Numbers and Operations: Naturals, Integers, Rationals, Powers and Roots
What you must be able to do
You need to move confidently between natural numbers (0, 1, 2, …), integers (… −2, −1, 0, 1, 2 …) and rational numbers (anything writable as a fraction of two integers, including terminating and repeating decimals). Master the order of operations: brackets first, then powers and roots, then multiplication and division left to right, then addition and subtraction left to right. Be fluent with fractions (common denominators, simplifying, mixed numbers) and decimals (place value, multiplying and dividing by powers of ten).
Understand powers: a^n means n factors of a; a^0 = 1 for a ≠ 0; a^(−n) = 1/a^n. Know that (−2)^2 = 4 but −2^2 = −4, because the power binds to the base only. Know square roots of perfect squares up to 15² = 225 and that √a is defined only for a ≥ 0, while √(a²) = |a|.
Traps that appear in questions
- Sign errors when subtracting a negative: 5 − (−3) = 8.
- Treating a fraction bar as if it were not a bracket: (2+4)/2 = 3, not 4.
- Cancelling terms instead of factors: (x+2)/2 is not x+2.
- Rounding too early in a multi-step calculation.
- Forgetting that dividing by a fraction means multiplying by its reciprocal.
Rule of thumb
When a calculation looks messy, convert everything to fractions first; fractions keep exact values, whereas decimals can hide repeating expansions.
Worked example
Compute −3² + (−2)³ · (1/2) − √81. −3² = −9; (−2)³ = −8; −8 · 1/2 = −4; √81 = 9. So −9 + (−4) − 9 = −22.