Rational Numbers

What are Rational Numbers?

Rational numbers are numbers that can be expressed as the ratio of two integers, where the denominator is not zero.

A rational number is any number that can be written in the form: p/q where:

• p is an integer (numerator)
• q is an integer (denominator)
• q ≠ 0

Examples:

1/2, -3/4, 5, 0, 0.75, -2.5 : All of these can be written as p/q, so they are rational numbers.

Types of Rational Numbers

1. Positive Rational Numbers

Numbers greater than zero.

Examples:

• 3/5
• 7/2
• 11

2. Negative Rational Numbers

Numbers less than zero.

Examples:

• −4/7
• −9
• −3/2

3. Zero as a Rational Number

Zero is also rational because:$$0 = \frac{0}{5}$$

where the denominator is non-zero.

Rational Numbers in Different Forms

Rational numbers can appear in different forms.

Fraction Form

Examples:

1/3
5/8
7/4

Integer Form

All integers are rational numbers because:$$n = \frac{n}{1}$$

Examples:

2 = 2/1
−5 = −5/1

Decimal Form

Rational numbers also include decimals that terminate or repeat.

Examples:

0.5 = 1/2
0.25 = 1/4
0.333… = 1/3

Important Rule

A decimal is rational if it:

• terminates OR
• repeats

Examples:

0.666…
0.121212…

Equivalent Rational Numbers

Two rational numbers are equivalent if they represent the same value.

Example

$$\frac{1}{2} = \frac{2}{4} = \frac{3}{6}$$

All these fractions represent the same point on the number line.

How to Find Equivalent Fractions

Multiply numerator and denominator by the same number.

Example:$$\frac{3}{5} = \frac{6}{10}$$

Standard Form of Rational Numbers

A rational number is said to be in standard form when it satisfies the following two conditions:

• The numerator and denominator have no common factors except 1 (they are coprime).
• The denominator is positive.

In other words, the fraction is fully simplified and the negative sign, if any, appears only in the numerator.

Definition:

A rational number p/q is in standard form if:

• p and q are integers
• q ≠ 0
• The HCF (or GCD) of p and q = 1
• q > 0

Examples of Rational Numbers in Standard Form are:

3/5 → Already simplified
-7/8 → Denominator is positive
4/9 → No common factors between numerator and denominator

These fractions already satisfy the conditions of standard form.

Examples of Rational Numbers Not in Standard Form:

6/8 → Not in standard form because 6 and 8 have a common factor 2.

-4/-5 → Not in standard form because the denominator should not be negative.

10/15 → Not in standard form because both numbers are divisible by 5.

These fractions need to be simplified to convert them into standard form.

Steps to Convert a Rational Number into Standard Form:

Step 1: Find the HCF (Highest Common Factor) of the numerator and denominator.

Step 2: Divide both numerator and denominator by the HCF.

Step 3: Make sure the denominator is positive. If it is negative, multiply both numerator and denominator by −1.

Example 1

Convert 6/8 into standard form.

Step 1: HCF of 6 and 8 = 2

Step 2: Divide both numbers by 2.

6/8 = (6 ÷ 2) / (8 ÷ 2)

6/8 = 3/4

So, the standard form is 3/4.

Example 2

Convert -12/18 into standard form.

Step 1: HCF of 12 and 18 = 6

Step 2: Divide numerator and denominator by 6.

-12/18 = (-12 ÷ 6) / (18 ÷ 6)

-12/18 = -2/3

So, the standard form is -2/3.

Example 3

Convert 4/-7 into standard form.

The denominator should be positive.

4/-7 = -4/7

So, the standard form is -4/7.

Important Points

• The fraction must be completely simplified.
• The denominator must always be positive.
• The negative sign should be written in the numerator.
• Standard form helps make mathematical calculations simpler and clearer.