multiplication.

Common mistakes

  • ✗ Keeping the sign when multiplying by a negative: −5 < −4 ⇒ 20 < 16.

    ✓ If c < 0, a < b gives ac > bc: 20 > 16.

  • ✗ Calling 2 + (3 + 5) = (2 + 3) + 5 commutative.

    ✓ The order is the same; only the brackets move. That is associative.

  • ✗ Saying 0 has a multiplicative inverse.

    ✓ Nothing times 0 gives 1, so 0 has none (p. 7).

MCQs

  1. 1. For all x ∈ R, x = x is called the:

    1. (a) symmetric property
    2. (b) reflexive property
    3. (c) transitive property
    4. (d) trichotomy property
    Show answer

    (b) Reflexive: a = a (p. 8; Review Exercise 1 v).

  2. 2. a > b and b > c ⇒ a > c is the:

    1. (a) trichotomy property
    2. (b) additive property
    3. (c) multiplicative property
    4. (d) transitive property
    Show answer

    (d) Transitive: the order passes through b (p. 8).

  3. 3. If a > 13 then a + 2 > 15 uses:

    1. (a) order property w.r.t. addition
    2. (b) distributive property
    3. (c) additive identity
    4. (d) reciprocal property
    Show answer

    (a) 2 is added to both sides (Example 6 i).

  4. 4. With a = 2/3, b = 3/2, c = 5/3, (a + b)c equals:

    1. (a) 19/9
    2. (b) 13/6
    3. (c) 65/18
    4. (d) 25/18
    Show answer

    (c) (13/6)(5/3) = 65/18 (Example 5 ii).

  5. 5. Which real number has no multiplicative inverse?

    1. (a) 1
    2. (b) 0
    3. (c) −1
    4. (d) 1/7
    Show answer

    (b) 0 × anything = 0, never 1 (p. 7).

Quick revision

  • Closure, commutative, associative, identity and inverse hold for + and ×; 0 has no × inverse.
  • a(b + c) = ab + ac and (a + b)c = ac + bc are left and right distributive.
  • Multiplying an inequality by a negative number flips its sign.

Before you start

What is Class 9 Maths Properties of real numbers?
Properties of real numbers is the open Class 9 Maths lesson for Matric Part 1, Punjab board. It is a step-by-step lesson, not a PDF of the book.
Which boards is this Class 9 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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