For any two integers a, b ; a×b is also an integer. For any two integers a, b ; a×b = b×a.

**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.

Tags : Number System | Term 1 Chapter 1 | 7th Maths , 7th Maths : Term 1 Unit 1 : Number System

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