The properties of multiplication of integers are discussed with examples. All properties of multiplication of whole numbers also hold for integers.
The multiplication of integers possesses the following properties:
The product of two integers is always an integer.
That is, for any two integers m and n, m x n is an integer.
For example:
(i) 4 × 3 = 12, which is an integer.
(ii) 8 × (-5) = -40, which is an integer.
(iii) (-7) × (-5) = 35, which is an integer.
For any two integer’s m and n, we have
m × n = n × m
That is, multiplication of integers is commutative.
For example:
(i) 7 × (-3) = -(7 × 3) = -21 and (-3) × 7 = -(3 × 7) = -21
Therefore, 7 × (-3) = (-3) × 7
(ii) (-5) × (-8) = 5 × 8 = 40 and (-8) × (-5) = 8 × 5 = 40
Therefore, (-5) × (-8) = (-8) × (-5).
The multiplication of integers is associative, i.e., for any three integers a, b, c, we have
a × ( b × c) = (a × b) × c
For example:
(i) (-3) × {4 × (-5)} = (-3) × (-20) = 3 × 20 = 60
and, {(-3) × 4} × (-5) = (-12) × (-5) = 12 × 5 = 60
Therefore, (- 3) × {4 × (-5)} = {(-3) × 4} × (-5)
(ii) (-2) × {(-3) × (-5)} = (-2) × 15 = -(2 × 15)= -30
and, {(-2) × (-3)} × (-5) = 6 × (-5) = -(6 × 5) = -30
Therefore, (- 2) × {(-3) × (-5)} = {-2) × (-3)} × (-5)
The multiplication of integers is distributive over their addition. That is, for any three integers a, b, c, we have
(i) a × (b + c) =a × b + a × c
(ii) (b + c) × a = b × a + c × a
For example:
(i) (-3) × {(-5) + 2} = (-3) × (-3) = 3 × 3 = 9
and, (-3) × (-5) + (-3) × 2 = (3 × 5 ) -( 3 × 2 ) = 15 - 6 = 9
Therefore, (-3) × {(-5) + 2 } = ( -3) × (-5) + (-3) × 2.
(ii) (-4) × {(-2) + (-3)) = (-4) × (-5) = 4 × 5 = 20
and, (-4) × (-2) + (-4) × (-3) = (4 × 2) + (4 × 3) = 8 + 12 = 20
Therefore, (-4) × {-2) + (-3)} = (-4) × (-2) + (-4) × (-3).
Note: A direct consequence of the distributivity of multiplication over addition is
a × (b - c) =a × b - a × c
For every integer a, we have
a × 1 = a = 1 × a
The integer 1 is called the multiplicative identity for integers.
For any integer, we have
a × 0 = 0 = 0 × a
For example:
(i) m × 0 = 0
(ii) 0 × y = 0
For any integer a, we have
a × (-1) = -a = (-1) × a
Note: (i) We know that -a is additive inverse or opposite of a. Thus, to find the opposite of inverse or negative of an integer, we multiply the integer by -1.
(ii) Since multiplication of integers is associative. Therefore, for any three integers a, b, c, we have
(a × b) × c = a × (b × c)
In what follows, we will write a × b × c for the equal products (a × b) × c and a × (b × c).
(iii) Since multiplication of integers is both commutative and associative. Therefore, in a product of three or more integers even if we rearrange the integers the product will not change.
(iv) When the number of negative integers in a product is odd, the product is negative.
(v) When the number of negative integers in a product is even, the product is positive.
If x, y, z are integers, such that x > y, then
(i) x × z > y × z, if z is positive
(ii) x × z < y × z , if z is negative.
These are the properties of multiplication of integers needed to follow while solving the multiplication of integers.
● Numbers - Integers
Properties of Multiplication of Integers
Examples on Multiplication of Integers
Properties of Division of Integers
Examples on Division of Integers
Examples on Fundamental Operations
● Numbers - Worksheets
Worksheet on Multiplication of Integers
Worksheet on Division of Integers
Worksheet on Fundamental Operation
7th Grade Math Problems
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