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Multiplicative Identity Let us observe the following products:
(a) 3 × 1 = 3 (b) (–9) × 1 = –9 (c) 1 × (–4) = –4
It is clear from the above products that if any integer is multiplied by one
or one is multiplied by an integer, the product will be the integer itself.
If a is an integer, then a × 1 = a = 1 × a
Thus, 1 is called the multiplicative identity.
Multiplication by Zero Let us observe the following products:
(a) 5 × 0 = 0 (b) (–9) × 0 = 0 (c) 0 × (–4) = 0

It is clear from the above products that if an integer is multiplied by
zero or zero is multiplied by an integer, the product will be zero. If a is
an integer, then a × 0 = 0 = 0 × a
Hence, the product of an integer and zero gives the result 0.

Distributive Property Let us take three integers (–2), (–3) and (–4).
Then, (–2) × [(–3) + (–4)] = (–2) × (–7) = 14

Or [(–2) × (–3)] + [(–2) × (–4)] = 6 + 8 = 14
\ (–2) × [(–3) + (–4)] = [(–2) × (–3)] + [(–2) × (–4)] = 14

Hence, if a, b, and c are any three integers, then a × (b + c) = (a × b) + (a × c).
It is called the distributive property of multiplication of integers over
addition.
Now, let us take another three integers (–5), (–6) and (–7).

Then, (–5) × [(–6) – (–7)] = (–5) × 1 = –5 Or (–5) × (–6) – (–5) × (–7) = 30 – 35 = –5
\ (–5) × [(–6) – (–7)] = [(–5) × (–6)] – [(–5) × (–7)] = –5

Hence, if a, b and c are any three integers, then a × (b – c) = (a × b) – (a × c).
It is called the distributive property of multiplication of integers over
subtraction.

Multiplicative Inverse
FACTS
Roma: Hey! Ruhi, do you know? Imran has told me that when we • Multiplicative inverse of 0
multiply an integer with its reciprocal, we get 1 as a result. is not defined.
• Multiplicative inverse of 1
Ruhi: Yes. It is called the inverse property of multiplication. Also, an 1
integer and its reciprocal are called multiplicative inverse of each is 1 as 1 × = 1.
1
other. For example: • Multiplicative inverse of –1
1 1
(a) (–2) × = 1 is –1 as (–1) × − ( 1) = 1.
− ( 2)

 1 
−

  Q is the multiplicative inverse (reciprocal) of (2)
 − ( 2) 
1  1 
(b) 8 × = 1 Q is the multiplicative inverse (reciprocal) of 8. 

8  8 

15 Integers