**Properties of multiplication :**

We have four binary operations in math. They are

Addition, Subtraction, Multiplication and Division

When we use each of the above binary operations, the result may be different.

It is because of the work done by the binary operation that we have between the two numbers. Knowing the work done by the binary operation is called the property of that operation.

Here, we are going to know the work done by the binary operation "Multiplication".

Let us consider the following example to understand the important property of multiplication.

**Example : **

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.

More clearly,

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

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

**Other properties of multiplication : **

(i) Commutative property

(ii) Multiplication by zero

(iii) Associative property

(iv) Multiplicative identity

(v) Multiplicative inverse

(vi) Distributive property of multiplication

Observe the following :

5 × (– 6) = – 30 and (– 6) × 5 = – 30

Therefore, 5 × (– 6) = (– 6) × 5

Therefore, multiplication is commutative for all real numbers.

In general, for any two real numbers a and b, a × b = b × a.

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

Observe the following:

5 × 0 = 0

– 8 × 0 = 0

In general, for any nonzero real number "a"

a × 0 = 0 × a = 0

Consider the real numbers 2, – 5, 6.

Look at

[2 x (-5)] x 6 = -10 x 6 = -60 and

2 x [(- 5) x 6] = 2 x (-30) = -60

Thus [2 x (-5)] x 6 = 2 x [(- 5) x 6]

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

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

(a × b) × c = a × (b × c)

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

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

**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.

After having gone through the stuff given above, we hope that the students would have understood "Properties of multiplication".

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