Significant Figures Calculator

About Significant Figures Calculator

The Five Rules for Significant Figures

Every digit in a number is handled by exactly one of five rules. The tool applies them in order, left to right, and tells you which rule it used for each digit.

These rules are codified in the IUPAC Green Book and in NIST SP 811, which is the U.S. reference for measurement precision and uncertainty reporting. UK A-level specifications (AQA, OCR, Edexcel) and GCSE combined science assessments use the same five rules.

Counting Examples

The most common mistake is counting leading zeros. They locate the decimal point, they do not add precision.

How Do Sig Figs Work in Arithmetic?

Addition and subtraction use decimal places; multiplication and division use sig figs. The rule is that a calculation cannot produce more precision than its least precise input.

Worked example (multiplication): 3.24 x 2.1

• 3.24 has 3 sig figs, 2.1 has 2 sig figs
• Raw result: 6.804
• Round to 2 sig figs (the smaller of 3 and 2): 6.8

Worked example (addition): 103.25 + 0.4

• 103.25 has 2 decimal places, 0.4 has 1 decimal place
• Raw result: 103.65
• Round to 1 decimal place: 103.7

Worked example (division): 14.7 / 3.1415

• 14.7 has 3 sig figs, 3.1415 has 5 sig figs
• Raw result: 4.678975...
• Round to 3 sig figs (the smaller of 3 and 5): 4.68

Chain calculations: Keep one or two extra "guard digits" through intermediate steps and only round to the correct sig figs at the very end. Rounding at each step compounds error. This is the standard practice taught in analytical chemistry courses and recommended by the ACS Style Guide.

How to Round to N Significant Figures

Count from the first non-zero digit, look at the digit immediately after the last one you're keeping, and apply standard half-up rounding.

To round 0.004735 to 2 sig figs:

1. Count from the first non-zero digit: 4, 7 (two sig figs)
2. Look at the next digit (3): it is less than 5, so round down
3. Result: 0.0047

To round 123,456 to 3 sig figs:

1. Count the first three digits: 1, 2, 3
2. Look at the next digit (4): less than 5, round down
3. Result: 123,000

To round 0.1975 to 2 sig figs:

1. Count the first two sig figs: 1, 9
2. Look at the next digit (7): 5 or more, so round up
3. 9 becomes 10, which carries over: 0.20
4. The trailing zero is kept to preserve the 2 sig figs (0.2 would only be 1)

What Is the Difference Between Sig Figs and Decimal Places?

Sig figs measure total precision; decimal places measure precision after the point. The number 0.0045 has 2 sig figs but 4 decimal places. The number 1200 has (by convention) 2 sig figs but 0 decimal places.

This matters because the arithmetic rules key on different things. When a chemist adds masses measured on the same balance, decimal places are what matter because the balance's precision is fixed per reading. When a chemist multiplies concentrations by volumes, sig figs are what matter because each measurement has its own relative precision.

Exact Numbers and Defined Constants

Exact numbers have unlimited sig figs and never limit the precision of a calculation. This includes:

• Counted integers - 12 eggs, 3 moles, 6 trials
• Unit conversion factors by definition - 1 inch = 2.54 cm (exact, per the 1959 International Yard agreement), 1 minute = 60 seconds
• Defined constants since 2019 - the speed of light (299,792,458 m/s), Avogadro constant (6.02214076 x 10²³), Planck constant (6.62607015 x 10⁻³⁴ J·s), and elementary charge are all exact by the SI redefinition that took effect 20 May 2019, per NIST

Measured constants like g (9.81 m/s²) or the gas constant R (8.314 J/mol·K) do have a finite precision and should be treated as inputs with their stated sig figs.

Common Mistakes Students Make

• Counting leading zeros. 0.00042 has 2 sig figs, not 5.
• Dropping trailing zeros after a decimal. 2.00 is 3 sig figs and must stay as 2.00, not 2.
• Rounding too early. Doing 3.4 x 2.1 x 1.9 one step at a time and rounding to 2 sig figs between steps gives a different answer than rounding once at the end.
• Mixing the arithmetic rules. Addition is by decimal places, multiplication is by sig figs. Swapping them is the single most common error on A-level and AP chemistry papers.
• Applying sig figs to exact counts. "I have 3 beakers" does not limit the result to 1 sig fig - 3 is exact.
• Half-even vs half-up. Most schools teach half-up (round 0.5 away from zero). Some engineering contexts and IEEE 754 use half-even ("banker's rounding"). This tool uses half-up, which matches JavaScript's Math.round for positive numbers.

Where Do Sig Fig Conventions Come From?

The modern convention was formalised in the mid-20th century as laboratory instruments became precise enough that "about five" was no longer acceptable in a scientific paper. The 1960 General Conference on Weights and Measures (CGPM) established the modern International System of Units, and NIST (then the National Bureau of Standards) published the first editions of its Special Publication 811 style guide to standardise how precision should be communicated. Most university textbooks in the UK and US now treat the five-rule system as fixed, with the ambiguous-trailing-zero convention resolved by scientific notation.

In published papers, you will also see two related conventions. The first is explicit uncertainty, written as 5.23 ± 0.04, which supersedes sig figs when the uncertainty is known. The second is the "underline" or "overbar" notation used in older chemistry texts to mark the last significant digit in an ambiguous number (e.g. 1500 with the second zero underlined to mean 3 sig figs). Modern practice is to switch to scientific notation instead.

Sig Figs in Lab Reports and Exams

Marks are lost for sig fig errors in A-level chemistry (AQA, OCR, Edexcel) and AP Chemistry exams. The most common deductions are:

• Too many sig figs in the answer. If the mass was measured to 3 sig figs (2.34 g) and the volume to 2 sig figs (25 mL), the concentration cannot be reported to 4 sig figs.
• Too few sig figs. Rounding 0.0234 mol/L to 0.02 mol/L throws away real precision from the measurement.
• Inconsistent sig figs across a table. If one row reports a density to 4 sig figs and another to 2, the reader can't tell which is the measurement precision and which is a typo.
• Rounding before averaging. When computing mean and standard deviation, keep guard digits in each value and round only at the final reported statistic.

A practical rule from the UK Royal Society of Chemistry lab handbook: report your final answer with the same number of sig figs as the least precise input measurement, no more and no less.

Scientific Notation and Sig Figs

Scientific notation is the cleanest way to show exactly how many sig figs a measurement has. It removes the ambiguity of trailing zeros in whole numbers entirely.

For calculating how far off a measurement is from a known value, the percent error calculator computes absolute and relative error with the right sig figs. For working with scientific notation directly, the scientific notation calculator handles conversions and arithmetic. When results involve logarithms, the log calculator follows the rule that the number of decimal places in a log equals the sig figs of the original number.

All calculations run in your browser. No data is sent anywhere.

Sources

• NIST SP 811 - Guide for the Use of the International System of Units
• IUPAC Green Book - Quantities, Units and Symbols in Physical Chemistry
• NIST - SI Redefinition (2019 exact constants)
• ACS Guide to Scholarly Communication - numerical data reporting