Number and Operations
In the Pancyprian Core Mathematics paper, Number and Operations items are short, exact and unforgiving: one sign, index or bracket slip costs the whole mark. The exam tests manipulation of real numbers - powers, roots, logarithms, ratio, percentage and controlled approximation - rather than long arithmetic, so the marks sit in the structure of an expression, not in the size of the numbers.
Real numbers, order and absolute value
- The chain N ⊂ Z ⊂ Q ⊂ R; numbers such as √2, π and e are irrational and cannot be written as p/q.
- |x| is the distance of x from 0, so |x − a| < d means a − d < x < a + d, and |x| = √(x²).
- Respect the order of operations: −3² = −9, while (−3)² = 9.
- Multiplying or dividing an inequality by a negative number reverses the sign.
Powers, roots and logarithms
- aᵐ·aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁻ⁿ = 1/aⁿ, a^(m/n) is the n-th root of aᵐ, and a⁰ = 1 for a ≠ 0.
- √(ab) = √a·√b only when a, b ≥ 0; in general √(a²) = |a|.
- logₐx is defined only for x > 0, a > 0, a ≠ 1, and logₐ(xy) = logₐx + logₐy, logₐ(x/y) = logₐx − logₐy, logₐxⁿ = n·logₐx.
- Change of base: logₐx = log_b x / log_b a. Also a^(logₐx) = x.
Ratio, percentage and approximation
- Percentage change = (new − old)/old × 100. Successive changes multiply: +20% then −20% leaves 96%, not 100%.
- Simple interest A = P(1 + nr); compound interest A = P(1 + r)ⁿ.
- Carry full accuracy through intermediate steps and round only the final answer, to the significant figures requested.
Worked example
Evaluate log₂48 − log₂3 + 2^(log₂5) without a calculator. Combine the logarithms: log₂48 − log₂3 = log₂(48/3) = log₂16 = 4, since 2⁴ = 16. For the second term use a^(logₐx) = x, giving 2^(log₂5) = 5. Total: 4 + 5 = 9.
Traps that cost marks
- Writing √(a + b) = √a + √b or log(a + b) = log a + log b - neither is true.
- Cancelling a term across a sum, e.g. turning (x + 3)/3 into x.
- Confusing percentage points with percentage change, and adding successive percentages instead of multiplying.
- Taking a logarithm of a negative number or of zero, or ignoring the base condition.