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 and division of 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?

Answer :

(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

multiplication and 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 StatementCorresponding Division Statements
(−8)×4 = −32(−32)÷4 = −8,   (−32)÷(−8) = 4
5×(−9) = −45(−45)÷(−9) = 5,   (−45)÷5 = −9
(−10)×(−5) = 5050÷(−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 ?

ADDITIONcloseda + b is an integer, for all integers a & b.
SUBTRACTIONcloseda − b is an integer, for all integers a & b.
MULTIPLICATIONcloseda × b is an integer, for all integers a & b.
DIVISIONnot closed1÷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 ?

ADDITIONcommutative   a + b = b + a 2+5 =5+2
SUBTRACTIONnot commutative2−5 ≠5−2
MULTIPLICATIONcommutative   a × b = b × a 2×5 =5×2
DIVISIONnot commutative4÷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)

⏪ IntegersFractions ⏩

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)

⏪ IntegersFractions ⏩

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