Significant figures are the digits in a number that carry meaningful information about its precision. Using the right number of significant figures tells your reader how well you actually know a value.
This guide covers the rules for counting significant figures, how to round, how precision carries through multiplication, addition and logarithms, and how to match a result to its uncertainty, with worked examples throughout.
The Significant Figure Rules in Brief
- All non-zero digits count, and so do zeros between them.
- Leading zeros never count; trailing zeros after a decimal point always do.
- When multiplying or dividing, round to the fewest significant figures in the data.
- When adding or subtracting, round to the fewest decimal places.
- Keep extra digits during working and round only the final answer.
How to Count Significant Figures
Five rules cover almost every number you will meet. The tricky cases are all about zeros.
| Rule | Example | Significant figures |
|---|---|---|
| Non-zero digits are significant | 247 | 3 |
| Zeros between non-zero digits are significant | 1002 | 4 |
| Leading zeros are not significant | 0.0045 | 2 |
| Trailing zeros after a decimal point are significant | 2.50 | 3 |
| Trailing zeros in a whole number are ambiguous | 1500 | 2, 3 or 4 |
Leading zeros only fix the position of the decimal point. Writing 0.0045 m as 4.5 mm makes this obvious: the precision has not changed, so neither has the count.
Trailing zeros after a decimal point show precision you measured. Writing 2.50 g says the balance read to the nearest 0.01 g; writing 2.5 g says it did not.
Scientific Notation Removes Ambiguity
A value such as 1500 m could be known to the nearest metre or the nearest hundred metres. Scientific notation makes the precision explicit, because every digit in the leading number counts.
| Written as | Significant figures |
|---|---|
| 1.5 × 10³ m | 2 |
| 1.50 × 10³ m | 3 |
| 1.500 × 10³ m | 4 |
| 3.40 × 10⁻⁴ m (0.000340 m) | 3 |
Use scientific notation whenever a large whole number has trailing zeros that matter, or a small number has many leading zeros.
Exact Numbers Have Unlimited Precision
Some numbers are not measurements, so they never limit the significant figures of a result.
- Counted values: 20 oscillations, 3 trials, 12 samples.
- Defined conversions: 1 km = 1000 m, 1 inch = 2.54 cm exactly.
- Pure numbers in formulas: the 2 in 2πr or the ½ in ½mv².
Physical constants are different. A value of g taken as 9.81 m/s² has three significant figures and does count in the rule for your answer.
When you divide a timed total by a count, the count does not limit the result. Timing 20 oscillations as 28.4 s gives a period of 28.4 / 20 = 1.42 s, with three significant figures from the timing, because 20 is exact.
How to Round to a Set Number of Significant Figures
Find the last digit you will keep, look at the digit after it, and round up if that digit is 5 or more. Replace dropped digits to the left of the decimal point with zeros.
| Number | Rounded to | Result |
|---|---|---|
| 2.3456 | 3 significant figures | 2.35 |
| 0.0078249 | 2 significant figures | 0.0078 |
| 145,500 | 3 significant figures | 146,000 (1.46 × 10⁵) |
| 9.996 | 3 significant figures | 10.0 |
The last row is a common trap. Rounding 9.996 carries all the way up to 10.0, and the trailing zero is written to show three significant figures.
Some fields round an exact half to the nearest even digit instead, so 6.025 becomes 6.02 rather than 6.03. Use whichever convention your course teaches, and apply it consistently.
Multiplying and Dividing
For multiplication and division, the result has as many significant figures as the least precise value used.
Worked example: density. A sample has a mass of 25.12 g and a volume of 9.8 cm³. Find its density.
- ρ = m / V = 25.12 g / 9.8 cm³ = 2.5633 g/cm³ on the calculator.
- 25.12 has four significant figures; 9.8 has two.
- Round to two: ρ = 2.6 g/cm³.
The precise mass cannot rescue the rough volume. To improve the result, measure the volume more precisely.
