Page 17 - Maths Skills - 8

Rational Numbers 15

ABSOLUTE VALUE OF A RATIONAL NUMBER
The absolute value of a rational number is the number with no regard of its sign. If ‘x’ is any rational number, then
its absolute value is denoted by | x | and is defined as
3
x, if x For example, absolute value of − 3 = 3 3 .
0
7 7 7 7
x , 0 if x 0
− 7 7 7 7
x, if x 0 absolute value of = .
− 5 5 5 5

PROPERTIES OF RATIONAL NUMBERS

Closure Property
1. Addition: When two rational numbers are added, their sum is always a rational number.

For example, 2 + 3 = 89+ = 17 , which is also a rational number.
3 4 12 12
Therefore, rational numbers are closed under addition. It means for any two rational numbers a and b, a + b,
is also a rational number.

2. Subtraction: When two rational numbers are subtracted, their difference is always a rational number.

For example, 5 4 1 , which is also a rational number.
6 5 30
Therefore, rational numbers are closed under subtraction. It means for any two rational numbers a and b,
a – b is also a rational number.
3. Multiplication: When two rational numbers are multiplied, their product is always a rational number.

2
For example, 5 10 , which is also a rational number.
3 7 21
Therefore, rational numbers are closed under multiplication. It means for any two rational numbers a and b,
a × b is also a rational number. Fact-o-meter

4. Division: As for any rational number a, a ÷ 0 is not defined, therefore
not all rational numbers are closed under division. You can say that Division by 0 is not defined.
except zero, all rational numbers are closed under division.

Commutative Properties

1. Addition: Addition is commutative for rational numbers. In general, for any two rational numbers a and b,

a + b = b + a.

The following example proves the commutativity of addition for rational numbers.
2 3 5 and 3 2 5
7 7 7 7 7 7
2. Subtraction: Subtraction is not commutative for rational numbers. In general, for any two rational numbers
a and b, a – b ≠ b – a

Look at the following example showing that subtraction of rational numbers is not commutative.
5 2 1 but 2 5 1
6 3 6 3 6 6