Question

Generalizes the properties of multiplication of intege
carries out the mathematical operations easily by us
Solve the following sums.
6. (-20) × (-1), Find the product.
7.
(-12) * ( * (-17), Find the answer.
Find the product.
8.
(-3) * (-4) * 2 * (-1)
Uses the decimal numbers in day to day activit
the following sums.​

Answer 1

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:

Property 1 (Closure property):

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.Property 2 (Commutativity property):

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).

Property 3 (Associativity property):

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)

Property 4 (Distributivity of multiplication over addition property):

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).

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