Significant figures

Five students measure a table and get 2.34, 2.35, 2.31, 2.32 and 2.38 m. They all agree on 2.3; only the last digit differs. Significant figures are the digits we are sure of plus the first uncertain digit. Here each reading has three.

How to count them:

• All non-zero digits count: 13.75 has four.
• Zeros between non-zero digits count: 10.05 has four.
• Zeros before the first non-zero digit do not count: 0.0035 has two.
• Trailing zeros count only if there is a decimal point: 4.300 has four, but 4300 has two.
• Powers of ten and changes of unit change nothing: 16.4 cm = 0.164 m = 1.64 × 10⁻⁴ km, three significant figures each time.

Rounding off: drop a digit below 5; raise the digit before it for above 5. For exactly 5, look at the digit before it: raise it if it is odd, leave it if it is even. So 18.35 → 18.4 (3 is odd), but 14.45 → 14.4 (4 is even).

Answers cannot be more exact than the data:

• Add or subtract: keep as few decimal places as the least precise number. 12.587 − 12.5 = 0.087, which becomes 0.1.
• Multiply or divide: keep as few significant figures as the least precise number. A sphere of radius 1.41 cm has volume (4/3)π(1.41)³ = 11.742…, which becomes 11.7 cm³.

The uncertainties combine in the same spirit:

• In a sum or difference, the absolute errors add: 100 ± 3 Ω and 200 ± 4 Ω in series give 300 ± 7 Ω.
• In a product or quotient, the relative (percentage) errors add: if V is measured to 5% and I to 2%, then R = V/I is known to 5% + 2% = 7%.
• A power multiplies the error: the radius of a sphere to 2% gives its volume (∝ r³) to 3 × 2% = 6%.