# 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:**

- Closure under multiplication:

For all integers a and b, a x b is an integer.

Example: (-20) x (-5) = 100, product is an integer. - Commutativity of Multiplication:

For any two integers a and b, a x b = b x a

Example: (-3) x 4 = -12 = 4 x (-3) - Multiplication by zero:

For any integer a, a x 0 = 0 x a = 0

Example: (-3) x 0 = 0 - Multiplicative Identity:

For any integer a, a x 1 = 1 x a = a

Example: (-3) x 1 = -3 - 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 - 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**

- 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 - 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 - 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 - 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