Number
Number
Number is the anchor of Paper 1: it tests whether you can move confidently between fractions, decimals, percentages, indices and standard form, use set language correctly, and apply ratio and proportion to short contexts such as sharing a harvest, converting kwacha, or interpreting a population figure.
Sets and number bases
- Use ∈, ∉, ⊂, ∪, ∩ and complement, with the universal set written ξ.
- Convert from any base to base 10 by place value: 1101₂ = 8 + 4 + 1 = 13.
Indices and standard form
- aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ.
- Standard form is A × 10ⁿ with 1 ≤ A < 10. Count the places the decimal point moves; very small numbers give a negative index.
Ratio, percentage and proportion
- To share an amount, add the ratio parts to get the total number of parts, then divide.
- Percentage change = change ÷ original × 100. An increase of 15% means multiplying by 1.15.
Surds
- Rationalise a denominator using the conjugate: 1/(√5 − 2) × (√5 + 2)/(√5 + 2) = (√5 + 2)/(5 − 4) = √5 + 2.
Worked example. Evaluate (3.2 × 10^4) × (2.5 × 10^−6), giving your answer in standard form. Multiply the number parts: 3.2 × 2.5 = 8.0. Add the indices: 10^4 × 10^−6 = 10^−2. The answer is 8.0 × 10^−2. Since 8.0 already lies between 1 and 10, no adjustment is needed.
Traps that cost marks.
- Writing 0.00045 as 45 × 10^−5 instead of 4.5 × 10^−4.
- Using the new value rather than the original as the base for a percentage change.
- Leaving a surd in the denominator of a final answer.
- Mixing up ∪ and ∩ in a set question.