Measurements and significant figures

Every measurement has some uncertainty, and the way a number is written says how much. A balance that reads 12.30 g is telling you it's good to the hundredth of a gram. The digits you can trust, plus one estimated digit, are the significant figures.

Counting them: all nonzero digits count. Zeros between digits count (1.05 has 3). Leading zeros don't, they only place the decimal point (0.0045 has 2). Trailing zeros count if there's a decimal point (2.50 has 3, 2.500 has 4). In 1200 it's unclear, so write 1.2 × 10³ or 1.20 × 10³ to say which you mean.

In a result, multiplying or dividing keeps as many significant figures as the least precise input: 12.0 × 3.1 = 37.2, written 37 (two figures, because 3.1 has two). Adding or subtracting keeps as many decimal places as the least precise input: 12.11 + 0.3 = 12.4. Round only at the end, so rounding errors don't pile up.

Accuracy is how close a result is to the true value; precision is how close repeated results are to each other. Darts all bunched in the corner are precise but not accurate. Percent error = |measured − true| ÷ true × 100: measuring 9.6 when the true value is 10.0 is 4% off.

Random error scatters results both ways (reading a scale by eye, electrical noise); averaging more measurements shrinks it. Systematic error pushes every result the same way (an uncalibrated balance, a contaminated reagent); averaging doesn't help, but calibrating against a known standard does.