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Rational Numbers

A rational number is any number that can be written as p/q, where p and q are integers and q is not zero.

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Student-friendly explanation

Rational numbers include integers, fractions, terminating decimals, and non-terminating repeating decimals. They can be represented on the number line by dividing the space between integers into equal parts.

How to write this in exams

  1. 1

    Start with the exact idea

    A rational number is any number that can be written as p/q, where p and q are integers and q is not zero.

  2. 2

    Then show how to use it

    Write the number as p/q if possible. For number-line placement, convert improper fraction to mixed form, locate the whole-number interval, divide it equally, and mark the required part.

  3. 3

    Add one concrete example

    -3, 0, 4/7, 2.5, and 0.333... are rational numbers because each can be written as p/q with q not equal to zero.

  4. 4

    Avoid this incomplete answer

    Marking 7/3 between 1 and 2 is wrong because 7/3 is 2⅓, so it lies between 2 and 3.

Definition

A rational number is any number that can be written as p/q, where p and q are integers and q is not zero.

Example

-3, 0, 4/7, 2.5, and 0.333... are rational numbers because each can be written as p/q with q not equal to zero.

Rule to remember

Rational form: p/q, where p and q are integers and q ≠ 0.

Memory hook

Rational means ratio-ready.

Examples and method

Worked example

Represent 7/3 on the number line. Since 7/3 = 2 + 1/3, mark the interval from 2 to 3, divide it into three equal parts, and take the first mark after 2.

Method to apply

Write the number as p/q if possible. For number-line placement, convert improper fraction to mixed form, locate the whole-number interval, divide it equally, and mark the required part.

Diagram support

Show 3/4 on a number line by dividing the segment from 0 to 1 into four equal parts and marking the third part.

How CBSE asks it

Students may be asked to identify rational numbers, convert decimals into fractions, or mark a rational number on a number line.

Avoid common mistakes

Common confusion

Students sometimes think only positive fractions are rational, but negative fractions and integers are rational too.

Common wrong answer

Marking 7/3 between 1 and 2 is wrong because 7/3 is 2⅓, so it lies between 2 and 3.

Exam tip

For number-line questions, first locate the two nearest integers and then divide the interval according to the denominator.

Quick check

Why is 2.75 a rational number?

2.75 is rational because it is a terminating decimal and can be written as 275/100, which simplifies to 11/4.

Study the rational numbers diagram carefully

Use the labelled diagram to keep rational numbers clear in short answers and revision.

What this diagram makes clear

This diagram keeps the labels and direction of rational numbers in the right order.

Where this helps in exams

Use this for labelled diagram work and short exam answers on rational numbers.

Revision cue

Revise rational numbers through the labels before writing the answer.

Answer writing and exam use

1-mark answer

A rational number is any number that can be written as p/q, where p and q are integers and q is not zero.

2-mark answer

A rational number is any number that can be written as p/q, where p and q are integers and q is not zero. Rational form: p/q, where p and q are integers and q ≠ 0. -3, 0, 4/7, 2.5, and 0.333... are rational numbers because each can be written as p/q with q not equal to zero.

3-mark answer

Rational numbers include integers, fractions, terminating decimals, and non-terminating repeating decimals. They can be represented on the number line by dividing the space between integers into equal parts. Rational form: p/q, where p and q are integers and q ≠ 0. Represent 7/3 on the number line. Since 7/3 = 2 + 1/3, mark the interval from 2 to 3, divide it into three equal parts, and take the first mark after 2. Students may be asked to identify rational numbers, convert decimals into fractions, or mark a rational number on a number line. Marking 7/3 between 1 and 2 is wrong because 7/3 is 2⅓, so it lies between 2 and 3.
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