Page 15 - Maths Skills - 7

Integers 13

3. Associative Property: The product of any three integers remains the same, regardless of the way in which the
integers are grouped.

In other words, if a, b and c are any three integers, then (a × b) × c = a × (b × c)
For example;
(i) [2 × (– 3)] × 4 = – 6 × 4 = – 24 (ii) [(– 3) × (– 4)] × (– 3) = 12 × (– 3) = – 36

and 2 × [(– 3) × 4] = 2 × (– 12) = – 24 and (– 3) × [(– 4) × (– 3)] = (– 3) × 12 = – 36
\ [2 × (– 3)] × 4 = 2 × [(– 3) × 4] \ [(– 3) × (– 4)] × (– 3) = (– 3) × [(– 4) × (– 3)]

4. Multiplicative Identity Property: The product of an integer and 1 is the integer itself. In other words, if a is
any integer, then a × 1 = a = 1 × a

For example; (i) (– 3) × 1 = – 3 (ii) 1 × (– 3) = – 3 (iii) 2 × 1 = 2 (iv) 1 × 2 = 2
Hence, 1 is called the multiplicative identity or identity element in multiplication.

5. Zero Property: The product of an integer and 0 is zero.
In other words, if a is an integer, then a × 0 = 0 × a = 0

For example; (i) 2 × 0 = 0 (ii) 0 × 2 = 0 (iii) (– 1) × 0 = 0 (iv) 0 × (– 1) = 0
6. Distributive Property of Multiplication over Addition and Subtraction: The multiplication of integers
distributes over addition as well as subtraction.
In other words if a, b and c are any three integers, then a × (b + c) = a × b + a × c, and a × (b – c) = a × b – a × c
For example;
(i) 5 × [(– 3) + 1] = 5 × (– 3) + 5 × 1 (ii) 4 × [(– 6) – (2)] = 4 × [(– 6) + (– 2)]

or 5 × (– 2) = – 15 + 5 or 4 × (– 6 – 2) = 4 × ( – 6) + 4 × (– 2)
or – 10 = – 10 or 4 × (– 8) = – 24 – 8
or – 32 = – 32
7. Multiplication by (–1): The product of an integer and –1 is the opposite or inverse of the integer itself.

In other words, if a is any integer, then a × (– 1) = – a
For example; (i) 2 × (– 1) = – 2 (ii) (– 3) × (– 1) = 3

Let’s Attempt

Example 1: Multiply the following.
(i) – 5 and – 9 (ii) 6 and – 7

Solution: (i) The integers are of same signs, so the sign of the product is positive.
\ ( – 5) × (– 9) = 45
(ii) The integers are of unlike signs, so the sign of the product is negative.

\ 6 × (– 7) = – 42 Fact-o-meter

Example 2: Evaluate: (– 5) × 2 × (– 1) × (– 3) × 4 The product of two or more integers
Solution: We have, [(– 5) × 2] × [(– 1) × (– 3)] × 4 is positive or negative according
= – 10 × 3 × 4 to the number of negative integers
= (– 10 × 3) × 4 = – 30 × 4 = – 120 being even or odd respectively.

Since the number of negative integers is odd (3), therefore the product is negative.