Properties of Multiplication of Integers

Properties of Multiplication of Integers

Recall that multiplication of whole numbers had the closure property. If we test this property for integers we find ( −7) × ( −2) = +14 , ( −6) × 5 = −30 , 4 × ( −9) = −36 . Thus, product of any two integers is always an integer. "Closure property" holds for integers.

Therefore, for any two integers a, b ; a×b is also an integer.

Consider the examples, 21 × ( −5) = −105 is the same as ( −5) × 21 = −105 . The product (−9) × ( −8) = +72 , ( −8) × ( −9) = 72 . In both the cases the product is the same integer. So, changing the order of multiplication does not change the value of the product. Hence, we conclude that multiplication of integers are "commutative".

Therefore, for any two integers a, b ; a×b = b×a.

To verify that multiplication of integers is associative, let us check whether ( −5) × [( −9) × ( −12)] and [( −5) × ( −9)] × ( −12) are equal.

 ( −5) × [( −9) × ( −12)] = ( −540) and [( −5) × ( −9)] × ( −12) = ( −540).

Hence, "associative" property is true for mutliplication of integers.

Therefore, for any three integers a, b, c ; (a×b)×c = a×(b×c).

Just as zero added to a integer leaves it unchanged, integer 1 multiplied with any integer leaves the integer unaltered. For example, 57 × 1 = 57 , and 1 × ( −62) = −62 . We say that the integer 1 is the identity for multiplication of integers or "multiplicative identity".

Therefore, for any integer a ; a×1 = 1×a=a.

Try these

1. Find the product and check for equality :

(i) 18 × ( −5) and (–5) × 18

18 × (–5) = –90

(–5) × 18 = –90

18 × (–5) = (–5) × 18

They are equal

(ii) 31 × (–6) and (–6) × 31

31 × (–6) = –186

(–6) × 31 = –186

31 × (–6) = (–6) × 31

They are equal

(iii) 4 × 51 and 51 × 4

4 × 51 = 204

51 × 4 = 204

4 × 51 = 51 × 4

They are equal 

11. Prove the following :

(i) (–20) × (13×4) = [(−20)×13] × 4

(–20) × (13 × 4) = (–20) × 52 = –1040

[(–20) × 13] × 4 = (–260) × 4 = –1040

(–20) × (13 × 4) = [(–20) × 13] × 4

Hence it is proved

(ii) [(−50)×(−2)] ×(−3) = (−50)× [(−2)×(−3)]

[(–50) × (–2)] × (–3) = (100) × (–3) = –300

 (–50) × [(–2) × (–3) = –50 × 6 = –300

[(–50) × (–2)] × (–3) = (–50) × [(–2) × (–3)]

Hence it is proved

(iii) [(−4)×(−3)] ×(−5) = (−4)× [(−3)×(−5)]

[(–4) × (–3)] × (–5) = 12 × (–5) = –60

(–4) × [(–3) × (–5)] = (–4) × [–5] = –60

[(–4) × (–3)] × (–5) = (–4) × [(–3) × (–5)]

Hence it is proved

Note

Consider an example, (–7) × (–6) × (–5) × (–4).

Let us try to do the above multiplication of integer,

(–7) × (–6) × (–5) × (–4) = [(–7) × (–6)] × [(–5) × (–4)]

= (+42) × (+20)

= + 840

From the above example, we see that the product of four negative integers is positive.

What will happen if we multiply odd number of negative integers.

Let us consider another example, (–7) × (–3) × (–2).

Multiplying the above integers, we get

(–7) × (–3) × (–2) = [(–7) × (–3)] × (–2)

= (+21) × (–2)

= – 42

From the above example, we see that the product of three negative integers is negative.

In general, if the negative integers are multiplied even number of times, the product is a positive integer, whereas negative integers are multiplied odd number of times, the product is a negative integer.