Every measurement has a last, uncertain digit
Measure a pencil with a millimetre ruler and you might read 14.2 cm. The 1 and the 4 are certain; that last digit, the 2, is your eye's best estimate between two marks. Read it again and you might get 14.3. That small wobble is the measurement uncertainty, and it is a real feature of the measurement, not carelessness.
The digits you can honestly stand behind — every certain one, plus the single estimated one — are the significant figures of the measurement. Writing 14.2 cm claims three of them; writing 14.20 cm claims four, and quietly asserts you owned a finer instrument.
Counting and keeping significant figures
The rules: all non-zero digits count; zeros between non-zeros count; leading zeros never count (0.0042 has two significant figures); trailing zeros after a decimal point do count (2.50 has three). Scientific notation removes every ambiguity — 2.50×10³ plainly shows three.
For example, 2.0 m × 3.14159 m = 6.3 m² — two significant figures, because the 2.0 has only two — not the false precision of 6.28318 m² that a calculator will happily display.
Reporting a result with its uncertainty
A complete measurement is a value, an uncertainty, and a unit — for example, a length of (5.00 \pm 0.02) cm. The \pm 0.02 cm is the absolute uncertainty; dividing it by the value gives the relative (fractional) uncertainty, often quoted as a percentage.
Absolute uncertainty carries units; relative uncertainty is a pure ratio.
When you combine measurements, uncertainties grow in predictable ways. For a sum or difference, add the absolute uncertainties. For a product or quotient, add the relative uncertainties. The second rule is the one you will lean on most.
For a product, relative uncertainties add.
Accuracy is not precision
These are two different virtues, endlessly confused. Accuracy is how close your result sits to the true value; precision is how tightly repeated measurements cluster together, regardless of where that cluster sits. Picture a dartboard: a tight cluster off to one side is precise but inaccurate; a scatter spread evenly around the bullseye is accurate on average but imprecise.
The cause matters. A systematic error pushes every reading the same way — a mis-zeroed scale, a tape stretched by heat — and it destroys accuracy; you defeat it by calibrating. A random error scatters readings unpredictably and destroys precision; you beat it down by averaging many trials. This is the heart of systematic and random errors.