Significant figures
Also called Sig figs
Significant figures are the digits of a measured value that carry real information about its precision. A product or quotient keeps the fewest significant figures of its inputs, and a sum keeps the fewest decimal places.
Counting rules, in order of how often they go wrong:
- Every nonzero digit counts.
- Zeros between nonzero digits count, so 2005 has four.
- Leading zeros never count, so 0.0042 has two.
- Trailing zeros count only when a decimal point is written, so 0.00420 has three and 4200 is ambiguous.
Then the two arithmetic rules. Multiplying or dividing, the answer takes the smallest number of significant figures among the inputs: . Adding or subtracting, it takes the smallest number of decimal places: . Round once, at the end, not at every step.
Where this bites on the AP exam is narrower than most students fear. The CED defines the task verb Calculate as arriving at a final answer "including algebraic expressions, properly substituted numbers, and correct labeling of units and significant figures", so the expectation is stated. The sample scoring guidelines printed in the CEDs do not award or withhold a point for significant figures on their own.
The real weight is in the laboratory question. Reporting a length as 0.4523 m from a metre rule marked in millimetres claims a precision the instrument does not have, and that is an uncertainty error, not a rounding one.