Year 8 Mathematics · AC9M8N01

Irrational Numbers in Context

Some exact lengths and ratios cannot be written as fractions. Learn how square roots and π fit into the real number system, how to locate them approximately, and how…

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Key conceptTeach from the board

Rational numbers can be written as fractions of integers. Irrational numbers cannot. Their decimals neither terminate nor eventually repeat, but each irrational number still has one exact position on the number line.

Keep exact notation when it carries structure. A unit-square diagonal is exactly √2; 1.414 is only a measurement approximation. Bounds such as 1.4<√2<1.5 and 3.141<π<3.142 locate exact irrational values without pretending the decimal has ended.

Preserve the full applied AC9 contexts. A-series paper keeps the long-side:short-side ratio 1:√2 when halved, linking geometry and design. The golden ratio φ=(1+√5)/2 is another irrational ratio investigated in art/design contexts. Historical societies developed useful approximations to π; an approximation can be practical without being equal to π.

Worked examplesWe do

Worked examples

AC9M8N01 - Irrational Numbers in Context
Example 1

Classify 3/7, 0.125, 0.272727…, √3 and π.

Example 2

Use 1.7² and 1.8² to bound √3.

Example 3

Find the exact diagonal of a 4 cm square and approximate it.

Example 4

Explain why 22/7 is useful but not equal to π.

Curriculum examplesCopied content

Australian Curriculum v9.0 — AC9M8N01: recognise irrational numbers in applied contexts, including square roots and π.

Victorian Curriculum F–10 Version 2.0 — Level 8, VC2M8N01:Exact.

NSW Mathematics K–10 Syllabus (2022) — Stage 4, MA4-IND-C-01; MAO-WM-01:Partial — roots/indices and Working mathematically support the reasoning, but NSW does not mirror this descriptor one-to-one.

FrameworkLevel/StageRelationshipMapping
Australian Curriculum v9.0Year 8CanonicalAC9M8N01
Victoria V2.0Level 8ExactVC2M8N01
NSW K–10 (2022)Stage 4PartialMA4-IND-C-01; MAO-WM-01
Questions and answersWith answers
  1. What makes a number irrational? It cannot be expressed as a ratio of integers; its decimal is non-terminating and non-recurring.
  2. Why is √2 important in a square and A-series paper? It appears as a unit-square diagonal and as the paper side ratio that preserves similarity on halving.
  3. Can irrational numbers be located? Yes, between increasingly close rational bounds.
  4. Why study historical approximations to π? They show how rational approximations can be practically useful while remaining distinct from exact π.
Practice and reviewReady for practice
  • An irrational number has no exact value. It is exact; its decimal representation does not terminate/repeat.
  • Every non-terminating decimal is irrational. Recurring decimals are rational.
  • A square-root symbol always means irrational. √9=3 is rational.
  • A good approximation equals the irrational number. Use ≈, not =, for finite decimal or fractional approximations.
Curriculum alignmentStart here
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