Rounding
Significant Figures Rules
The sig fig rules decide which digits in a measurement are meaningful. Non-zero digits always count, zeros between them count, leading zeros never count, and trailing zeros count only when there is a decimal point. For calculations, multiplying and dividing keep the fewest significant figures, while adding and subtracting keep the fewest decimal places. This guide covers all of these rules with worked examples, a counting table and the mistakes to avoid.
What significant figures are, and why the rules exist
A measurement is only as precise as the instrument that made it. If a ruler is marked in millimeters, you can read a length as 12.4 cm but not as 12.4371 cm. The significant figures (sig figs) in a number are the digits that carry real information about the measurement: all the digits you are sure of, plus one final estimated digit.
Writing 12.4 cm says “accurate to the nearest tenth.” Writing 12.40 cm says “accurate to the nearest hundredth,” a claim ten times more precise. The two numbers have the same value but tell a different story about how carefully they were measured. Sig fig rules are the shared convention that lets a reader decode that story.
To round a number to a given count, or check how many sig figs a number has, the significant figures calculator does it instantly and flags ambiguous trailing zeros. The rules below explain what it is doing.
The five rules for counting significant figures
Rule 1: Non-zero digits are always significant
Every digit from 1 to 9 counts, wherever it sits. 48,923 has 5 sig figs and 2.7 has 2.
Rule 2: Zeros between non-zero digits are significant
These are sometimes called captive or sandwiched zeros. 3.005 has 4 sig figs, and 1,070 has at least 3 (the 1, the 0 and the 7). A zero trapped between measured digits was itself measured.
Rule 3: Leading zeros are never significant
Zeros before the first non-zero digit only show where the decimal point goes. 0.00821 has 3 sig figs, not 6. Change the unit from grams to milligrams and the same mass becomes 8.21 mg, with no zeros at all; the precision did not change, so those zeros were never carrying information.
Rule 4: Trailing zeros after a decimal point are significant
A zero at the end of a decimal is written only because it was measured. 5.6900 has 5 sig figs, and 0.0500 has 3 (the 5 and the two zeros after it; the leading zeros still do not count).
Rule 5: Trailing zeros in a whole number with no decimal point are ambiguous
1200 could be a measurement good to the nearest hundred (2 sig figs), the nearest ten (3) or the nearest unit (4). By the usual classroom convention, these zeros are not counted, so 1200 is treated as 2 sig figs. To show they are significant, write a decimal point after the number (1200. has 4) or, better, use scientific notation, which removes the ambiguity completely.
Counting sig figs: a reference table
| Number | Sig figs | Rule that decides it |
|---|---|---|
| 48,923 | 5 | All non-zero (rule 1) |
| 3.005 | 4 | Captive zeros count (rule 2) |
| 0.00821 | 3 | Leading zeros do not count (rule 3) |
| 0.00450 | 3 | Leading zeros out, trailing zero in (rules 3 and 4) |
| 5.6900 | 5 | Trailing zeros after a decimal count (rule 4) |
| 1.050 | 4 | Captive and trailing zeros both count |
| 100.0 | 4 | Decimal point makes every zero count |
| 1200 | 2 (ambiguous) | Trailing zeros, no decimal point (rule 5) |
| 1200. | 4 | Decimal point shown |
| 30,040 | 4 (ambiguous) | Captive zeros count; the final zero is ambiguous |
| 100 | 1 (ambiguous) | Trailing zeros, no decimal point |
| 6.02 × 10²³ | 3 | Only the digits before × 10 count |
| 7.0 × 10⁻⁴ | 2 | Trailing zero after a decimal counts |
Notice that the size of a number has nothing to do with its sig figs. 0.00450 and 6.02 × 10²³ both have 3. Sig figs measure precision, not magnitude. The place value guide is useful here: sig figs start at the first non-zero place value and run to the last measured one.
How to round to a number of significant figures
- Start at the first non-zero digit and count from the left until you have the number of sig figs you need.
- Look at the next digit to the right. If it is 5 or more, round the last kept digit up; if it is 4 or less, leave it.
- Replace dropped digits to the left of the decimal point with zeros so the number keeps its size. Drop any dropped digits to the right of it.
| Number | Round to | Result |
|---|---|---|
| 45,678 | 3 sig figs | 45,700 |
| 0.004567 | 2 sig figs | 0.0046 |
| 7.0349 | 3 sig figs | 7.03 |
| 1.2345678 | 4 sig figs | 1.235 |
| 123.45 | 2 sig figs | 120 |
| 2.9951 | 3 sig figs | 3.00 |
The last row is the one people get wrong. Rounding 2.9951 to 3 sig figs carries all the way up to 3.00, and the two zeros must stay: writing just “3” would show only 1 sig fig. The rounding calculator shows a number rounded at every place side by side, and the guide to rounding to decimal places covers the place-based version of the same idea.
Sig fig rules for calculations
A calculated answer cannot be more precise than the measurements that went into it. Two different rules enforce that, depending on the operation.
Multiplying and dividing
Count the sig figs in each measurement. The answer gets as many as the least precise one.
