Page 13 - Start Up Mathematics_7

Properties of Addition of Integers
Closure
The sum of any two integers is again an integer.

Since 3 and 5 are integers, ∴ 3 + 5 = 8 is also an integer.
Since 2 and –4 are integers, ∴ 2 + (–4) = 2 – 4 = –2 is again an integer.
Commutative

If a and b are any two integers, then a + b = b + a.
Since 3 and 6 are integers, ∴ 3 + 6 = 9 = 6 + 3.
Since –3 and 5 are integers, ∴ (–3) + 5 = 2 = 5 + (–3).
Associative

If a, b and c are any three integers, then (a + b) + c = a + (b + c).
Since 2, 3 and 5 are integers, ∴ (2 + 3) + 5 = 10 = 2 + (3 + 5).

Since 2, –3 and 5 are integers, ∴ {2 + (–3)} + 5 = 4 = 2 + (–3 + 5).
Existence of additive identity
If a is any integer, then a + 0 = a = 0 + a. We say 0 is an additive identity for integers.
Since 5 is an integer, ∴ 5 + 0 = 5 = 0 + 5.

Since –5 is an integer, ∴ (–5) + 0 = –5 = 0 + (–5).
Hence 0 is the additive identity for integers.

Existence of additive inverse
If a is any integer, then a + (–a) = 0 = (–a) + a. We say –a is an additive inverse of a.
Since 6 + (–6) = 0 = (–6) + 6, ∴ additive inverse of 6 is –6.

Properties of Subtraction of Integers
Closure
If we subtract any two integers the result is again an integer.

Since 5 and 3 are integers, ∴ 5 – 3 = 2 and 3 – 5 = –2 are also integers.
Since –2 and 4 are integers, ∴ –2 – 4 = –6 is an integer. Also, 4 – (–2) = 6 is also an integer.
Not commutative

If a and b are any two integers, then a – b ≠ b – a, i.e., commutative property does not hold for
subtraction of integers.
For integers –3 and 5 we have –3 – (+5) = –(3 + 5) = –8 and 5 – (–3) = 5 + 3 = 8,
∴ –3 – (+5) ≠ 5 – (–3)
Not associative

If a, b and c are integers, then (a – b) – c ≠ a – (b – c). Associative property does not hold good
for subtraction of integers.
For integers –1, +6 and 7 we have (–1 – 6) – 7 = –7 – 7 = –14
and –1 – (6 – 7) = –1 – (–1) = –1 + 1 = 0

∴ (–1 – 6) – 7 ≠ –1 – (6 – 7)

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