Year 9 Mathematics · AC9M9N01

Real Numbers: Rational, Irrational and Exact Values

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…

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Learning goalsSay it simply

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
Key conceptTeach from the board

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.

Worked examplesWe do

Worked examples

AC9M9N01 - Real Numbers: Rational, Irrational and Exact Values
Example 1

Classify numbers −7 is integer and rational; 0.125=1/8 is rational; 0.272727… is rational because it recurs; √7 and π are irrational.

Example 2

Bound a surd 49<58<64, so 7<√58<8; a calculator check gives about 7.616.

Example 3

Keep a circle result exact For radius 4 cm, area=16π cm² exactly, or about 50.27 cm² to 2 decimal places.

Example 4

Classify −3, 2/7, √16, √18 and 0.1010010001… as rational or irrational, justifying each.

Curriculum examplesCopied content

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.

Lesson componentAustralian CurriculumVictoriaNSW
Concept teaching + worked examplesAC9M9N01VC2M9N01 — Level 9 NumberStage 5 Core — Number and finance, supported by Working mathematically
Guided + independent practiceApplies the descriptor through progressively less-scaffolded problemsBuilds the corresponding Level 9 mathematical knowledge and fluencySupports Stage 5 Core/Path application and Working mathematically
Reasoning + assessment + masteryChecks transfer, justification, interpretation and model limitsChecks Level 9 reasoning at the mapped content depthChecks relevant Stage 5 reasoning without claiming a false Year 9 equivalent
Questions and answersWith answers
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.
Practice and reviewReady for practice
  • 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. [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.

Curriculum alignmentStart here

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
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