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.
6th Grade Page
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