Significant figures are the digits in a measured number that carry real information about its precision. The core rules are: all nonzero digits are significant; zeros between them are significant; leading zeros are not; and trailing zeros count only when there is a decimal point. So 0.00320 has three significant figures (3, 2 and the final 0), while 4,507 has four.
When you calculate with measurements, the answer can be no more precise than the least precise input. Significant figures are a quick way to respect that. They are not a full uncertainty analysis, which metrology bodies such as NIST handle with explicit error estimates, but they stop you from reporting a calculator’s ten digits when your ruler only gave you three.
The rules for counting significant figures
- Nonzero digits are always significant. 7.46 has 3.
- Zeros between nonzero digits are significant. 4,507 has 4; 10.03 has 4.
- Leading zeros are never significant. They only locate the decimal point. 0.0045 has 2.
- Trailing zeros after a decimal point are significant. 3.200 has 4; 7.0 has 2. Someone wrote those zeros on purpose to show precision.
- Trailing zeros in a whole number without a decimal point are ambiguous. 1,200 might have 2, 3 or 4. Use scientific notation to remove the doubt.
| Number | Significant figures | Why |
|---|---|---|
| 4,507 | 4 | Zero sits between nonzero digits |
| 0.00320 | 3 | Leading zeros don’t count; trailing zero after the point does |
| 0.0005 | 1 | Only the 5 counts |
| 10.050 | 5 | Captive zeros and a trailing zero after the point |
| 7.0 | 2 | Trailing zero after the point |
| 100.0 | 4 | Decimal point makes every zero significant |
| 1,200 | 2 (ambiguous) | Trailing zeros, no decimal point |
| 1,200. | 4 | Trailing decimal point signals all digits count |
| 1.20 × 10³ | 3 | Scientific notation shows exactly 3 |
The significant figures counter applies these rules to any number you type and highlights which digits count.
Scientific notation removes ambiguity
In scientific notation, every digit in the coefficient is significant. That makes it the clearest way to state precision: 1.2 × 10³ has 2 significant figures, 1.20 × 10³ has 3 and 1.200 × 10³ has 4, though all three equal 1,200. See how to write scientific notation for the conversion steps.
How to round to significant figures
- Count from the first nonzero digit to the number of significant figures you want.
- Look at the next digit. If it is 5 or more, round up; if it is less than 5, leave the last kept digit alone.
- Replace dropped digits to the left of the decimal point with zeros; drop them entirely to the right.
0.048763 to 3 sig figs → first nonzero digit is 4 → keep 4, 8, 7 → next digit 6 rounds up → 0.0488
92,451 to 2 sig figs → keep 9, 2 → next digit 4 rounds down → 92,000 (better: 9.2 × 10⁴)
A note on exact fives. Some laboratories and standards, including ASTM E29, round a digit followed by exactly 5 (with nothing after it) to the nearest even digit, so 2.45 becomes 2.4 and 2.55 becomes 2.6. This “round half to even” convention prevents a slight upward bias over many roundings. School courses usually round 5 up. Follow whichever rule your class or lab specifies; the rounding to significant figures calculator handles the standard method.
Significant figures in calculations
Multiplication and division: fewest significant figures
The result keeps as many significant figures as the input with the fewest significant figures.
- 4.56 × 1.4 = 6.384 → 6.4 (1.4 has 2)
- 152.06 ÷ 0.24 = 633.58… → 630, better written 6.3 × 10² (0.24 has 2)
Addition and subtraction: fewest decimal places
Here precision is about place value, not digit count. The result keeps as many decimal places as the input with the fewest.
12.11 + 18.0 + 1.013 = 31.123
18.0 is known only to the tenths place → 31.1
This is why subtracting two close numbers can destroy precision: 12.6 − 11.38 = 1.22 is reported as 1.2, two significant figures, even though both inputs had three or four.
Mixed operations: round once at the end
Keep at least one or two extra digits through intermediate steps and round only the final answer, but track how many significant figures each step allows.
(12.6 − 11.38) × 3.1415
Step 1: 12.6 − 11.38 = 1.22 → limited to the tenths place, so 2 sig figs (carry 1.22)
Step 2: 1.22 × 3.1415 = 3.8326 → 2 sig figs → 3.8
Rounding 1.22 to 1.2 before multiplying would give 3.77, which rounds to the same 3.8 here, but early rounding can shift the last digit in longer chains.
Logarithms and pH
For a logarithm, the number of decimal places in the result equals the number of significant figures in the input. The digits before the decimal point only reflect the power of ten.
[H⁺] = 2.5 × 10⁻⁴ M (2 sig figs)
pH = −log(2.5 × 10⁻⁴) = 3.602… → 2 decimal places → 3.60
Going the other way, a pH of 7.40 (two decimal places) gives [H⁺] = 10−7.40 = 4.0 × 10⁻⁸ M, with two significant figures. The pH calculator and how to calculate pH apply this rule.
Exact numbers
Some numbers have no measurement uncertainty and therefore unlimited significant figures:
- Counted quantities: 12 eggs, 3 trials, 25 students.
- Defined conversion factors: 1 inch = 2.54 cm exactly, 1 foot = 12 inches, 1 kg = 1,000 g. NIST lists these definitions in its guide to SI units.
- Pure numbers in formulas: the 2 in 2πr, the ½ in ½mv².
Exact numbers never limit the result. Converting a measured 6.25 inches to centimeters gives 6.25 × 2.54 = 15.875, reported as 15.9 cm because 6.25 has 3 significant figures and 2.54 is exact. Similarly, the average of three readings, 2.31, 2.35 and 2.36, is 7.02 ÷ 3 = 2.34; the 3 is a count and does not reduce the answer to one significant figure.
Quick reference
| Situation | Rule |
|---|---|
| Counting | Nonzero and captive zeros count; leading zeros never; trailing zeros only with a decimal point |
| × and ÷ | Fewest significant figures |
| + and − | Fewest decimal places |
| log(x) | Decimal places in the answer = sig figs in x |
| 10ˣ | Sig figs in the answer = decimal places in x |
| Exact numbers | Ignore when deciding precision |
The significant figures calculator performs arithmetic with these rules built in, and the general rounding calculator rounds to decimal places instead. For very large or small results, the scientific notation calculator keeps the significant digits visible.
Frequently asked questions
Are zeros significant figures?
It depends on where they are. Zeros between nonzero digits are always significant (4,507 has 4). Leading zeros never are (0.0032 has 2). Trailing zeros are significant when the number has a decimal point (3.200 has 4) and ambiguous when it does not (3,200 could have 2, 3 or 4).
How many significant figures does 100 have?
As written, 100 is ambiguous: it has at least 1 significant figure and could have up to 3. Writing 100. with a trailing decimal point, or 1.00 × 10² in scientific notation, shows that all three digits are significant.
What is the rule for multiplying with significant figures?
Round the answer to the same number of significant figures as the measurement with the fewest. For example, 4.56 × 1.4 = 6.384, which is reported as 6.4 because 1.4 has only two significant figures.
What is the rule for adding with significant figures?
Round the answer to the same number of decimal places as the measurement with the fewest decimal places. For example, 12.11 + 18.0 + 1.013 = 31.123, which is reported as 31.1 because 18.0 is known only to the tenths place.
Do exact numbers limit significant figures?
No. Counted quantities (12 eggs) and defined conversion factors (1 inch = 2.54 cm exactly) have unlimited significant figures, so they never reduce the precision of a result.