Multiplication & Division 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) + −(−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 with
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 :

(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.

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 with
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 : 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 : Are results with different grouping 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 a integer with sum of two integers will give same result when the number 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
	Fraction ➤