Properties of Multiplication on Integers:
Property 1: (Closure Property)
The product of two integers is always an integer.
Verification: We have,
1) 6 x 5 = 30, 30 is an integer
2) 4 x (-5) = -20, -20 is an integer
3) (-9) x (8) = -72, -72 is an integer
2) (-3) x (-2) = 6, 6 is an integer
Property 2: (Commutative Law for Multiplication)
For any two integers a and b, we have a x b = b x a.
Verification: We have,
5 x (-7) = -35 and (-7) x 5 = -35
Therefore, 5 x (-7) = (-7) x 5.
Property 3: (Associative Law for Multiplication)
If a, b, c are any three integers then (a x b) x c = a x (b x c).
Verification: Consider the Integers -2, —4 and -6.
We have,
((—2) x (—4)) x (-6) = -48.
And, (—2) x ((—4) x (-6)) = (-2) x (24) = —48.
Therefore, ((—2) x (—4)) x (-6) = (—2) x ((—4) x (-6))
Property 4: (Distributive Law for Multiplication)
If a, b, c are any three integers then a x (b + c) = a x b + a x c.
Verification: Consider the Integers 7, —5 and -2.
We have,
7 x ((—2) + (—4)) = 7 x (-6) = -42.
And, 7 x (—2) + 7 x (-4) = (-14) + (-28) = —42.
Therefore, 7 x ((—2) + (—4)) = 7 x (—2) + 7 x (-4)
Property 5:
For any integer a, we have a x 1 = a. The integer 1 is called multiplicative identity for integers.
Verification: We have,
1) (-6) x 1 = -6
2) 5 x 1 = 5
Property 6:
For any integer a, we have a x 0 = 0.
Verification: We have,
1) 7 x 0 = 0
2) 5 x 0 = 0