Page 18 - ICSE Math 7

Properties of subtraction of integers

• Closure property
If a and b are any two integers, then a – b is also an integer.
For example, 3 – (–7) = 3 + 7 = 10 which is an integer.

• Commutative property
If a and b are any two integers, then a – b ≠ b – a. For example, 11 – 7 = 4 but 7 – 11 = –4
\ 11 – 7 ≠ 7 – 11

Thus, integers are not commutative under subtraction.
• Associative property
If a, b and c are any three integers, then (a – b) – c ≠ a – (b – c).

For example, (9 – 7) – 3 = 2 – 3 = –1 but 9 – (7 – 3) = 9 – 4 = 5
\ (9 – 7) – 3 ≠ 9 – (7 – 3)
Thus, integers are not associative under subtraction. Try These
• Existence of identity
1. Write True or False.
Let a be any integer other than zero, then a – 0 = a but 0 – a = –a. (a) 6 – (–11) = 17
⇒ a – 0 ≠ 0 – a (b) (–15) + (–3) = 18
Thus, identity element for subtraction of integer does not exist. (c) 3 + (–8) + 0 = –5
2. Find x, if 7 – 2 = x – 2.
• Existence of inverse
As there is no identity element for subtraction of integers, hence inverse does not exist.
• Cancellation law

If a, b and c are any three integers, then a – c = b – c ⇒ a = b.
For example, x – (–3) = –22 – (–3) ⇒ x = –22

Example 5: Evaluate the following.
(a) 64 + 121 – (–75) (b) –48 – (+175) – (–26)
(c) –85 + 670 – (–19) + (–21) (d) 21 – (–167) + 68 + (–112) – 24
Solution: (a) 64 + 121 – (–75) = 64 + 121 + 75 = 260

(b) –48 – (+175) – (–26) = –48 – 175 + 26 = –(48 + 175) + 26
= –223 + 26 = –197
(c) –85 + 670 – (–19) + (–21) = –85 + 670 + 19 – 21 = –(85 + 21) + (670 + 19)

= –106 + 689 = 583
(d) 21 – (–167) + 68 + (–112) – 24 = 21 + 167 + 68 – 112 – 24
= (21 + 167 + 68) – (112 + 24)

= 256 – 136 = 120

Example 6: The sum of two integers is –265. If one of the integers is 161, find the other integer.
Solution: Let the required integer be x.
So, 161 + x = –265

⇒ x = –265 – 161 = –426

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