Significant figures represent the meaningful precision in a measured or calculated quantity. When a scientist records a measurement as 4.56 grams, they are communicating not just the value but also its precision: the measurement is reliable to the nearest 0.01 g. Writing 4.5600 g would falsely imply precision to the nearest 0.0001 g — a claim the measuring instrument may not support.
This is why sig figs are not just a school exercise: they convey genuine information about measurement quality. In chemistry, physics, and engineering, using too many significant figures implies false precision; using too few loses real information. Significant figures are a communication tool as much as a mathematical rule.
Leading zeros (zeros before the first non-zero digit) are merely placeholders that position the decimal point. They contribute no information about precision. Consider: 0.0045 and 4.5 × 10−3 represent the same quantity. In scientific notation, the leading zeros disappear, revealing that the number has only 2 significant figures (4 and 5). The zeros in 0.0045 exist because we measure in grams rather than milligrams — a choice of units, not a claim about precision.
Trailing zeros after a decimal point do carry information. If you measure a mass as 3.00 g (rather than 3 g or 3.0 g), the two trailing zeros tell the reader that the measurement was made to the nearest 0.01 g, not the nearest gram. The extra zeros are deliberate: they communicate precision. This is why a well-calibrated balance reads "3.00" rather than "3" when the true mass is 3 grams — the zeros confirm the precision of the instrument.
In addition and subtraction, the rule is based on decimal places, not significant figures. The answer should have the same number of decimal places as the measurement with the fewest decimal places. Example:
The reasoning: 349.0 is uncertain in the tenths place; adding it to more precise numbers cannot improve that uncertainty. The sum is only as precise as the least precise addend's decimal position.
In multiplication and division, the answer should have the same number of significant figures as the factor with the fewest significant figures. Example:
Many students make the error of applying decimal-place rules (from addition) to multiplication — these are distinct rules. Multiplication/division uses sig fig count; addition/subtraction uses decimal place count.
Chemistry is perhaps where sig figs matter most in practice, because measurements underpin all quantitative analysis.
Titration: A burette is typically read to ±0.05 cm³, giving readings to 2dp and volumes to 2dp after subtraction. If your titre is 24.35 cm³ (4 sig figs) and your concentration standard is 0.100 mol/dm³ (3 sig figs), your calculated moles should be rounded to 3 sig figs: 24.35 × 0.100 ÷ 1000 = 0.002435 mol → 0.00244 mol (3 sig figs).
Molarity calculations: Weighing 5.00 g of NaCl (3 sig figs, molar mass 58.44 g/mol) dissolved in 250.0 cm³ (4 sig figs): moles = 5.00/58.44 = 0.08554 mol (round to 3 sf: 0.0855 mol); concentration = 0.0855/0.2500 = 0.342 mol/dm³ (3 sig figs).
In GCSE Science practicals (required AQA/Edexcel practical skills), students must record measurements to appropriate sig figs and express calculated quantities to the same precision as the least precise measurement. Common errors in GCSE and A-Level physics mark schemes include:
A-Level Physics mark schemes typically accept answers to 2 or 3 sig figs unless a specific precision is required. Always match the precision of the given data.
These two terms are often confused. Accuracy refers to how close a measurement is to the true value. Precision refers to how reproducible and detailed the measurement is — which is what significant figures capture.
Example: A balance that reads 4.9812 g for a true mass of 5.0000 g is precise (many sig figs) but not very accurate (there is a systematic error). A balance that reads 5.0 g is more accurate but less precise. The ideal instrument is both accurate and precise: consistent readings close to the true value with appropriate significant figures.
Scientific notation (standard form) is the unambiguous way to express significant figures. In a × 10n, the number of significant figures is determined entirely by the digits in a. For example:
For the ambiguous number 5000 (whole number, no decimal), you would write 5 × 103 (1sf), 5.0 × 103 (2sf), 5.00 × 103 (3sf), or 5.000 × 103 (4sf) to make the precision explicit. This is particularly important in data reporting for scientific publications and engineering specifications.
Significant figures (sig figs) are the meaningful digits in a number that carry information about its measurement precision. They start from the first non-zero digit and include all digits that convey genuine precision, with specific rules for zeros. For example, 4.500 has 4 significant figures (all trailing zeros after a decimal are significant), while 0.0045 has only 2 significant figures (leading zeros are not significant).
The rules: (1) All non-zero digits are significant. (2) Zeros between non-zero digits are significant (405 has 3 sig figs). (3) Leading zeros are NOT significant (0.0025 has 2 sig figs). (4) Trailing zeros after a decimal point ARE significant (2.300 has 4 sig figs). (5) Trailing zeros in whole numbers without a decimal are ambiguous — use scientific notation to clarify.
0.00150 has 3 significant figures. The leading zeros (0.00) are not significant — they only indicate the position of the decimal point. The digits 1, 5, and the trailing 0 are significant. The trailing zero is significant because it appears after the decimal point and after non-zero digits, indicating the measurement's precision reaches that decimal place.
It depends entirely on position. Zeros between non-zero digits: always significant (1002 → 4 sig figs). Trailing zeros after a decimal point: significant (3.00 → 3 sig figs). Leading zeros: NEVER significant (0.005 → 1 sig fig). Trailing zeros in whole numbers without a decimal (1500): AMBIGUOUS — write 1.5 × 10³ or 1.50 × 10³ to make sig figs explicit.
In multiplication and division, round the result to the same number of significant figures as the factor with the fewest sig figs. Example: 4.56 × 1.4 = 6.384; since 1.4 has 2 sig figs, the answer is 6.4 (2 sig figs). Note: this is different from addition/subtraction, where you use the fewest decimal places, not sig figs.
Decimal places count digits after the decimal point. Significant figures count meaningful digits starting from the first non-zero digit. For 0.0045: 4 decimal places, but only 2 sig figs. For 1234: 0 decimal places, 4 sig figs. For 1.230: 3 decimal places, 4 sig figs. They measure different aspects of a number: decimal places measure position, significant figures measure precision.