What the digits in a number say
When you write a number, every digit you include makes a claim. Writing 12.3 mm suggests you know the length to about a tenth of a millimetre. Writing 12.3456 mm suggests you know it to a hundred-thousandth of a millimetre. If the measuring tool cannot support that, the extra digits are false precision: they look careful but carry no real information, and they can mislead a reader into trusting the result more than they should.
Significant figures are the digits that carry meaning. They are a simple, if imperfect, way of showing how well a value is known.
Counting significant figures
- All non-zero digits are significant: 4.82 has three.
- Zeros between non-zero digits are significant: 5.007 has four.
- Leading zeros are not significant. They only place the decimal point: 0.0042 has two.
- Trailing zeros after a decimal point are significant: 2.50 has three, and the zero shows that the value is known to the hundredths.
- Trailing zeros in a whole number are ambiguous: 1,500 might have two, three or four. Scientific notation removes the doubt, as 1.5 × 10^3, 1.50 × 10^3 or 1.500 × 10^3.
Exact numbers, such as a count of 12 bolts or a defined conversion factor, have no uncertainty and do not limit the result.
Combining numbers
Multiplication and division. The result should have about as many significant figures as the least precise input.
12.5 × 3.2 = 40.0 on a calculator. The factor 3.2 has two significant figures, so report 40, written better as 4.0 × 10^1 to make the two figures clear.
Addition and subtraction. Here the rule concerns decimal places, not total digits. The result is limited to the least precise decimal place.
12.11 + 0.3 = 12.41. The value 0.3 is known to one decimal place, so report 12.4.
These are rules of thumb. They are a quick way to avoid claiming too much, but a formal treatment of uncertainty, as outlined in measurement uncertainty, is more reliable when the answer matters.
Rounding rules
Round to the digit you want to keep and look at the next one. If it is below 5, leave the kept digit; if it is above 5, increase it by one. A trailing 5 with nothing after it is a special case: many people round up, while some conventions round to the nearest even digit to avoid bias across many values. Whichever you choose, apply it consistently and follow any rule your organisation or standard specifies.
Examples: 3.14159 to three significant figures is 3.14. 0.0045678 to two significant figures is 0.0046. 2,749 to two significant figures is 2,700, or 2.7 × 10^3.
Round at the end, not in the middle
Rounding intermediate values introduces errors that can grow through a calculation. Keep a few extra digits, often called guard digits, until the final step.
Illustration: compute (7.6 ÷ 3.0) × 3.0.
- Without intermediate rounding: 7.6 ÷ 3.0 = 2.5333…, and × 3.0 gives 7.6, as expected.
- With rounding to two figures partway: 2.5333 becomes 2.5, and 2.5 × 3.0 = 7.5.
The result is off by 0.1 purely because of the early rounding. In longer chains the effect can be larger. In engineering spreadsheets, avoid copying displayed, rounded values into other cells; let the cell hold the full value and change only the display format.
Display versus stored value
Spreadsheets and calculators show a limited number of digits but store many more. This leads to puzzling cases where displayed numbers do not seem to add up: three values displayed as 0.3 might sum to display as 1.0, while each is stored as 0.33. That is not an error, but it can look like one in a table. If a table must sum visibly, say so in a note, or round the stored values deliberately and consistently.
How many digits should I report?
A practical approach:
- Estimate the uncertainty in the result. Even a rough figure helps.
- Round the uncertainty to one or two significant figures.
- Round the value to the same decimal place as the uncertainty.
For example, a measured length of 25.3478 mm with an uncertainty of about 0.2 mm is best reported as 25.3 ± 0.2 mm.
When no uncertainty is known, use the least precise input as a guide. An average speed from a distance measured to the nearest 10 m over a time measured to a tenth of a second should not be quoted to the nearest millimetre per second.
Where false precision causes trouble
- Unit conversions. Converting 5 inches to millimetres gives 127 exactly, because the conversion is exact, but 5 miles is not “8046.72 m” if the 5 was approximate. Say “about 8 km”. The approach to unit conversion matters, but the digits you keep matter too.
- Averages. An average of values measured to the nearest unit should not usually be quoted with many decimals. Quoting a mean and standard deviation to sensible digits is part of good practice.
- Calculator output. A calculator showing eight digits is not telling you that all eight mean something.
- Technical reports. Readers use the digits to judge how much to trust a figure. In technical reports, matching digits to the real precision builds credibility.
- Specifications and tolerances. A tolerance of ±0.1 on a dimension implies a quoted value like 25.0, not 25.00000.
Not too few, either
Throwing away digits too soon is also a mistake. If a design margin is thin, rounding a result up or down by a few percent could hide the problem. Keep enough digits to make the decision, and show the rounded figure only at the end. Where a value feeds into safety or compliance calculations, follow the rounding rules of the relevant standard, which vary by industry and country.
Quick summary of habits
- Match the digits to the evidence.
- Use scientific notation to make trailing zeros unambiguous.
- Keep extra digits during calculation; round once at the end.
- State an uncertainty when you can.
- Be consistent, and say which convention you used.