Significant Figures Rules: How to Count, Round and Calculate with Sig Figs
Short answer: Significant figures are the digits in a measured number that carry meaning. Non-zero digits always count, zeros between them count, leading zeros never count and trailing zeros count only when there is a decimal point. Add and subtract by decimal places; multiply and divide by significant figures. A pH has as many decimal places as the [H⁺] has significant figures. The free significant figures calculator applies all these rules and shows why.
What significant figures are
A measurement is only as good as the instrument. A balance that reads 2.500 g tells you the mass to the nearest milligram. The digits 2, 5, 0 and 0 are the significant figures: every digit you are sure of plus the first uncertain one.
Sig figs keep a calculated answer from claiming more precision than the data.
The rules
Counting
- Non-zero digits are always significant: 4.56 has 3.
- Zeros between non-zero digits are significant: 1.007 has 4.
- Leading zeros are never significant: 0.0045 has 2.
- Trailing zeros are significant when there is a decimal point: 2.500 has 4 and 1500. has 4.
- Trailing zeros in a whole number with no decimal point are ambiguous: 1500 has 2, 3 or 4. Use scientific notation: 1.50 × 10³ has 3.
- In scientific notation only the mantissa counts: 6.022 × 10²³ has 4.
- Exact numbers (counts and defined values) have unlimited significant figures.
Rounding
Keep n digits from the first non-zero digit. If the first dropped digit is 5 or more, round up. 0.0045678 to 3 sig figs is 0.00457. 149.5 to 3 sig figs is 150., written 1.50 × 10² to be clear.
Calculations
- Addition and subtraction: round to the fewest decimal places.
- Multiplication and division: round to the fewest significant figures.
- Logarithms: the answer has as many decimal places as the number has significant figures.
- Antilogs (10^x, e^x): the answer has as many significant figures as x has decimal places.
- Carry extra digits through the steps and round only the final answer.
How to use the significant figures calculator
- Open the significant figures calculator. The Count sig figs tab is open.
- Type a number such as 0.004050, 1.20e3 or 1.20 × 10^3. Green digits are significant, grey digits are not and amber digits are ambiguous. The rules used are listed below.
- Tick This is an exact number (counted or defined) for values like 12 eggs.
- Open Round, type a number and the Significant figures you want.
- Open Calculate and type a whole calculation such as
(25.0 - 23.42) / 4.18or-log(2.5e-3). Each step shows the rule used and the rounded value. Tick Treat whole numbers without a decimal point as exact for counted numbers. - Open Scientific notation for scientific, E and engineering forms.
- Open Unit converter, pick a quantity (pressure, temperature, energy, length, concentration and more) and the From and To units. The answer keeps the significant figures of your value.
Worked examples
1. Counting zeros
0.004050: three leading zeros do not count. 4, 0, 5 and the final 0 do. Answer: 4 significant figures, or 4.050 × 10⁻³.
2. Addition
12.11 g + 18.0 g + 1.013 g = 31.123 g. 18.0 has one decimal place, so the answer is 31.1 g.
3. Multiplication
3.0 cm × 4.56 cm = 13.68 cm². 3.0 has two significant figures, so the answer is 14 cm².
4. Mixed operations
(25.0 − 23.42) ÷ 4.18. The subtraction gives 1.58, good to one decimal place, so it has 2 significant figures (1.6). Then 1.58 ÷ 4.18 = 0.378, rounded to 2 significant figures: 0.38.
5. pH from [H⁺]
[H⁺] = 2.5 × 10⁻³ M has 2 significant figures. pH = −log(2.5 × 10⁻³) = 2.602, written 2.60 with 2 decimal places. The 2 before the decimal point only gives the power of ten.
6. [H⁺] from pH
pH = 4.75 has 2 decimal places. [H⁺] = 10⁻⁴·⁷⁵ = 1.778 × 10⁻⁵, written 1.8 × 10⁻⁵ M.
7. Unit conversions
1.00 atm × 101.325 kPa/atm = 101.325, written 101 kPa (3 sig figs, because the factor is exact). 25.0 °C + 273.15 = 298.15, written 298.2 K (one decimal place, because 273.15 is exact).
Summary table
| Situation | Rule | Example | Answer |
|---|---|---|---|
| Counting | Leading zeros never count | 0.0045 | 2 sig figs |
| Counting | Trailing zeros count with a decimal point | 2.500 | 4 sig figs |
| Rounding | 5 or more rounds up | 2.675 to 3 s.f. | 2.68 |
| + and − | Fewest decimal places | 12.11 + 18.0 + 1.013 | 31.1 |
| × and ÷ | Fewest sig figs | 3.0 × 4.56 | 14 |
| log | Decimal places = sig figs of x | −log(2.5 × 10⁻³) | 2.60 |
| Antilog | Sig figs = decimal places of x | 10⁻⁴·⁷⁵ | 1.8 × 10⁻⁵ |
| Exact numbers | Do not limit the answer | 2 × 3.45 g (2 samples) | 6.90 g |
Common mistakes
- Counting leading zeros: 0.0025 has 2 significant figures, not 5.
- Dropping measured trailing zeros: 2.500 g is not the same as 2.5 g.
- Using the multiplication rule for addition: 100.0 + 0.123 = 100.1.
- Rounding after every step, which adds rounding error.
- Counting the digit before the decimal point in a pH as significant.
- Letting exact numbers, such as the 2 in 2H₂O, limit the answer.
- Writing a rounded large number as 150 000 instead of 1.50 × 10⁵.
Exam tips
- AP Chemistry: show units on every value and round only the final answer. The pH decimal place rule is a common question.
- A-level: give answers to the same number of sig figs as the least precise data, usually 3. Never give fewer than the data justify.
- JEE and NEET: multiple-choice options often differ only in rounding. Keep extra digits until the last step.
- In titration results, burette readings to 0.05 mL (two decimal places) set the precision of the titre.
Questions and answers
How many significant figures does 100 have?
It is ambiguous: 1, 2 or 3. Without a decimal point the trailing zeros may only be placeholders. Write 1 × 10² for one, 1.0 × 10² for two, or 100. or 1.00 × 10² for three.
Are zeros after a decimal point significant?
Trailing zeros after a decimal point are significant, so 2.50 has 3 significant figures. Leading zeros after a decimal point, as in 0.0025, are not significant.
What is the rule for significant figures in addition?
Round the answer to the same number of decimal places as the value with the fewest decimal places. 12.11 + 18.0 + 1.013 = 31.123, which rounds to 31.1.
What is the rule for significant figures in multiplication?
Round the answer to the same number of significant figures as the value with the fewest. 3.0 × 4.56 = 13.68, which rounds to 14 because 3.0 has two significant figures.
How many decimal places should a pH have?
As many as there are significant figures in the hydrogen ion concentration. [H+] = 2.5 × 10⁻³ M has 2 significant figures, so pH = 2.60.
Do exact numbers affect significant figures?
No. Counted values and defined conversion factors, such as 1 L = 1000 mL or 1 atm = 101.325 kPa, are exact and never limit the significant figures of the answer.
When should I round in a multi-step calculation?
Only at the end. Keep at least one or two extra digits in the intermediate values, and apply the rules step by step to know how many figures the final answer gets.