Real Numbers: Rational, Irrational and Exact Values — AC9M9N01

Rational and irrational numbers together form the real number system. Exact forms such as √2 and π preserve value, while decimals may be terminating, recurring or approximate.

By the end of this lesson, you should be able to:

• classify rational and irrational numbers within the real number system
• distinguish exact values from decimal approximations
• locate and estimate irrational numbers on the real number line
• solve contextual problems involving real numbers and check calculator output

Recall fractions, decimals, percentages, integers, square numbers and square roots. Be comfortable ordering positive and negative numbers on a number line.

The real number system is nested

Natural numbers sit inside integers; integers sit inside rational numbers; rational and irrational numbers together make the real numbers. A rational number can be written as a/b with integers a,b and b≠0.

Decimal form reveals structure

Terminating and recurring decimals are rational. A non-terminating, non-recurring decimal is irrational. A finite calculator display for an irrational number is an approximation.

Exact and approximate forms have different jobs

Keep √13 or 5π exact when later reasoning depends on the full value. Approximate only when the context requests a decimal, and state the accuracy.

Irrational numbers still have exact positions

Use nearby square numbers to bound roots. Since 25<30<36, 5<√30<6. Geometric construction can locate √2 exactly on the number line.

• Every non-terminating decimal is irrational: Recurring decimals are rational.
• √9 is irrational because it has a root sign: √9=3, so it is rational.
• Calculator decimals are exact: For irrational values, calculator displays are rounded approximations.
• 3.14 equals π: 3.14 is only an approximation to π.

1. Classify −3, 2/7, √16, √18 and 0.1010010001… as rational or irrational, justifying each.
2. Place √12 between consecutive integers, then estimate it to 1 decimal place.
3. A circle has diameter 7 m. Give circumference exactly and to the nearest centimetre.

Guided method: Name the mathematical structure first, show the calculation or representation, then verify with an estimate, inverse operation, second representation or digital check.

1. Explain why every integer is rational.
2. Is 0.123123123… rational or irrational? Justify.
3. Between which integers does √95 lie?
4. Order 3.14, π and 22/7 from least to greatest using a digital check.
5. A square has side √20 cm. Give its perimeter exactly and approximately to 2 decimal places.
6. Explain why rounding √2 before a repeated calculation can change the final result.

A student claims √50 is irrational but 5√2 is rational because it contains an integer coefficient. Evaluate the claim and prove your conclusion by simplifying √50.

Reasoning standard: Make a claim, show the relevant mathematical evidence, explain why it supports the conclusion and state any condition or limitation.

How can I recognise a rational decimal?

It terminates or eventually repeats.

Why keep exact values?

Exact notation preserves information and avoids accumulated rounding error.

Can an irrational number be located exactly?

Yes. Irrational numbers have exact positions even though their decimals do not terminate or repeat.

1. [4 marks] Classify six given numbers into natural, integer, rational and irrational sets, allowing membership in more than one set.
2. [5 marks] Show that √72=6√2, estimate it to 3 significant figures and identify which form is exact.
3. [6 marks] Compare repeated calculations using an exact irrational value and a rounded version, explaining why the discrepancy grows.

Review hint: A full-mark response shows the method, keeps units and restrictions visible, interprets the result in context and checks whether the answer is reasonable.

• I can distinguish rational and irrational numbers.
• I can keep exact values exact until approximation is required.
• I can bound and estimate irrational square roots.
• I can explain why recurring decimals are rational.
• I can use digital tools to verify rather than replace reasoning.

Exit ticket: Solve one unfamiliar example and explain the key decision in words, not just symbols.

Australian Curriculum v9.0 — AC9M9N01: recognise that the real number system includes the rational numbers and the irrational numbers, and solve problems involving real numbers using digital tools

Victoria: VC2M9N01 — Level 9 Number. The mapping names direct Level 9 content where available and explicitly identifies supporting content where the Victorian structure separates an idea differently.

NSW: Stage 5 Core — Number and finance, supported by Working mathematically. NSW organises Years 7–10 Mathematics through Stage 5 Core content groups and Paths rather than a one-code-per-Year-9 structure, so this lesson does not force a false one-to-one outcome.

Use these links to move through the Year 9 learning sequence rather than studying the code in isolation.

• AC9M9A01 — Exponent Laws with Integer Exponents
• AC9M9M02 — Scientific Notation for Very Large and Very Small Measurements

• Australian Curriculum v9.0 — Mathematics
• Victorian Curriculum F–10 Version 2.0 — Mathematics
• NSW Mathematics K–10 Syllabus (2022)

Curriculum equivalents for That the real number system includes the rational numbers and...

Mapped skill: recognise that the real number system includes the rational numbers and the irrational numbers, and solve problems involving real numbers using digital tools

These references identify matching or closely related learning. Curriculum sequence, terminology and depth vary, so teachers should use the mapped skill and lesson difficulty to confirm suitability.

Australian Curriculum v9.0 is the canonical source for this SkillrHub lesson. Victoria and NSW entries name the closest published state codes or outcomes; international entries are planning references rather than claims of identical curricula.