Notes On Properties of Integers - CBSE Class 7 Maths
Closure property Closure property under addition: Integers are closed under addition, i.e. for any two integers a and b, a + b is an integer.  Ex: 3 + 4 = 7; (– 9) + 7 = – 2. Closure property under subtraction: Integers are closed under subtraction, i.e. for any two integers a and b, a – b is an integer. Ex: (– 21) – (– 9) = (– 12); 8 – 3 = 5. Closure property under multiplication: Integers are closed under multiplication, i.e. for any two integers a and b, ab is an integer. Ex: 5 × 6 = 30; (– 9) × (– 3) = 27. Closure property under division: Integers are not closed under division, i.e. for any two integers a and b, $\frac{\text{a}}{\text{b}}$may not be an integer. Ex:(– 2) ÷ (– 4) = $\frac{\text{1}}{\text{2}}$ Commutative property Commutative property under addition: Addition is commutative for integers.  For any two integers a and b, a + b = b + a. Ex: 5 + (– 6) = 5 – 6 = – 1; (– 6) + 5 = – 6 + 5 = –1 ∴ 5 + (– 6) = (– 6) + 5. Commutative property under subtraction: Subtraction is not commutative for integers.  For any two integers a and b, a – b ≠ b – a. Ex: 8 – (– 6) = 8 + 6 = 14; (– 6) – 8 = – 6 – 8 = – 14 ∴ 8 – (– 6) ≠ – 6 – 8. Commutative property under multiplication: Multiplication is commutative for integers.  For any two integers a and b, ab = ba. Ex: 9 × (– 6) = – (9 × 6) = – 54; (– 6) × 9 = – (6 × 9) = – 54 ∴ 9 × (– 6) = (– 6) × 9. Commutative property under division: Division is not commutative for integers.  For any two integers a and b, a ÷ b ≠ b ÷ a. Ex: (– 14) ÷ 2 = – 7 2 ÷ (–14) = – $\frac{\text{1}}{\text{7}}$ (– 14) ÷ 2 ≠ 2 ÷ (–14). Associative property Associative property under addition: Addition is associative for integers. For any three integers a, b and c, a + (b + c) = (a + b) + c Ex: 5 + (– 6 + 4) = 5 + (– 2) = 3; (5 – 6) + 4 = (– 1) + 4 = 3 ∴ 5 + (– 6 + 4) = (5 – 6) + 4. Associative property under subtraction: Subtraction is associative for integers.  For any three integers a, b and c, a – (b – c) ≠ (a – b) – c Ex: 5 – (6 – 4) = 5 – 2 = 3; (5 – 6) – 4 = – 1 – 4 = – 5 ∴ 5 – (6 – 4) ≠ (5 – 6) – 4. Associative property under multiplication: Multiplication is associative for integers. For any three integers a, b and c, (a × b) × c = a × (b × c) Ex: [(– 3) × (– 2)] × 4 = (6 × 4) = 24 (– 3) × [(– 2) × 4] = (– 3) × (– 8) = 24 ∴ [(– 3) × (– 2)] × 4 = [(– 3) × (– 2) × 4]. Associative property under division: Division is not associative for integers. Distributive property Distributive property of multiplication over addition: For any three integers a, b and c, a × (b + c) = (a × b) + (a × c). Ex: – 2 (4 + 3) =  –2 (7)  = –14                       = (– 2 × 4) + (– 2 × 3)                       = (– 8) + (– 6)                       = – 14. Distributive property of multiplication over subtraction: For any three integers, a, b and c, a × (b - c) = (a × b) – (a × c). Ex: – 2 (4 – 3) = – 2 (1) = – 2                      = (–2 × 4) – (– 2 × 3)                      = (– 8) – (– 6)                      = – 2. The distributive property of multiplication over the operations of addition and subtraction is true in the case of integers. Identity under addition: Integer 0 is the identity under addition.  That is, for any integer a, a + 0 = 0 + a = a. Ex: 4 + 0 = 0 + 4 = 4. Identity under multiplication: The integer 1 is the identity under multiplication. That is, for any integer a, 1 × a = a × 1 = a. Ex: (– 4) × 1 = 1 × (– 4) = – 4. When an integer is multiplied by –1, the result is the integer with sign changed i.e. the additive identity of the integer. For any integer a, a × –1 = –1 × a = –a.

