## Multiplication and Division of Integers

Here we will learn the methods for multiplication and division of integers. Multiplication and Division of integers can be explained through following three cases.
(i) Multiplication and Division of two or more positive integers.
(ii) Multiplication and Division of two or more negative integers.
(iii) Multiplication and Division of positive and negative integers.

# MULTIPLICATION OF INTEGERS

## Multiplication of Positive & Negative Integers

We know that multiplication of whole numbers is repeated addition.
e.g., 5 + 5 + 5 = 5 × 3 = 15

Can we represent multiplication of integers in the same way?

(−5) + (−5) + (−5) = (−5) × 3 = −15

While, multiplying a positive integer and a negative integer, we multiply them as whole numbers and put a minus sign (–) before the product. We thus get a negative integer.
e.g.,
15×(−16) = −240 ,
21×(−32) = −672

a×(−b) = (−a)×b = −(a×b)

## Multiplication of two Negative Integers

(−5) × (−3) = (-5) minus three times = −(−5) + −(−5) + −(−5) = 5 + 5 + 5 = 15
or, (−5) × (−3) = 15

The product of two negative integers is a positive integer. We multiply the two negative integers as whole numbers and put the positive sign before the product.
e.g.,(−15)×(−16) = 240 ,
(−21)×(−32) = 672

(−a)×(−b) = a×b

#### Product of Integers

Same sign = Positive
Different sign = Negative

## Multiplication 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.
e.g.,
(−5)×(−6) = 30,
(−2)×(−3)×(−5) = −30,
(−1)×(−3)×(−4)×(−7) = 84

#### Time to think : [ Multiplication and division of integers ]

(i) The product (–9) × (–5) × (– 6)×(–3) is positive whereas the product
(–9) × ( –5) × 6 × (–3) is negative. Why?

(ii) What will be the sign of the product if we multiply together:
(a) 8 negative integers and 3 positive integers?
(b) 5 negative integers and 4 positive integers?
(c) (–1), twelve times?
(d) (–1), 2m times, m is a natural number?

(i) First expression has four (even) negative integers, thus its product is positive whereas second expression has three (odd) negative integers and so its product is negative.

(ii) (a)Positive (b) Negative (c)Positive (d)Positive,  since m is a natural number, 2m will be even.

Multiplication and division of integers

# DIVISION OF INTEGERS

We know that division is the inverse operation of multiplication.

Since 3 × 5 = 15, so 15 ÷ 5 = 3 and 15 ÷ 3 = 5
Similarly, 4 × 3 = 12 gives 12 ÷ 4 = 3 and 12 ÷ 3 = 4

We can say for each multiplication statement of numbers there are two division statements,

 Multiplication Statement Corresponding Division Statements (−8)×4 = −32 (−32)÷4 = −8,   (−32)÷(−8) = 4 5×(−9) = −45 (−45)÷(−9) = 5,   (−45)÷5 = −9 (−10)×(−5) = 50 50÷(−5) = −10,   50÷(−10) = −5

By observing the pattern we can say that,

• (−32) ÷ 4 = −8,
• (−45) ÷ 5 = −9,
• 50 ÷ (−5) = −10,
• 50 ÷ (−10) = −5
• When we divide a negative integer by a positive integer or divide a positive integer by a negative integer, we divide them as whole numbers and then put a minus sign (–) before the quotient. We, thus, get a negative integer.

(−a) ÷ b = a ÷ (−b) = −(a ÷ b)

where b≠0

•  −32 ÷ (−8) = 4,
• (−45) ÷ (−9) = 5
• When we divide a negative integer by a negative integer, we first divide them as whole numbers and then put a positive sign (+). That is, we get a positive integer.

(-a) ÷ (-b) = a ÷ b

where b≠0

#### Division of Integers

Same sign = Positive

Different sign = Negative

# PROPERTIES OF INTEGERS

## CLOSURE PROPERTY

#### Is the result also an integer ?

 ADDITION closed a + b is an integer, for all integers a & b. SUBTRACTION closed a − b is an integer, for all integers a & b. MULTIPLICATION closed a × b is an integer, for all integers a & b. DIVISION not closed 1÷2 = 1/2, 2÷0 = not defined, etc.

## COMMUTATIVE PROPERTY

#### Is result obtained after reversing the order is same? / Are (a?b) and (b?a) same ?

 ADDITION commutative   a + b = b + a 2+5 =5+2 SUBTRACTION not commutative 2−5 ≠5−2 MULTIPLICATION commutative   a × b = b × a 2×5 =5×2 DIVISION not commutative 4÷2 ≠2÷4

## ASSOCIATIVE PROPERTY

#### Is the result obtained after regrouping same ? / Are (a?b)?c and a?(b?c) same?

 ADDITION associative(a+b) + c = a + (b+c) (2+5)+3 =2+(5+3) SUBTRACTION not associative (2−5)−3 ≠2−(5−3) MULTIPLICATION associative   (a×b) × c = a × (b×c) (2×5)×3 =2×(5×3) DIVISION not associative (8÷4)÷2 ≠8÷(4÷2)

## DISTRIBUTIVE PROPERTY

Multiplying an integer with sum of two integers will give same result when the integer is first multiplied with each addend & then adding the products.

e.g.,

(−2)×(4+5) = (−2)×9 = −18,

(−2×4)+(−2×5) = (−8)+(−10) = −18

For all integers a, b and c,

a × (b + c) = (a × b) + (a × c)

a × (b − c) = (a × b) − (a × c)

 ⏪ Integers Fractions ⏩

e.g.,

(−2)×(4+5) = (−2)×9 = −18,

(−2×4)+(−2×5) = (−8)+(−10) = −18

For all integers a, b and c,

a × (b + c) = (a × b) + (a × c)

a × (b − c) = (a × b) − (a × c)

 ⏪ Integers Fractions ⏩