1.9 Rational and irrational numbers
Classify numbers, convert recurring decimals and preserve exact values using pi and surds.
How to study WJEC GCSE Mathematics and Numeracy
Move from concept and notation to a justified method, show a complete mathematical chain, check scale and units, and interpret the answer in context.
Core concepts
Concept 1
Recognise rational, irrational, terminating and recurring values and convert recurring decimals to fractions.
Exam cue: Confirm the unit, entered tier, calculator condition and available formula list before choosing a method.
Concept 2
Manipulate and simplify surds at the required tier.
Exam cue: Keep pi or a surd symbolic when exact form is requested and approximate only at the final stated stage.
Concept 3
Use pi and surds in exact calculations, including the Version 5 Unit 2 exact-surds scope.
Exam cue: Show a complete mathematical chain, use correct notation and units, and interpret or check the result in its original context.
Concept 4
Current matrix coverage: Unit 1, Unit 2, Unit 3. Read the exact subpoint scope in each applicable unit column.
Concept 5
Foundation content is shown in standard type in the specification; bold additions apply only to Higher Tier.
Risk pitfalls and guardrails
Replacing an exact value with a rounded decimal before it is used in a later calculation.
Guardrail: The Unit 1 and Unit 3 lists are limited and Unit 2 supplies none; preserve exact values and know all other required relationships.
Assuming that every subpoint in a cross-unit strand appears in all three papers, rather than reading the correct matrix column.
Guardrail: Read the precise subpoint in the applicable unit column; a cross-unit strand does not mean every subpoint appears in every paper.
Using a Higher-only method or expectation for a Foundation entry, or importing the outgoing Mathematics/Mathematics-Numeracy structure.
Guardrail: Standard type applies to Foundation and Higher; bold specification additions are Higher-only. There is no Intermediate Tier.
Memory anchors
Rational number
A rational number can be written as a fraction of integers with non-zero denominator.
Irrational number
An irrational number cannot be written as a fraction of integers.
Recurring decimal
A recurring decimal repeats a fixed digit pattern indefinitely.
Surd
A surd is an irrational root kept in exact symbolic form.
Exact pi
A multiple of π remains exact until a decimal approximation is explicitly required.
Checkpoint rule
Do the check-up only after you can summarize each concept in one sentence and identify one dangerous pitfall from memory.
Knowledge Check (after reading)
Short check-up to confirm understanding of this module.
Check-up Questions
Add 3/4 and 5/6, expressing the exact result as an improper fraction.
What is the product of the mixed numbers 2 1/3 and 1 1/2?
Answer all questions to submit.
Next step personalized recommendations
Continue learning
Move forward only after this module is stable.
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