Numbers and Calculation
What the domain tests
This domain checks whether you can move confidently between natural numbers (0, 1, 2, …), integers, rational numbers (anything writable as a fraction a/b with integers a and b ≠ 0, which includes terminating and repeating decimals) and irrational real numbers such as √2 and π. Exam tasks usually combine several of these: simplify a numeric expression with powers, compare a fraction with a decimal, round a large result to a sensible number of significant figures, or decide whether a divisibility claim is true.
Rules worth having automatic
- Order of operations: brackets first, then powers and roots, then multiplication and division left to right, then addition and subtraction left to right.
- Index laws: a^m · a^n = a^(m+n), a^m : a^n = a^(m−n), (a^m)^n = a^(mn), a^0 = 1 for a ≠ 0, a^(−n) = 1/a^n.
- Signs: (−3)² = 9, but −3² = −9, because the power binds only to the 3. Two negatives multiply to a positive; three give a negative.
- Fractions: a/b · c/d = ac/bd, and division means multiplying by the reciprocal: a/b : c/d = a/b · d/c. Addition needs a common denominator.
- Absolute value |a| is the distance from zero and is never negative: |−7| = 7.
- Divisibility: by 3 or 9 use the digit sum; by 4 use the last two digits; by 5 the last digit is 0 or 5.
- Estimation: round each number to one significant figure to predict the size of the answer, then compare with your exact result.
Worked example
Calculate (−3)² − 2 · (5 − 7)³ : 4. Brackets: 5 − 7 = −2. Powers: (−3)² = 9 and (−2)³ = −8. Multiplication and division, left to right: 2 · (−8) = −16, then −16 : 4 = −4. Subtraction: 9 − (−4) = 13. Check the order of magnitude: about 9 plus about 4 is about 13, so the result is plausible.
Traps that cost marks
- Writing −3² = 9. Only brackets protect the sign.
- Cancelling inside a sum: (a + b)/a is not 1 + b.
- Rounding in the middle of a calculation and only then adding - round at the end, or state that you are rounding.
- Mixing percentages of different bases: a 20% rise followed by a 20% fall does not return the original value.
- Forgetting that division by zero is undefined, so any expression with a variable denominator carries a condition.