Properties of Subtracting Integers

The properties of subtracting integers are explained here along with the examples.

1. Closure Property: The difference (subtraction) of any two integers is always an integer.

i.e., The difference of two integers is always an integer.

Hence, integers are closed under subtraction.

If x and y are any two integers, then x - y always an integer

For Examples:

(i) (+7) – (+4) = 7 - 4 = 3, which is an integer.

(ii) (-8) – (+3) = -8 – 3 = -11, which is an integer.

(iii) 6 - 14 = 6 + (-14) = - 8, which is an integer.

(iv) (- 18) - 10 = (- 18) + (- 10) = - 28, which is an integer.

2. Commutative Property: Subtraction is not commutative for integers.

For any two different integers ‘x’ and ‘y’,

x - y ≠ y - x

For Examples:

(i) 4 - 8 = 4 + (-8) = -4 and 8 - 4 = 8 + (- 4) = 4

Therefore, 4 - 8 ≠ 8 - 4

(ii) (- 4) - 7 = (- 4) + (- 7) = - 11 and 7 - (-4) = 7 + 4 = 11

Therefore, (- 4) - 7 ≠ 7 - (-4)

3. Associative Property: Subtraction is not associative for integers.

For any three integers ‘x’, ‘y’ and ‘z’, 

x - (y - z) ≠ (x - y) - z

For Example:

[2 - (- 3)] - (- 6) = [2 + (3)] + 6 = 5 + 6 = 11 and

2 - [(- 3) - (- 6)] = 2 - [(- 3) + 6] = 2 - 3 = -1

Therefore, [2 - (- 3)] - (- 6) ≠ 2 - [(- 3) - (- 6)]

4. Subtraction Property of Zero: The result of subtracting zero from an integer is the integer itself.

For any three integers ‘x’,

x - 0 = x

For Example:

(i) 5 - 0 = 5

(ii) 8 - 0 = 8

(iii) 100 - 0 = 100

(iv) 9999 - 0 = 9999

(v) 99999999 - 0 = 99999999

5. For any integer ‘x’, x - 0 ≠ 0 - x

For Example:

(i) 7 - 0 = 7      and      0 - 7 = -7

Therefore, 7 - 0 ≠ 0 - 7

(ii) 10 - 0 = 10     and      0 - 10 = -10 

Therefore, 10 - 0 ≠ 0 - 10

6. For any three integers ‘x’, ‘y’ and ‘z’, and x > y, then

x - z > y - z

To evaluate an expression containing various integers with plus and minus sign:

1. Evaluate:

(i) (+15) + (-11) - (+5) - (-7)

= 15 - 11 - 5 + 7

= 22 – 16, [Adding all integers with plus (+) sign together and with minus (-) sign respectively together]

= +6 or simply 6.

(ii) (-72) + (-93) - (-85) + (+78)

= -72 -93 + 85 + 78

= -165 + 163, [Adding all integers with plus (+) sign together and with minus (-) sign respectively together]

= - 2

2. Evaluate the expression (-45) + (-32) – (-69) + (87)

Solution:

(-45) + (-32) – (-69) + (87)

= -45 – 32 + 69 +87

Add all the positive terms and add all the negative terms

= -(45 + 32) + (69 + 87)

= -77 + 156

= +79

= 79

3. Simplify: 32 – 13 + 35 + 18 - 60

Solution:

32 – 13 + 35 + 18 – 60

Add all the positive terms and add all the negative terms

= (32 + 35 + 18) – (13 + 60)

= 85 – 73

= +12 or simply 12

4. The sum of two integers is -17. If one of them is -7, find the other.

Solution:

Other integer = Sum of two integers - the given integer

= (-17) – (-7)

= -17 + 7

= -10

Therefore the other number is -10.

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