Summary

Closure property Closure property under addition: Integers are closed under addition, i.e. for any two integers a and b, a + b is an integer.  Ex: 3 + 4 = 7; (– 9) + 7 = – 2. Closure property under subtraction: Integers are closed under subtraction, i.e. for any two integers a and b, a – b is an integer. Ex: (– 21) – (– 9) = (– 12); 8 – 3 = 5. Closure property under multiplication: Integers are closed under multiplication, i.e. for any two integers a and b, ab is an integer. Ex: 5 × 6 = 30; (– 9) × (– 3) = 27. Closure property under division: Integers are not closed under division, i.e. for any two integers a and b, $\frac{\text{a}}{\text{b}}$may not be an integer. Ex:(– 2) ÷ (– 4) = $\frac{\text{1}}{\text{2}}$ Commutative property Commutative property under addition: Addition is commutative for integers.  For any two integers a and b, a + b = b + a. Ex: 5 + (– 6) = 5 – 6 = – 1; (– 6) + 5 = – 6 + 5 = –1 ∴ 5 + (– 6) = (– 6) + 5. Commutative property under subtraction: Subtraction is not commutative for integers.  For any two integers a and b, a – b ≠ b – a. Ex: 8 – (– 6) = 8 + 6 = 14; (– 6) – 8 = – 6 – 8 = – 14 ∴ 8 – (– 6) ≠ – 6 – 8. Commutative property under multiplication: Multiplication is commutative for integers.  For any two integers a and b, ab = ba. Ex: 9 × (– 6) = – (9 × 6) = – 54; (– 6) × 9 = – (6 × 9) = – 54 ∴ 9 × (– 6) = (– 6) × 9. Commutative property under division: Division is not commutative for integers.  For any two integers a and b, a ÷ b ≠ b ÷ a. Ex: (– 14) ÷ 2 = – 7 2 ÷ (–14) = – $\frac{\text{1}}{\text{7}}$ (– 14) ÷ 2 ≠ 2 ÷ (–14). Associative property Associative property under addition: Addition is associative for integers. For any three integers a, b and c, a + (b + c) = (a + b) + c Ex: 5 + (– 6 + 4) = 5 + (– 2) = 3; (5 – 6) + 4 = (– 1) + 4 = 3 ∴ 5 + (– 6 + 4) = (5 – 6) + 4. Associative property under subtraction: Subtraction is associative for integers.  For any three integers a, b and c, a – (b – c) ≠ (a – b) – c Ex: 5 – (6 – 4) = 5 – 2 = 3; (5 – 6) – 4 = – 1 – 4 = – 5 ∴ 5 – (6 – 4) ≠ (5 – 6) – 4. Associative property under multiplication: Multiplication is associative for integers. For any three integers a, b and c, (a × b) × c = a × (b × c) Ex: [(– 3) × (– 2)] × 4 = (6 × 4) = 24 (– 3) × [(– 2) × 4] = (– 3) × (– 8) = 24 ∴ [(– 3) × (– 2)] × 4 = [(– 3) × (– 2) × 4]. Associative property under division: Division is not associative for integers. Distributive property Distributive property of multiplication over addition: For any three integers a, b and c, a × (b + c) = (a × b) + (a × c). Ex: – 2 (4 + 3) =  –2 (7)  = –14                       = (– 2 × 4) + (– 2 × 3)                       = (– 8) + (– 6)                       = – 14. Distributive property of multiplication over subtraction: For any three integers, a, b and c, a × (b - c) = (a × b) – (a × c). Ex: – 2 (4 – 3) = – 2 (1) = – 2                      = (–2 × 4) – (– 2 × 3)                      = (– 8) – (– 6)                      = – 2. The distributive property of multiplication over the operations of addition and subtraction is true in the case of integers. Identity under addition: Integer 0 is the identity under addition.  That is, for any integer a, a + 0 = 0 + a = a. Ex: 4 + 0 = 0 + 4 = 4. Identity under multiplication: The integer 1 is the identity under multiplication. That is, for any integer a, 1 × a = a × 1 = a. Ex: (– 4) × 1 = 1 × (– 4) = – 4. When an integer is multiplied by –1, the result is the integer with sign changed i.e. the additive identity of the integer. For any integer a, a × –1 = –1 × a = –a.

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