PROPERTIES OF MULTIPLICATION

A teacher wants to gift $20 cash prize to one of the students for his performance in the class. 

In case, if the teacher wants to gift the same cash prize $20 to six students, how much does he have to spend ?

To get answer for the above question, we will be writing 20 six times and add them all.

That is, 

20 + 20 + 20 + 20 + 20 + 20

Adding 20 six times can be written as (6x20).

Here, the fact what we have to understand is, the result of adding 20 six times is equal to multiplying 6 and 20.

That is, 

20 + 20 + 20 + 20 + 20 + 20  =  6x20  =  120

Therefore,  to gift $20 cash prize to six students, the teacher has to spend $120. 

Following are the properties of multiplication : 

(i)  Commutative property

(iii)  Associative property

(iv)  Identity property of multiplication

(v)  Inverse property of multiplication

(vi)  Multiplication by zero 

(vi)  Distributive property of multiplication

Commutative Property of Multiplication

Observe the following :

5 x (–6)  =  –30 and (–6) x 5  =  –30

Therefore,

5 x (– 6)  =  (– 6) x 5

Therefore, multiplication is commutative for all real numbers.

In general, for any two real numbers a and b,

a x b  =  b x a

Associative Property of Multiplication

Consider the real numbers 2,  5, 7.

Look at

2 x (5 x 7)  =  2 x 35  =  70

(2 x 5) x 7  =  10 x 7  =  70

Thus

2 x (5 x 7)  =  (2 x 5) x 7

So we can say that real numbers are associative under multiplication.

In general, for any real numbers a, b, c,

a x (b x c)  =  (a x b) x c

Identity Property of Multiplication

Observe the following:

5 x 1  =  5

1 x (- 7)  =  -7

This shows that ‘1’ is the multiplicative identity for real numbers.

In general, for any real number 'a' we have

a x 1  =  1 x a  =  a

Inverse Property of Multiplication

For any real number, say 'a', the multiplicative inverse is its reciprocal.

That is, multiplicative inverse of 'a' is 1/a.

For example, 

The multiplicative inverse of 5 is 1/5

The multiplicative inverse of 3 is 1/3

Note : 

Multiplication of a number and its multiplicative inverse is always 1. 

That is, 

5 x 1/5  =  1

Multiplication by Zero

The product of any nonzero real number with zero is zero.

Observe the following:

5 x 0  =  0

–8 x 0  =  0

In general, for any nonzero real number 'a',

a x 0  =  0 x a  =  0

Distributive Property of Multiplication

Distributive Property of Multiplication Over Addition :

Consider the real numbers 12, 9, 7.

Look at

12 x (9 + 7)  =  12 x 16  =  192

12 x (9 + 7)  =  12 x 9  +  12 x 7  =  108 + 84  =  192

Thus

12 x (9 + 7)  =  (12 x 9) + (12 x 7)

In general, for any real numbers a, b, c.

a x (b + c)  =  (a x b) + (a x c)

Therefore, multiplication is distributive over addition.

Distributive Property of Multiplication Over Subtraction :

Consider the real numbers 12, 9, 7.

Look at

12 x (9 - 7)  =  12 x 2  =  24

12 x (9 - 7)  =  12 x 9  -  12 x 7  =  108 - 84  =  24

Thus

12 x (9 - 7)  =  (12 x 9) - (12 x 7)

In general, for any real numbers a, b, c.

a x (b - c)  =  (a x b) - (a x c)

Therefore, multiplication is distributive over subtraction.

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