CBSE Class 7 Mathematics/Integers/Chapter 1/Notes/Part 2

Multiplication of Integers
Multiplication of a positive and a negative integer:
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.
Examples:
(-2) x 5 = -10
6 x (-3) = -18
Multiplication of two negative integers:
We multiply the two negative integers as whole numbers and put the positive sign before the product. The product of two negative integers is a positive integer.
Examples:
(-10) x (-12) = 120
(-5) x (-6) = 30

Properties of multiplication of integers:

1. Closure under multiplication:
   For all integers a and b, a x b is an integer.
   Example: (-20) x (-5) = 100, product is an integer.
2. Commutativity of Multiplication:
   For any two integers a and b, a x b = b x a
   Example: (-3) x 4 = -12 = 4 x (-3)
3. Multiplication by zero:
   For any integer a, a x 0 = 0 x a = 0
   Example: (-3) x 0 = 0
4. Multiplicative Identity:
   For any integer a, a x 1 = 1 x a = a
   Example: (-3) x 1 = -3
5. Associativity for Multiplication:
   For any three integers a, b and c, (a x b) x c = a x (b x c)
   Example: Consider a = -3, b = -2 and c = 5
   (a x b) x c = [(-3) x (-2)] x 5 = 6 x 5 = 30
   a x (b x c) = (-3) x [(-2) x 5] = (-3) x (-10) = 30
6. Distributive property:
   For any integers a, b and c, a x (b + c) = a x b + a x c
   Example: Consider a = 4, b = -3 and c = -5
   a x (b + c) = 4 x [(-3) + (-5)] = 4 x (-8) = -32
   a x b + a x c = 4 x (-3) + 4 x (-5) = (-12) + (-20) = -32

Exercise 1.2

1. Find each of the following products:
   a) 3 x (-1)
   b) (-1) x 225
   c) (-21) x (-30)
   d) (-316) x (-1)
   e) (-15) x 0 x (-18)
   Solution:
   a) -3
   b) -225
   c) 630
   d) 316
   e) 0
2. Verify the following:
   a) 18 x [7 + (-3)] = [18 x 7] + [18 x (-3)]
   b) (-21) x [(-4) + (-6)] = [(-21) x (-4)] + [(-21) x (-6)]
   Solution:
   a) LHS = 18 x [7 + (-3)] = 18 x 4 = 72
   RHS = [18 x 7] + [18 x (-3)] = 126 + (-54) = 72
   LHS = RHS, verified.
   b) LHS = (-21) x [(-4) + (-6)] = (-21) x (-10) = 210
   RHS = [(-21) x (-4)] + [(-21) x (-6)] = 84 + 126 = 210
   LHS = RHS, verified
3. i) For any integer a, what is (-1) x a equal to?
   ii) Determine the integer whose product with (-1) is
   a) -22 b) 37 c) 0
   Solution:
   i) –a
   ii) a) 22
   b) -37
   c) 0
4. Starting from (-1) x 5, write various products showing some pattern to show (-1) x (-1) = 1.
   Solution:
   (-1) x 5 = -5
   (-1) x 4 = -4 = -5 + 1
   (-1) x 3 = -3 = -4 + 1
   (-1) x 2 = -2 = -3 + 1
   (-1) x 1 = -1 = -2 + 1
   (-1) x 0 = 0 = -1 + 1
   (-1) x (-1) = 1 = 0 + 1