Integers

Properties of multiplication of integers

Closure under Multiplication

We have learnt that product of two whole numbers is again a whole number, which is known as the closure property for multiplication of the whole numbers. Since product of integers gives integers too, we say integers are closed under multiplication.

In other words, for any two integers a and b, a × b is an integer.

Look at the following table.

Statement	Value	Observation
17 + 23	= 40	Result is an integer
(–10) + 3	= –7	Result is an integer
19 + (– 25)	= – 6	Result is an integer
27 + (– 27)	= 0	Result is an integer
(– 20) + 0	= – 20	Result is an integer
(– 35) + (– 10)	= – 45	Result is an integer

Commutativity of Multiplication

We know that the whole numbers can be multiplied in any order, so that multiplication is commutative for whole numbers. In the same way, we say that multiplication is commutative for integers.

In general, for any two integers a and b, we can say a × b = b × a.

Look at the following table.

Statement 1	Statement 2	Inference
3 × (– 5) = –15	(– 5) × 3 = –15	3 × (– 5) = (– 5) × 3
(–30) × 12 = – 360	12 × (–30) = – 360	(–30) × 12 = 12 × (–30)
(–15) × (–10) = 150	(–10) × (–15) = 150	(–15) × (–10) = (–10) × (–15)
(–17) × 0 = 0	0 × (–17) = 0	(–17) × 0 = 0 × (–17)

Multiplication by Zero

We know that any whole number when multiplied by zero gives zero. Also, the product of a negative integer and zero is zero.

In general, for any integer a, we have a × 0 = 0 × a = 0.

Look at the following table.

Statement	Value	Observation
3 × 0	= 0	Result is zero
(–3) × 0	= 0	Result is zero
0 × (– 4)	= 0	Result is zero
– 5 × 0	= 0	Result is zero
0 × 0	= 0	Result is zero

Multiplicative Identity

We know that 1 is the multiplicative identity for whole numbers. Likewise, 1 is the multiplicative identity for integers also.

In general, for any integer a we have, a × 1 = 1 × a = a.

We get additive inverse of an integer a when we multiply (–1) to a, i.e., a × (–1) = (–1) × a = – a.

Look at the following table.

Statement	Value	Observation
3 × 1	= 3	1 is the additive identity
(–3) × 1	= –3	1 is the additive identity
1 × (– 4)	= – 4	1 is the additive identity
1 × 0	= 0	1 is the additive identity
0 × 1	= 0	1 is the additive identity

Associativity for Multiplication

Like whole numbers, the product of three integers does not depend upon the grouping of integers and this is called the associative property for multiplication of integers.

In general, for any three integers a, b and c, we can say (a × b) × c = a × (b × c).

Look at the following table.

Operation 1	Operation 2	Observation
[(–3) × (–2)] × 5 = 6 × 5 = 30	(–3) × [(–2) × 5] = (–3) × (–10) = 30	[(–3) × (–2)] × 5 = (–3) × [(–2) × 5]
[7 × (– 6)] × 4 = (–42) × 4 = –168	7 × [(– 6) × 4] = 7 × (–24) = –168	[7 × (– 6)] × 4 = 7 × [(– 6) × 4]

Distributive Property

We know that the distributivity of multiplication over addition is true for whole numbers. Likewise, it is true for integers also.

In general, for any integers a, b and c, we have

 a × (b + c) = a × b + a × c and a × (b – c) = a × b – a × c.

Look at the following table.

Operation 1	Operation 2	Observation
15 × (10 + 2) = 15 × 12 = 180	(15 × 10) + (15 × 2) = 150 + 30 = 180	15 × (10 + 2) = (15 × 10) + (15 × 2)
(–10) × [(–2) + (–1)] = (–10) × (–3) = 30	[(–10) × (–2)] + [(–10) × (–1)] = 20 + 10 = 30	(–10) × [(–2) + (–1)] = [(–10) × (–2)] + [(–10) × (–1)]
(–4) × (3 + 5) = –4 × 8 = –32	[(–4) × 3] + [(–4) × 5] = (–12) + (–20) = –32	(–4) × (3 + 5) = [(–4) × 3] + [(–4) × 5]
(– 4) × [(–2) + 7] = (– 4) × 5 = –20	[(– 4) × (–2)] + [(– 4) × 7] = 8 + (–28) = –20	(– 4) × [(–2) + 7] = [(– 4) × (–2)] + [(– 4) × 7]
6 × (3 – 8) = 6 × (–5) = –30	6 × 3 – 6 × 8 = 18 – 48 = –30	6 × (3 – 8) = 6 × 3 – 6 × 8
(–2) × [(– 4) – (– 6)] = (–2) × 2 = –4	[(–2) × (– 4)] – [ (–2) × (– 6)] = 8 – 12 = –4	( –2) × [(– 4) – (– 6)] = [(–2) × (– 4)] – [ (–2) × (– 6)]