Integers Class 7 Notes Maths Chapter 1

1. Integers

Operations on Integers

3 x 2 = 6
 2 x 2 = 4 = 6 - 2
 1 x 2 = 2 = 4- 2
 0 x 2 = 0 = 2 - 2
 -1 x 2 = 0- 2 = - 2
 -2 x 2 = -2-2 = - 4 

When two positive integers are added, we get a positive integer.

When two positive integers are added, we get a positive integer.

Eg: 8 + 2 = 10 When two negative integers are added, we get a negative integer. Eg: -6 + (-3) = -9

When a positive and a negative integer are added, the sign of the sum is always the sign of the bigger number of the two, without considering their signs.

Eg: 45 + -25 = 20 and -45 + 20 = -25

The additive inverse of any integer a is - a, and the additive inverse of (- a) is a.

Eg: Additive inverse of (-12) = - (-12) = 12 Subtraction is the opposite of addition, and, therefore, we add the additive inverse of the integer that is being subtracted, to the other integer.

Eg: 23 - 43 = 23 + Additive inverse of 43 = 23 + (- 43) = - 20

The product of a positive and a negative integer is a negative integer.

The product of two negative integers is a positive integer. If the number of negative integers in a product is even, then the product is a positive integer. Similarly, if the number of negative integers in a product is odd, then the product is a negative integer.

Division is the inverse operation of multiplication. The division of a negative integer by a positive integer results in a negative integer. The division of a positive integer by a negative integer results in a negative integer. The division of a negative integer by a negative integer results in a positive

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integer. For any integer p, p multiplied by zero is equal to zero multiplied by p, which is equal to zero. For any integer p, p divided by zero is not defined, and zero divided by p is equal to zero, where p is not equal to zero.

Properties of Integers

Integers are closed under addition, i.e. for any two integers,a and b, a+b is an integer.

1. Closure property:
 2. Closure property under addition:

Integers are closed under addition, i.e. for any two integers,a and b, a+b is an integer. Eg: 3+4=7;-9+7=2

3. Closure property under subtraction:

Integers are closed under subtraction, i.e. for any two integers,a and b, a-b is an integer. Eg: -21-(-9)=-12;8-3=5

4. Closure property under multiplication:

Integers are closed under multiplication, i.e. for any two integers,a and b, ab is an integer. Eg: 5×6=30; -9×-3=27

5. Closure property under division:

Integers are NOT closed under division, i.e. for any two integers, 

 may not be integer

Eg:

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6. Commutative property:

7. Commutative property under addition:

Addition is commutative for integers. For any two integers, a and b, a+b=b+a Eg:5+(-6)=5-6=-1; (-6)+5=-6+5=-1 ∴5+(-6)=(-6)+5

8. Commutative property under subtraction:

Subtraction is NOT commutative for integers. For any two integers, a-b≠b-a Eg:8-(-6)=8+6=14; (-6)-8=-6-8=-14 ∴8-(-6)≠-6-8

9. Commutative property under multiplication:

Multiplication is commutative for integers. For any two integers, a and b, ab=ba Eg:9×(-6)=-(9×6)=-54; (-6)×9=-(6×9)=-54 ∴9×(-6)=(-6)×9

10. Commutative property under division:

Division is NOT commutative for integers. For any two integers, Eg:3/6=1/2;

11. Associative property:

12. Associative property under addition:

Addition is associative for integers. For any three integers, a, b and c, a+(b+c)=(a+b)+c Eg:5+(-6+4)=5-2=3; (5-6)+4=-1+4=3 ∴5+(-6+4)=(5-6)+4

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13. Associative property under subtraction:

Subtraction is associative for integers. For any three integers, a-(b-c)≠(a-b)-c Eg:5—(6-4)=5-2=3; (5-6)-4=-1-4=-5 ∴5—(6-4)≠(5-6)-4

14. Associative property under multiplication:

Multiplication is associative for integers. For any three integers, a, b and c, (a×b)×c=a×(b×c) Eg: [(-3)×(-2))×4]=(6×4)=24 [(-3)×(-2×4) ]=(-3×-8)=24 ∴[(-3)×(-2))×4]=[(-3)×(-2×4) ]

15. Associative property under division:

Division is NOT associative for integers.

16. Distributive property:

17. Distributive property of multiplication over addition:

For any three integers, a, b and c, a×(b+c) = a×b+a×c Eg: -2 (4 + 3) = -2 (7) = -14 -2(4+3)=(-2×4)+(-2×3) =(-8)+(-6) =-14

18. Distributive property of multiplication over subtraction:

For any three integers, a, b and c, a×(b-c)= a×b-a×c Eg: -2 (4- 3) = -2 (1) = -2 -2(4-3)=(-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.

19. Identity under addition:

Integer 0 is the identity under addition. That is, for an integer a, a+0=0+a=a. Eg: 4+0=0+4=4

20. Identity under multiplication:

The integer 1 is the identity under multiplication. That is, for an integer a, 1×a=a×1=a Eg: (-4)×1=1×(-4)=-4