Number and Measure
Number and Measure tests number systems, prime factorisation, surds, scientific notation, percentage error, measurement formulae, units, and similarity. Higher Level questions often embed these skills in a practical or proof context.
Core rules
- Number sets: (\mathbb N\subset\mathbb Z\subset\mathbb Q\subset\mathbb R\subset\mathbb C). A rational number can be written as (\frac pq) with integers (p,q), (q\ne0).
- Irrationality proof: (\sqrt2) is irrational; prove by contradiction by assuming (\sqrt2=\frac pq) in lowest terms and deriving a common factor.
- Prime factorisation gives HCF and LCM. For surds, rationalise denominators and simplify products such as (\sqrt a\sqrt b=\sqrt{ab}).
- Percentage error: (\frac{|\text{error}|}{\text{true value}}\times100%).
- Measurement: cylinder (V=\pi r^2h); sphere (V=\frac43\pi r^3); cone (V=\frac13\pi r^2h). Use consistent units.
- Similar figures with scale factor (k): lengths scale by (k), areas by (k^2), volumes by (k^3).
Worked example
A cylinder has radius 3 cm and height 10 cm. Its volume is (V=\pi(3^2)(10)=90\pi\approx282.7\text{ cm}^3). If the radius and height are both doubled, the new volume is (8) times the old volume, because volume scales by (k^3) with (k=2).
Traps
Converting (\text{cm}^3) to litres incorrectly; applying a length scale factor to area or volume; rounding too early in multi-step calculations; using the wrong base in percentage change; forgetting to rationalise a surd denominator; mixing units in a formula.