What significant figures are
Significant figures, or sig figs, are the digits of a measured value that carry real information about it. A scale that shows 2.50 g tells you the mass to the hundredth of a gram; writing 2.5 g would throw that information away, and writing 2.500 g would claim more than the scale can tell. Counting significant figures is how you keep track of that precision when you write a number down and when you calculate with it.
The rules for counting
These are the rules as the general chemistry course on LibreTexts, listed in the sources below, states them:
- Nonzero digits always count. 4.52 has 3 significant figures.
- Captive zeros, zeros between nonzero digits, count. 1005 has 4.
- Leading zeros, in front of the first nonzero digit, never count. They only place the decimal point, so 0.0045 has 2.
- Trailing zeros, at the end of a number, count only if the number has a decimal point. 2.50 has 3, but 1500 has 2.
| Number | Sig figs | Rule |
|---|---|---|
| 4.52 | 3 | Nonzero digits always count |
| 1005 | 4 | Zeros between nonzero digits count |
| 0.0045 | 2 | Leading zeros do not count |
| 2.50 | 3 | Trailing zeros after a decimal point count |
| 1500 | 2 | Trailing zeros without a decimal point do not count |
| 1500. | 4 | A decimal point makes the trailing zeros count |
| 1.50e3 | 3 | Scientific notation shows exactly the counted digits |
In scientific notation every digit in front of the power of ten follows the same rules, and the power of ten itself never counts. That is why scientific notation is the clearest way to show precision: 4.50 × 10⁻³ has exactly the 3 digits you see. The scientific notation converter keeps your significant figures when it converts.
Why 1500 is ambiguous
A whole number with zeros at the end does not show whether those zeros were measured. 1500 could be a count rounded to the nearest hundred, with 2 significant figures, or a reading to the nearest unit, with 4. The usual rule plays it safe and counts the zeros as placeholders. There are three ways to say otherwise:
- A decimal point at the end: 1500. means all four digits count, a convention described in the Columbia University course notes in the sources.
- A bar over the last significant digit: 1500 has 3 significant figures; any zeros after the bar are placeholders. A physics handout from King Fahd University of Petroleum and Minerals describes this marking.
- Scientific notation: 1.50 × 10³ has 3 and 1.500 × 10³ has 4. This one cannot be misread.
The calculator understands the trailing decimal point and a typed bar, the combining overline character after a digit. For a plain 1500 it counts 2 and lets you pick how many zeros were measured.
Rounding to significant figures
To round to a number of significant figures, count from the first nonzero digit and look at the first digit you drop. If it is 5 or more, the last kept digit goes up by one; if it is less than 5, it stays. 0.004567 to 2 significant figures is 0.0046. A carry can move the leading digit: 9.996 to 3 figures is 10.0, still 3 figures. When the result is a whole number that ends in placeholder zeros, like 1499.7 to 3 figures, the calculator writes 1500 with a bar and 1.50 × 10³, so the precision is still visible. The rounding rule menu also offers half to even, which sends an exact 5 to the even neighbor; the rounding calculator explains all the rules.
Rounding can never add precision. If you round 2.5 to 4 significant figures, the calculator writes 2.500 but warns you, because those extra zeros would claim a precision the original number did not have.
Significant figures in calculations
The answer of a calculation cannot be more precise than the numbers that went into it. Two rules cover the four basic operations:
- Addition and subtraction: round the answer to the rightmost decimal place that every number has significant digits in. 12.11 + 0.3 = 12.41, and since 0.3 is only known to the tenths, the answer is 12.4. The number of significant figures can grow or shrink: 99.5 + 1.2 = 100.7 has 4.
- Multiplication and division: give the answer as many significant figures as the number with the fewest. 2.50 × 1.2 = 3, and since 1.2 has 2 significant figures, the answer is 3.0.
Exact numbers, counted or defined rather than measured, have no uncertainty and count as having infinitely many significant figures, as the Northwestern Polytechnic chemistry guide puts it, so they never limit the answer. For a calculation with several steps, keep every digit until the end and round once, since rounding at every step can change the final digit.
How to use the significant figures calculator
- 1Choose Count, Round or Calculate.
- 2Type the number exactly as it was written or measured, including zeros at the end and the decimal point. Scientific notation such as 6.02e23 works.
- 3Read the count and see each digit marked as significant or not, with the rule that applies. To round, pick the number of significant figures; to calculate, enter two numbers and an operation.
- 4Copy the result, which includes scientific notation, or copy a link that opens the calculator with your number.
Sources
- Expressing Numbers: Significant Figures, CHE 124 General Chemistry, University of Illinois Springfield, Chemistry LibreTexts: counting rules, rules for calculations and rounding
- Significant Figures, Chemistry guide, Northwestern Polytechnic Learning Commons: exact numbers
- Significant Figures, Frontiers of Science, Columbia University: a trailing decimal point marks significant zeros
- Identifying significant digits, physics class handout, King Fahd University of Petroleum and Minerals: a bar over the last significant digit
Frequently asked questions
How many significant figures does 1500 have?
By the usual rule, 2: trailing zeros in a whole number without a decimal point do not count. The number itself cannot tell whether the zeros were measured, though. If they were, write 1500. with a decimal point for 4 figures, or use scientific notation: 1.50 × 10³ has 3 and 1.500 × 10³ has 4.
Do leading zeros count as significant figures?
No. Zeros in front of the first nonzero digit only place the decimal point. 0.0045 has 2 significant figures, the 4 and the 5, which you can see in scientific notation: 4.5 × 10⁻³.
Do zeros after the decimal point count?
Trailing zeros after a decimal point count, because writing them says the value was measured to that place. 2.50 has 3 significant figures and 2.500 has 4. Zeros between the decimal point and the first nonzero digit, as in 0.05, are leading zeros and do not count.
How do significant figures work in addition and multiplication?
For addition and subtraction, round the answer to the last decimal place that every number has: 12.11 + 0.3 = 12.41, rounded to the tenths gives 12.4. For multiplication and division, give the answer as many significant figures as the number with the fewest: 2.50 × 1.2 = 3, written as 3.0 with 2 figures.
Do exact numbers limit significant figures?
No. Counted numbers, like 3 trials or 12 eggs, and defined values are exact, so they have no uncertainty and do not limit the result. Mark an operand as an exact number in the calculator and the rule only looks at the other one.
When should I round in a calculation with several steps?
Keep all digits through the intermediate steps and round only the final answer. Rounding after every step can change the last digit of the answer. The calculator shows the unrounded result next to the rounded one for that reason.