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; (â€“ 9) + 7 = â€“ 2. Closure property under subtraction: Integers are closed under subtraction, i.e. for any two integers a and b, a â€“ b is an integer. Ex: (â€“ 21) â€“ (â€“ 9) = (â€“ 12); 8 â€“ 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 Ã— 6 = 30; (â€“ 9) Ã— (â€“ 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:(â€“ 2) Ã· (â€“ 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 + (â€“ 6) = 5 â€“ 6 = â€“ 1; (â€“ 6) + 5 = â€“ 6 + 5 = â€“1 âˆ´ 5 + (â€“ 6) = (â€“ 6) + 5. Commutative property under subtraction: Subtraction is not commutative for integers.  For any two integers a and b, a â€“ b â‰  b â€“ a. Ex: 8 â€“ (â€“ 6) = 8 + 6 = 14; (â€“ 6) â€“ 8 = â€“ 6 â€“ 8 = â€“ 14 âˆ´ 8 â€“ (â€“ 6) â‰  â€“ 6 â€“ 8. Commutative property under multiplication: Multiplication is commutative for integers.  For any two integers a and b, ab = ba. Ex: 9 Ã— (â€“ 6) = â€“ (9 Ã— 6) = â€“ 54; (â€“ 6) Ã— 9 = â€“ (6 Ã— 9) = â€“ 54 âˆ´ 9 Ã— (â€“ 6) = (â€“ 6) Ã— 9. Commutative property under division: Division is not commutative for integers.  For any two integers a and b, a Ã· b â‰  b Ã· a. Ex: (â€“ 14) Ã· 2 = â€“ 7 2 Ã· (â€“14) = â€“ $\frac{\text{1}}{\text{7}}$ (â€“ 14) Ã· 2 â‰  2 Ã· (â€“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 + (â€“ 6 + 4) = 5 + (â€“ 2) = 3; (5 â€“ 6) + 4 = (â€“ 1) + 4 = 3 âˆ´ 5 + (â€“ 6 + 4) = (5 â€“ 6) + 4. Associative property under subtraction: Subtraction is associative for integers.  For any three integers a, b and c, a â€“ (b â€“ c) â‰  (a â€“ b) â€“ c Ex: 5 â€“ (6 â€“ 4) = 5 â€“ 2 = 3; (5 â€“ 6) â€“ 4 = â€“ 1 â€“ 4 = â€“ 5 âˆ´ 5 â€“ (6 â€“ 4) â‰  (5 â€“ 6) â€“ 4. Associative property under multiplication: Multiplication is associative for integers. For any three integers a, b and c, (a Ã— b) Ã— c = a Ã— (b Ã— c) Ex: [(â€“ 3) Ã— (â€“ 2)] Ã— 4 = (6 Ã— 4) = 24 (â€“ 3) Ã— [(â€“ 2) Ã— 4] = (â€“ 3) Ã— (â€“ 8) = 24 âˆ´ [(â€“ 3) Ã— (â€“ 2)] Ã— 4 = [(â€“ 3) Ã— (â€“ 2) Ã— 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 Ã— (b + c) = (a Ã— b) + (a Ã— c). Ex: â€“ 2 (4 + 3) =  â€“2 (7)  = â€“14                       = (â€“ 2 Ã— 4) + (â€“ 2 Ã— 3)                       = (â€“ 8) + (â€“ 6)                       = â€“ 14. Distributive property of multiplication over subtraction: For any three integers, a, b and c, a Ã— (b - c) = (a Ã— b) â€“ (a Ã— c). Ex: â€“ 2 (4 â€“ 3) = â€“ 2 (1) = â€“ 2                      = (â€“2 Ã— 4) â€“ (â€“ 2 Ã— 3)                      = (â€“ 8) â€“ (â€“ 6)                      = â€“ 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 Ã— a = a Ã— 1 = a. Ex: (â€“ 4) Ã— 1 = 1 Ã— (â€“ 4) = â€“ 4. When an integer is multiplied by â€“1, the result is the integer with sign changed i.e. the additive identity of the integer. For any integer a, a Ã— â€“1 = â€“1 Ã— a = â€“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; (â€“ 