medium difficulty5 min

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.

In one line: Rational means ratio-ready.

Understand it

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.

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.

Formula and rule

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

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

1Represent 7/3 on the number line.
2Since 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 you can reuse

1Write the number as p/q if possible.
2For number-line placement, convert improper fraction to mixed form, locate the whole-number interval, divide it equally, and mark the required part.

Sketch it in your answer

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

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Why is 2.75 a rational number?

Think first, then show answer

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

Common wrong answer

Marking 7/3 between 1 and 2.

Why it loses marks: 7/3 is 2⅓, so it lies between 2 and 3.

Don't mix this up

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

Exam tip

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