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.

StatementValueObservation
17 + 23= 40Result is an integer
(–10) + 3= –7Result is an integer
19 + (– 25)= – 6Result is an integer
27 + (– 27)= 0Result is an integer
(– 20) + 0= – 20Result is an integer
(– 35) + (– 10)= – 45Result 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 1Statement 2Inference
3 × (– 5) = –15(– 5) × 3 = –153 × (– 5) = (– 5) × 3
(–30) × 12 = – 36012 × (–30) = – 360(–30) × 12 = 12 × (–30)
(–15) × (–10) = 150(–10) × (–15) = 150(–15) × (–10) = (–10) × (–15)
(–17) × 0 = 00 × (–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.

StatementValueObservation
3 × 0 = 0Result is zero
(–3) × 0 = 0Result is zero
0 × (– 4) = 0Result is zero
– 5 × 0 = 0Result is zero
0 × 0 = 0Result 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.

StatementValueObservation
3 × 1 = 31 is the additive identity
(–3) × 1 = –31 is the additive identity
1 × (– 4) = – 41 is the additive identity
1 × 0 = 01 is the additive identity
0 × 1 = 01 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 1Operation 2Observation
[(–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 1Operation 2Observation
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)]

6 comments

  1. The post become most attractive due to additional exercise…or skill development…super sir!!
    Thank you

  2. Sir , I am from class 8
    Maths was really easy
    I solved all in first attempt
    Plz upload more interesting maths for us , as I have shared the post with my friends

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