9) + 7 = â€“ 2. Closure property under subtraction: Integers are closed under subtraction, i.e. for any two integers a and b, a â€“ b is an integer. Ex: (â€“ 21) â€“ (â€“ 9) = (â€“ 12); 8 â€“ 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 Ã— 6 = 30; (â€“ 9) Ã— (â€“ 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:(â€“ 2) Ã· (â€“ 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 + (â€“ 6) = 5 â€“ 6 = â€“ 1; (â€“ 6) + 5 = â€“ 6 + 5 = â€“1 âˆ´ 5 + (â€“ 6) = (â€“ 6) + 5. Commutative property under subtraction: Subtraction is not commutative for integers.  For any two integers a and b, a â€“ b â‰  b â€“ a. Ex: 8 â€“ (â€“ 6) = 8 + 6 = 14; (â€“ 6) â€“ 8 = â€“ 6 â€“ 8 = â€“ 14 âˆ´ 8 â€“ (â€“ 6) â‰  â€“ 6 â€“ 8. Commutative property under multiplication: Multiplication is commutative for integers.  For any two integers a and b, ab = ba. Ex: 9 Ã— (â€“ 6) = â€“ (9 Ã— 6) = â€“ 54; (â€“ 6) Ã— 9 = â€“ (6 Ã— 9) = â€“ 54 âˆ´ 9 Ã— (â€“ 6) = (â€“ 6) Ã— 9. Commutative property under division: Division is not commutative for integers.  For any two integers a and b, a Ã· b â‰  b Ã· a. Ex: (â€“ 14) Ã· 2 = â€“ 7 2 Ã· (â€“14) = â€“ $\frac{\text{1}}{\text{7}}$ (â€“ 14) Ã· 2 â‰  2 Ã· (â€“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 + (â€“ 6 + 4) = 5 + (â€“ 2) = 3; (5 â€“ 6) + 4 = (â€“ 1) + 4 = 3 âˆ´ 5 + (â€“ 6 + 4) = (5 â€“ 6) + 4. Associative property under subtraction: Subtraction is associative for integers.  For any three integers a, b and c, a â€“ (b â€“ c) â‰  (a â€“ b) â€“ c Ex: 5 â€“ (6 â€“ 4) = 5 â€“ 2 = 3; (5 â€“ 6) â€“ 4 = â€“ 1 â€“ 4 = â€“ 5 âˆ´ 5 â€“ (6 â€“ 4) â‰  (5 â€“ 6) â€“ 4. Associative property under multiplication: Multiplication is associative for integers. For any three integers a, b and c, (a Ã— b) Ã— c = a Ã— (b Ã— c) Ex: [(â€“ 3) Ã— (â€“ 2)] Ã— 4 = (6 Ã— 4) = 24 (â€“ 3) Ã— [(â€“ 2) Ã— 4] = (â€“ 3) Ã— (â€“ 8) = 24 âˆ´ [(â€“ 3) Ã— (â€“ 2)] Ã— 4 = [(â€“ 3) Ã— (â€“ 2) Ã— 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 Ã— (b + c) = (a Ã— b) + (a Ã— c). Ex: â€“ 2 (4 + 3) =  â€“2 (7)  = â€“14                       = (â€“ 2 Ã— 4) + (â€“ 2 Ã— 3)                       = (â€“ 8) + (â€“ 6)                       = â€“ 14. Distributive property of multiplication over subtraction: For any three integers, a, b and c, a Ã— (b - c) = (a Ã— b) â€“ (a Ã— c). Ex: â€“ 2 (4 â€“ 3) = â€“ 2 (1) = â€“ 2                      = (â€“2 Ã— 4) â€“ (â€“ 2 Ã— 3)                      = (â€“ 8) â€“ (â€“ 6)                      = â€“ 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 Ã— a = a Ã— 1 = a. Ex: (â€“ 4) Ã— 1 = 1 Ã— (â€“ 4) = â€“ 4. When an integer is multiplied by â€“1, the result is the integer with sign changed i.e. the additive identity of the integer. For any integer a, a Ã— â€“1 = â€“1 Ã— a = â€“a.

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