In this blog, we will learn about the "Properties of Multiplication of Integers"

Properties of Multiplication of Integers

We will learn about various properties of integers such as closure property, commutative property, and others.

1. Closure property for multiplication

Let us first revise the closure property for multiplication on whole numbers.

Here, we have two numbers, four and seven belonging to the same group of whole numbers.

4 x 7 = 28

We get twenty-eight by multiplying four and seven; here, twenty-eight is also a whole number. 

So we can say that when we multiply any two whole numbers the result is also a whole number.

This is the closure property of multiplication

Let's see whether the closure property is true for the multiplication of integers.

Consider two integers -5 and 4.

On multiplying (-5) x 4, we get -20, which is also an integer.

Similarly, the multiplication of any two integers is always an integer. 

Thus we can say that integers are also closed under multiplication. In general, a into b is an integer, for all integers a and b.

Let's learn more about the other properties of multiplication of integers

2. Commutative property of multiplication

Now let us move on to the commutative property of multiplication.

We know that multiplication is commutative for whole numbers. 

For example, 4 x 8 =32

On changing the order we get 8 x 4 = 32

So we observe that on changing the order we get the same result.

Can we say multiplication is also commutative for integers?

For multiplication let us have,

5 x (-4) = -20

On changing the order, 

(-4) x 5 = -20

Here also changing the order did not change the result. In both cases, we have the same result i.e., minus 20. This shows that integers are commutative under multiplication. In general, for any two integers a and b, a x b = b x a

3. Let us see multiplication by zero.

We know that any whole number when multiplied by zero gives zero.

For example, 4 x 0 = 0, and 0 x 4 = 0

Let us now check for integers:

• (−3)×0=0
• 0÷(−4)=0
• −5×0=0
• 0÷(−6)=0

These examples show that the product of an integer and zero is zero. In general, for any integer a x 0 = (0 x a) = 0

Let us now see Multiplicative Identity. We know that 1 is the multiplicative identity for whole numbers.

4 x 1 = 4

1 x 4 = 4

Let us check that 1 is the multiplicative identity for integers as well. 

Observe the following products of integers with 1.

• (-3) x 1 = -3
• 1 x (-4) = -4
• (-5) x 1 = -5

These examples show that 1 is the multiplicative identity for integers.

In general, for any integer a we have, (a x 1) = (1 x a) = a

4. Let us now check Associativity for Multiplication.

Consider 3 integers, -2, - and 5.

(-2) x [(-3) x 5]      

= (-2) x (-15) = 30

By multiplying 2^nd and 3^rd and then multiplying the result with 1^st, we get 30

On changing the order, first multiplying 1^st and 2^nd and then multiplying the result by 3^rd, we also get 30.

[(-2) x (-3)] x 5

= 6 x 5 = 30

We see that on changing the order there is no change in the result. 

This implies that integers are associative under multiplication. In general, for any three integers, a, b, and c, (a x b) x c = a x (b x c)

5. Let us now learn about Distributive Property.

Let us check the distributivity of multiplication over addition for integers.

4 x [(−3)+(−7)]=4 x (−10)=−40

Splitting  a product as a sum of two different products i.e.,

[4 x (-3)] + [4 x (-7)] = (-12) + (-28) = -40

So we can say the distributivity of multiplication over addition for integers is true.

In general, for any integers a, b and c, [a x (b + c)] = [(a x b) + (a x c)

Let us now check the distributivity of multiplication over subtraction for integers.

Consider three integers, 5, -3, and -4.

5 x [(-3) x (-4)] = 5 x 1 = 5

Splitting a product as a sum of two different products i.e.,

[5 x (-3)] x [5 x (-4)] = (-15) x (-20) = 5

In both cases, we find that the answer is 5.

Hence, the distributive property of multiplication over subtraction is true for integers.
In general, for any integers a, b and c, [a x(b -c) = (a x b) - (a x c)

Read More: Probability Class 9: Definition and Examples - Maths Note