INTEGERS 171.5.5 Associativity for Multiplication
Consider –3, –2 and 5.
Look at [(–3) × (–2)] × 5 and (–3) × [(–2) × 5].
In the first case (–3) and (–2) are grouped together and in the second (–2) and 5 are
grouped together.
We see that [(–3) × (–2)] × 5 = 6 × 5 = 30
and (–3) × [(–2) × 5] = (–3) × (–10) = 30
So, we get the same answer in both the cases.
Thus, [(–3) × (–2)] × 5 = (–3) × [(–2) × 5]
Look at this and complete the products:
[(7) × (– 6)] × 4 = × 4 =
7 × [(– 6) × 4] = 7 × =
Is [7 × (– 6)] × (– 4) = 7 × [(– 6) × (– 4)]?
Does the grouping of integers affect the product of integers? No.
In general, for any three integers a,b and c
(a × b) × c=a × (b×c)Take any five values for a,b and c each and verify this property.
Thus, 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.
1.5.6 Distributive Property
We know
16 × (10 + 2) = (16 × 10) + (16 × 2) [Distributivity of multiplication over addition]
Let us check if this is true for integers also.
Observe the following:
(a) (–2) × (3 + 5) = –2 × 8 = –
and [(–2) × 3] + [(–2) × 5] = (– 6) + (–10) = –
So, (–2) × (3 + 5) = [(–2) × 3] + [(–2) × 8]
(b) (– 4) × [(–2) + 7] = (– 4) × 5 = –
and [(– 4) × (–2)] + [(– 4) × 7] = 8 + (–28) = –
So, (– 4) × [(–2) + 7] = [(– 4) × (–2)] + [(– 4) × 7]
(c) (– 8) × [(–2) + (–1)] = (– 8) × (–3) = 24
and [(– 8) × (–2)] + [(– 8) × (–1)] = 16 + 8 = 24
So, (– 8) × [(–2) + (–1)] = [(– 8) × (–2)] + [(– 8) × (–1)]