Unit conversions follow the same logic. Converting 72 km/h to m/s means dividing by 3.6, which is exact (1000 m per 3600 s), so 72 km/h becomes 20 m/s with two significant figures; write it as 2.0 × 10¹ m/s if that precision must be unmistakable. The conversion factor never reduces the precision of your measurement.
Adding and Subtracting
For addition and subtraction, the rule changes: the result keeps as many decimal places as the value with the fewest decimal places.
Worked example: total mass. 12.11 g + 18.0 g + 1.013 g = 31.123 g on the calculator.
18.0 has one decimal place, the fewest, so the sum is 31.1 g.
Subtraction can destroy precision. 10.52 m − 10.48 m = 0.04 m: both values have four significant figures, but the difference has only one. This is why small differences between large measurements are so uncertain.
Mixed Operations and Logarithms
In a calculation with several steps, apply the rules step by step to track precision, but keep the unrounded values in your calculator. Round only the final answer.
Worked example: mixed operations. Calculate (12.5 − 10.2) × 3.14159.
- Subtraction: 12.5 − 10.2 = 2.3, which has one decimal place and so two significant figures.
- Multiplication: 2.3 × 3.14159 = 7.2257 on the calculator.
- The least precise factor has two significant figures, so the answer is 7.2.
For logarithms, the number of decimal places in the log equals the number of significant figures in the original value. So [H⁺] = 4.2 × 10⁻⁴ mol dm⁻³ (two significant figures) gives pH = 3.38 (two decimal places). The digit before the decimal point only reflects the power of ten.
Significant Figures and Uncertainty
In lab work, uncertainty sets the precision, not a counting rule. Round the uncertainty to one significant figure (or two if it starts with a 1), then round the value to the same decimal place.
Worked example. A calculation gives g = 9.7183 m/s² with an uncertainty of 0.0721 m/s².
- Round the uncertainty to one significant figure: 0.07 m/s².
- Round the value to the same decimal place, the hundredths: 9.72 m/s².
- Report g = 9.72 ± 0.07 m/s².
Writing 9.7183 ± 0.07 claims precision you do not have. Writing 9.7 ± 0.07 throws precision away. Value and uncertainty should always end at the same decimal place.
Rounding in Software
Tools do not always round the way you expect. Python's round() rounds halves to the nearest even digit, so round(2.5) gives 2. To format a value to significant figures in Python, use f"{x:.3g}", which may switch to scientific notation for large or small values. In Excel, =ROUND(A1, n-1-INT(LOG10(ABS(A1)))) rounds A1 to n significant figures.
Common Significant Figure Mistakes
- Copying every digit from the calculator into the final answer.
- Rounding at each step, which builds up error.
- Counting leading zeros as significant.
- Using the multiplication rule for a sum, or the reverse.
- Letting exact numbers or counts limit the answer.
- Reporting a value and its uncertainty to different decimal places.
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Frequently Asked Questions
Three. The leading zeros do not count, but the trailing zero after the decimal point does.
It depends on position. Zeros between non-zero digits and trailing zeros after a decimal point are significant; leading zeros are not; trailing zeros in a whole number are ambiguous unless you use scientific notation.
For multiplication and division, match the least precise value. For addition and subtraction, match the fewest decimal places. In lab work, let the uncertainty decide.
No. Counted values and defined conversions, such as 1 km = 1000 m, are exact and never limit your answer.
No. Keep full precision in your calculator and round only the final answer, so rounding errors do not build up.
The number of decimal places in the logarithm equals the number of significant figures in the original value.
Significant figures count meaningful digits from the first non-zero digit. Decimal places count digits after the decimal point. 0.0450 has three significant figures but four decimal places.
They tell the reader how precise a measurement is. Reporting 2.5633 g/cm³ from a volume read to two significant figures claims a precision you never had, and markers deduct for it.
Use scientific notation: 2.00 × 10³. Written as 2000 or 2,000, the number of significant figures is unclear.