Numbers and computation: the ladder of number sets, powers, roots, proportions and percentages
What you must command
Numbers form a ladder: naturals (counting), integers (adding zero and negatives), rationals (fractions p/q with q ≠ 0, including terminating and repeating decimals), irrationals (non-repeating infinite decimals such as √2, π), and the reals that close the ladder. Every computation question tests which rung you are standing on.
Core skills: order of operations, the laws of exponents for integer and rational exponents, simplification of radicals (√(ab) = √a·√b only for non-negative factors, and √(a²) = |a|), rationalising denominators, and proportional reasoning. A percentage is a proportion with denominator 100, so a change of p% multiplies by (1 + p/100).
Traps to expect. (1) √(a²) = |a|, not a - decisive when a is a variable or negative. (2) (−2)⁴ = 16 but −2⁴ = −16; brackets decide. (3) Successive percentages do not add. (4) A percentage "of" a base that already changed is computed on the new base. (5) Repeating decimals are rational: 0.333… = 1/3.
Worked rule of thumb. To combine percentage changes, multiply factors, never add percentages. A price rises 20% then falls 20%: 1.20 × 0.80 = 0.96, so the final price is 96% of the original - a 4% decrease, not 0%. Always check whether the question asks for absolute change or relative change, and whether the denominator is the original or the new value.