Page 16 - Start Up Mathematics_7

• A positive integer multiplied by another positive integer gives a positive integer.
• A positive integer multiplied by a negative integer gives a negative integer.
• A negative integer multiplied by another negative integer gives a positive integer.

Properties of Multiplication of Integers
Closure

The product of any two integers is again an integer.
For integers –2 and 4, we have –2 × 4 = –8 which is again an integer.
Commutative

If a and b are any two integers, then a × b = b × a.
For integers 3 and –7, we have 3 × (–7) = –21 = –7 × 3.
Associative

If a, b and c are any three integers, then (a × b) × c = a × (b × c).
For integers 2, –3 and –4, we have {2 × (–3)} × (–4) = –6 × (–4) = 24
and 2 × {–3 × (–4)} = 2 × 12 = 24.

Hence, {2 × (–3)} × (–4) = 2 × {–3 × (–4)}.
Distributive

If a, b and c are any three integers, then a × (b + c) = (a × b) + (a × c).
For integers –7, –3 and 2, we have –7 × (–3 + 2) = –7 × (–1) = 7
and –7 × (–3) + (–7) × 2 = 21 – 14 = 7.
Hence, –7 × (–3 + 2) = –7 × (–3) + (–7) × 2.

Existence of multiplicative identity
If a is any integer, then a × 1 = a = 1 × a. The number 1 is known as the multiplicative identity
of a.
For integer 4, we have 4 × 1 = 4 = 1 × 4 and for integer –3, we have –3 × 1 = –3 = 1 × –3.

Hence, 1 is the multiplicative identity for integers.
Property of zero

For any integer a, we have (a × 0) = 0 = (0 × a).
For integer –3, we have –3 × 0 = 0 = 0 × –3 and for integer 4, we have 4 × 0 = 0 = 0 × 4.
Example 13: Find each of the following products:

(a) (–1) × 227 (b) (–416) × (–1) (c) (–13) × 0 × (–18)
(d) (–10) × (–5) × (–4) (e) (–3) × (–5) × (–2) × (–1)

Solution: (a) (–1) × 227 = –227
(b) (–416) × (–1) = 416

(c) –13 × 0 × (–18) = (–13 × 0) × (–18) = 0 × –18 = 0
(d) (–10) × (–5) × (–4) = {(–10) × (–5)} × (–4) = 50 × (–4) = –200

(e) (–3) × (–5) × (–2) × (–1) = {(–3) × (–5)} × {(–2) × (–1)} = 15 × 2 = 30

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