Rational and irrational numbers
A clear starting point, worked steps and space to practise.
Start here
A rational number can be written as a fraction of integers with a non-zero denominator. An irrational number cannot.
A short calculator display is rounded; it does not prove an irrational number terminates.
1 Work through these steps
- Look for an exact fraction representation.
- A terminating or recurring decimal is rational.
- Use exact facts for roots, not a rounded calculator display.
- Remember that irrational decimals neither terminate nor repeat.
2 Follow a worked example
Classify 0.125 and √2 as rational or irrational.
- 0.125 = 125/1,000 = 1/8, so it is rational.
- √2 has no exact fraction of integers.
- √2 is irrational.
3 Try it yourself
Show each step on paper or open Working out. Then check the answer.
-
Is √49 rational?
Check answer 1
Answer: Yes; √49 = 7.
Seven can be written as 7/1.
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Working out
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Check your understanding
Explain the worked example in your own words. Say why the method works, not just what the answer is.
Skills and explanation links
Key ideas taught here
- A rational number can be written as a fraction of integers with a non-zero denominator. An irrational number cannot.
- A short calculator display is rounded; it does not prove an irrational number terminates.
Method and checking
Optional recap: related Maths skills
These earlier examples revisit background skills. Start with the topic example and practice above.
Useful vocabulary
- digit
- A symbol from 0 to 9 used to write numbers.
- place value
- The value of a digit because of its position in a number.
- round
- Change a number to a nearby easier number.
- estimate
- A sensible approximate answer used to check reasonableness.
Rules for these related skills
Compare numbers from the largest place value, then move right only if the digits are the same.
For a positive whole number using the school halfway-up rule, check the next digit: 5–9 rounds the chosen place up; 0–4 keeps it unchanged.
Expanded form shows the value of every digit, such as 4 582 = 4 000 + 500 + 80 + 2.
Write 4 582 in expanded form.
- 4 thousands + 5 hundreds + 8 tens + 2 ones.
- 4 582 = 4 000 + 500 + 80 + 2.
Round 36 748 to the nearest thousand.
- Look at the thousands digit: 6.
- Check the hundreds digit: 7.
- Because 7 is 5 or more, round up to 37 000.
Order 8 905, 8 590 and 9 058 from smallest to largest.
- Compare thousands first.
- 8 590 and 8 905 both have 8 thousands, so compare hundreds.
- Smallest to largest: 8 590, 8 905, 9 058.
A school raised 12 486 points. About how many points is this to the nearest thousand?
- The hundreds digit is 4.
- Round down to 12 000.
- About 12 000 points were raised.
More visual models and everyday examples
Each everyday example has its own information and model. Use its stated values rather than copying values from another question.
Concept example. The story examples below have their own diagrams.
Say the value of each digit aloud before comparing or rounding.
For parents and teachers
- Ask your child to explain the method aloud before writing the answer.
- Use a real-life context at home, such as shopping, cooking, sport scores, maps or timetables.
- Praise clear working and checking, not only speed.
- Ask your child to write the formula or rule first, then substitute values carefully.
Practise choosing a method from the information in the question. The examples here are Pi Leo Academy learning activities. Official NAPLAN testing takes place in Years 3, 5, 7 and 9; a topic appearing in these notes does not mean it is tested in NAPLAN.
These notes provide guided examples and practice; they do not certify completion of every curriculum requirement.