Calculations

Standard form — a number written as a × 10ⁿ with a between 1 and 10. It keeps very large and very small numbers manageable and makes the number of significant figures unambiguous.

Multiplying and dividing in standard form — multiply or divide the numbers, and add or subtract the powers.

(a × 10ᵐ)(b × 10ⁿ) = ab × 10^(m+n)

Example: (3.0 × 10⁸) × (2.0 × 10⁻⁶) = 6.0 × 10² = 600. Adding the powers gives 8 + (−6) = 2, and the leading numbers multiply to 6.0.

Adding and subtracting — the powers must match first. 3.0 × 10⁵ + 4.0 × 10⁴ becomes 3.0 × 10⁵ + 0.4 × 10⁵ = 3.4 × 10⁵. Adding the digits without matching the powers is a common and expensive error.

Prefixes — each stands for a power of ten, and converting means substituting that power.

The trap with areas and volumes — the prefix is raised to the same power as the unit. 1 cm = 10⁻² m, so 1 cm² = 10⁻⁴ m² and 1 cm³ = 10⁻⁶ m³. Forgetting to raise the prefix is one of the most frequent mistakes in the whole course.

Significant figures — count from the first non-zero digit. Give a calculated answer to the smallest number of significant figures used in the calculation, since the answer cannot be more precise than its least precise input.

Using a calculator without being misled by it — a calculator shows arithmetic, not measurement. It will happily display ten digits from inputs known to two, and every digit past the second is fiction.

Estimating to check — before accepting an answer, round every input to one significant figure and redo the calculation in your head. If the two disagree by orders of magnitude, a prefix or a power has been dropped. This takes seconds and catches the errors that cost the most marks.