Number and Numeration
Number and Numeration carries the arithmetic foundations of the NECO SSCE paper: place value in bases other than ten, the laws of indices and logarithms, surds, standard form, sets, ratio and proportion, percentages, and number theory such as HCF and LCM. Objective questions here are short but unforgiving - a single carry or sign error removes the mark.
Number bases
- A numeral in base b uses only the digits 0 to b − 1.
- To convert from base b to base 10, expand in powers of b: 241₅ = 2×5² + 4×5 + 1 = 71.
- To convert from base 10 to base b, divide repeatedly by b and read the remainders upwards.
- Add, subtract and multiply column by column using b as the carry/borrow value, not 10.
Indices and logarithms
- aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ; a⁰ = 1; a⁻ⁿ = 1/aⁿ; a^(1/n) = ⁿ√a.
- log MN = log M + log N; log(M/N) = log M − log N; log Mⁿ = n log M.
- Solve aˣ = b by taking logarithms of both sides.
Surds and standard form
- √(ab) = √a × √b; rationalise a denominator by multiplying by its conjugate.
- Standard form is A × 10ⁿ with 1 ≤ A < 10, so 0.000305 = 3.05 × 10⁻⁴.
Worked example
Rationalise 1/(√7 − √5). Multiply numerator and denominator by the conjugate √7 + √5. The denominator becomes (√7)² − (√5)² = 7 − 5 = 2, so the expression is (√7 + √5)/2. The conjugate works because (a − b)(a + b) = a² − b² eliminates the surd.
Traps that cost marks
- Leaving a standard-form coefficient outside 1 ≤ A < 10, e.g. 30.5 × 10⁻⁵ instead of 3.05 × 10⁻⁴.
- Using an illegal digit, such as writing 5 in base 5.
- Confusing HCF (splitting into equal groups) with LCM (events recurring together).
- Forgetting that log 0 is undefined and that the log of a number below 1 is negative.