# 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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