Example: 4.56 × 1.4
Example: (8.314 × 298) ÷ 101.3
Adding and subtracting
Here sig figs are not counted at all. Look at decimal places instead: the answer stops at the same decimal place as the measurement with the fewest.
Example: 12.11 + 18.0 + 1.013
18.0 is only known to the nearest tenth, so the hundredths and thousandths of the total are meaningless.
Example: 104.6 − 98.27
Both inputs had 4 sig figs, but the answer has only 2. Subtracting nearly equal measurements loses precision, and the decimal-place rule shows exactly how much.
Exact numbers have unlimited sig figs
Counted quantities (12 eggs, 3 trials) and defined conversion factors (1 inch = 2.54 cm exactly, 12 inches = 1 foot) are not measurements, so they never limit the answer.
Example: converting 6.38 inches to centimeters
The multiplication calculator and division calculator give the full unrounded result to apply these rules to.
Multi-step calculations: round once, at the end
When a calculation has several steps, keep every digit (or at least two extra) until the final answer, and track how many sig figs each step would allow. Rounding in the middle builds up error.
Example: volume of a 1.24 cm × 1.24 cm × 4.75 cm block
The early rounding moved the final digit by 2. Each step’s rounding error carries into the next one.
For mixed operations, apply the rule for each step in order. For (12.11 + 18.0) × 1.4, the sum 30.11 is good to one decimal place (3 sig figs, 30.1), then multiplying by 1.4 limits the final answer to 2 sig figs: 30.11 × 1.4 = 42.154, which rounds to 42.
In a lab report, sig fig errors often show up next to percent error. A percent error can never be stated more precisely than the measurements behind it.
Common sig fig mistakes
- Counting leading zeros. 0.0042 has 2 sig figs, not 5.
- Dropping meaningful trailing zeros. A balance reading 3.50 g has 3 sig figs. Writing 3.5 g throws away a measured digit.
- Using the sig fig rule for addition.
12.11 + 18.0 + 1.013is limited by decimal places (answer 31.1), not by the 3 sig figs in 18.0. - Rounding at every step. Carry extra digits and round once, as the block-volume example shows.
- Letting exact numbers limit the answer. Multiplying by 2 in
2πror by 100 to make a percentage never reduces the sig figs. - Losing the size of the number. 45,678 to 2 sig figs is 46,000, not 46.
- Trusting the calculator display. A calculator shows every digit it can. Ten digits on the screen does not mean ten digits of precision in your answer.
Frequently asked questions
What are the 5 rules of significant figures?
Non-zero digits are always significant. Zeros between non-zero digits are significant. Leading zeros are never significant. Trailing zeros after a decimal point are significant. Trailing zeros in a whole number with no decimal point are ambiguous and are usually not counted, so 1200 is treated as 2 sig figs unless written as 1200. or in scientific notation.
How many sig figs are in 100?
By the usual convention, 100 has 1 significant figure, because trailing zeros in a whole number without a decimal point are not counted. The value is ambiguous, though. Written as 100. it has 3 sig figs, and as 100.0 it has 4. Scientific notation such as 1.00 × 10² makes the intended precision clear.
How many significant figures are in 0.00450?
0.00450 has 3 significant figures: the 4, the 5 and the final 0. The three zeros at the start are leading zeros, which only place the decimal point and never count. The trailing zero after the 5 counts because it sits after a decimal point. In scientific notation it is 4.50 × 10⁻³.
Do zeros count as significant figures?
It depends on where they are. Zeros between non-zero digits always count, as in 3.005. Zeros at the end of a decimal count, as in 5.60. Zeros at the start of a number never count, as in 0.008. Zeros at the end of a whole number, as in 1200, are ambiguous and usually not counted.
What is the sig fig rule for multiplication?
The answer has as many significant figures as the measurement with the fewest. For 4.56 × 1.4, 4.56 has 3 sig figs and 1.4 has 2, so the product 6.384 is rounded to 2 sig figs: 6.4. The same rule applies to division. Exact numbers, such as counted items or defined conversions, do not limit the answer.
What is the sig fig rule for addition and subtraction?
Round the answer to the fewest decimal places among the numbers you added or subtracted. For 12.11 + 18.0 + 1.013 = 31.123, 18.0 has only one decimal place, so the answer is 31.1. You count decimal places here, not significant figures, which is why subtraction can leave an answer with fewer sig figs than its inputs.
How many sig figs does 1.050 have?
1.050 has 4 significant figures. The 1 and the 5 are non-zero, the zero between them is a captive zero, and the final zero is a trailing zero after a decimal point. All four digits count, so the number claims a precision of one thousandth.
Should I round in the middle of a calculation?
No. Keep all digits, or at least two extra, through every step and round only the final answer. Rounding early adds error: computing 1.24 × 1.24 × 4.75 in one go gives 7.30, but rounding the first product to 1.54 before multiplying gives 7.32.
Calculators for this
Every one of these shows the working, so you can check the method as well as the answer.
More guides
https://fourfunctioncalculator.com/significant-figures-rules/