Multiplication of Integers

Multiplication of integers on Number Line

Multiplying 2 positive integers

2 x 3

To represent this on the number line, we start at 0 and put 2 groups of 3 of the number line.

We end up at 6. So, the product is 6.

Therefore, 2 x 3 = 6 (Positive × Positive = Positive)

Multiplying a positive integer and a negative integer

2 x (-2)

To represent this on the number line, just start at 0 and put 2 groups of -2 of the number line.

We end up at (-4). So, the product is (-4).

Therefore, 2 x (-2) = 4 [(Positive × Negative) or (Negative x Positive) = Negative]

Product of a positive integer and a negative integer without using number line

Steps

     1.   Multiply them as whole numbers.

     2.   Put a minus sign (–) before the product.

Example: 12 × (–15)

Solution: First find the product of whole numbers i.e. 12 x 15 = 180

Now, put a minus sign (–) before the product = -180

Product of two negative integers without using number line

Steps

     1.   Multiply the two negative integers as whole numbers.

    2.   Put the positive sign before the product because product of two negative integers is a positive integer.

In general, for any two positive integers a and b, (– a) × (– b) = a × b

Example: (-9) × (–11)

Solution: First multiply the two negative integers as whole numbers i.e. 9 x 11= 99

Now, put a minus sign (+) before the product = +99

Product of three or more Negative Integers

If the number of negative integers in a product is even, then the product is a positive integer; if the number of negative integers in a product is odd, then the product is a negative integer.

This means,

(a) The product of two negative integers is a positive integer.

(b) The product of three negative integers is a negative integer.

(c) Product of four negative integers is a positive integer.

Let’s understand this with following examples.

Example: (a) (– 4) × (–3)

(b) (– 4) × (–3) × (–2)

(c) (– 4) × (–3) × (–2) × (–1)

Solution: (a) (– 4) × (–3) = 12 (number of negative integers in a product is even, so the product is a positive integer)

(b) (– 4) × (–3) × (–2) = [(– 4) × (–3)] × (–2) = 12 × (–2) = – 24 (the number of negative integers in a product is odd, so the product is a negative integer)

(c) (– 4) × (–3) × (–2) × (–1) = [(– 4) × (–3) × (–2)] × (–1) = (–24) × (–1) (number of negative integers in a product is even, so the product is a positive integer)

Properties of Multiplication

Closure under Multiplication

Closure property under multiplication states the product of two integers will always be an integer.

Let’s consider the following pairs of integers.

(-5) x (-6) = 30 (Result is an integer)

15 x (-10) = -150 (Result is an integer)

(-7) x (-8) = 56 (Result is an integer)

Since multiplication of integers gives integers, we say integers are closed under multiplication.

In general, a × b is an integer, for all integers a and b.

Commutativity for Multiplication

Commutative property of multiplication states that swapping of terms will not change the product.

Let’s consider the following examples.

(-8) x (-12) = (-12) x (-8)

(-11) x 100 = 100 x (-11)

(-19) x 0 = 0 x (-19)

So, we can say that multiplication is commutative for integers.

In general, for any two integers a and b, we can say a x b = b x a

Associativity for Multiplication

Associative property of multiplication states that the way of grouping of numbers will not change the result.   

Let’s consider the following example.

(–6) x [(–4) x (–3)] and [(–6) x (–4)] x (–3)

In the first case (–4) and (–3) are grouped together and in the second (–6) and (–4) are grouped together.

(–6) x [(–4) x (–3)] = (-6) x 12 = (-72)

[(–6) x (–4)] x (–3) = 24 x (-3) = (-72)

In both the cases, we get –72.

So, Multiplication is associative for integers.

In general, for any three integers a, b and c (a × b) × c = a × (b × c)

Distributive Property

Distributive property of multiplication explains the distributing ability of an operation over another mathematical operation within a bracket. It can be either distributive property of multiplication over addition or distributive property of multiplication over subtraction.  

Let’s consider the following example.

(–2) × (3 + 5) = –2 × 8 = –16 and;

[(–2) × 3] + [(–2) × 5] = (– 6) + (–10) = –16

So, (–2) × (3 + 5) = [(–2) × 3] + [(–2) × 5]

In general, for any integers a, b and c, a × (b + c) = a × b + a × c

Multiplication by Zero

This property of multiplication states that the product of any integer (positive or negative) and zero is zero.

Let’s consider the following examples.

(-98) x 0 = 0

0 x 67 = 0

So, we can say that multiplication of any integer and zero gives zero.

In general, for any integer a, a × 0 = 0 × a = 0

Multiplicative Identity

Multiplicative identity property states that when we multiply one to any integer, we will get the integer itself as the product.  

Let’s observe the following examples:

(– 16) x 1 = – 16

1 x (– 81) = – 81

The above examples show that 1 is the multiplicative identity for integers also.

In general, for any integer a we have, a × 1 = 1 × a = a

Making Multiplication Easier

Example: Find 15 × 17

Solution: 15 × 17 can be written as 15 × (10 + 7).

= 15 × (10 + 7) = 15 × 10 + 15 × 7 = 150 + 105 = 255

Example: Find (-22) × 98

Solution: (-22) x 98 can be written as (-22) x (100 - 2)

= [(-22) × 100] – [(–22) × 2] = (–2200) – (– 44)

= –2200 + 44

= -2156

Example: Find (–16) × (–10) × 8

Solution: [(–16) × (–10)] × 8

           = 160 x 8

           = 1280

Example: Verify (–40) × [11 + (–3)] = [(–40) × 11] + [(–40) × (–3)]

Solution: L.H.S = (–40) × [11 + (–3)]

                      = (-40) x 8

                      = -320

              R.H.S = [(–40) × 11] + [(–40) × (–3)]

                       = -440 + 120

                       = -320

L.H.S = R.H.S

Hence verified

Example: [(– 10) × 8] + (– 6) is equal to (a) 100 (b) –90 (c) – 86 (d) 86

Solution: [(– 10) × 8] + (– 6)

            = (-80) + (-6)

            = -86

Correct answer is (c) = -86

Example: (– 15) × 40 = – 40 × _______.

Solution: (– 15) × 40 = – 40 × 15.