A measurement is never exact. A ruler marked in millimeters cannot tell you that a rod is 7.4623 cm long, and writing that number would claim a precision the instrument does not have. Significant figures are the digits of a number that carry real information: the digits you are sure of plus one estimated digit. The rules below tell you how to count them, how to round, and how many digits to keep after a calculation.
Contents
- What significant figures are
- Rules for counting significant figures
- How to round
- Significant figures in calculations
- Worked examples
- Significant figures and scientific notation
- Common mistakes
- Practice problems
- Frequently asked questions
- Related calculators
- See also
What significant figures are
When you read an instrument, you read every digit that is clearly marked and then estimate one more by eye. Those digits together are the significant figures. In a reading of 7.46 cm on a millimeter ruler, the 7 and the 4 are certain, and the 6 is an estimate between two marks. The reading has 3 significant figures.
Reading a ruler: certain digits and one estimate
Change the length of the rod with the slider, and switch between a ruler marked in millimeters and one marked only in centimeters. You read every digit the marks give you, then estimate one more.
Rules for counting significant figures
- Nonzero digits are always significant. In 3,472 all four digits count.
- Zeros between nonzero digits are significant. In 4,508 the zero counts, so there are 4.
- Leading zeros are not significant. They only show where the decimal point is. In 0.00450 the first three zeros do not count.
- Trailing zeros after a decimal point are significant. In 0.00450 the last zero counts, because writing it means you measured that place. 0.00450 has 3 significant figures.
- Trailing zeros in a whole number without a decimal point are ambiguous. In 1,200 the zeros may or may not be measured. To remove the doubt, write 1,200. (with a point, 4 figures) or use scientific notation: $latex 1.2\times 10^3$ has 2 and $latex 1.200\times 10^3$ has 4.
- Exact numbers have unlimited significant figures. A count (12 eggs) or a definition (100 cm in 1 m) does not limit the precision of an answer.
| Number | Significant figures | Why |
|---|---|---|
| 4,508 | 4 | Zero between nonzero digits |
| 0.0306 | 3 | Leading zeros do not count, the inner zero does |
| 2.500 | 4 | Trailing zeros after the point count |
| 6.0 | 2 | The zero shows that tenths were measured |
| 1,200 | 2 (ambiguous) | Trailing zeros without a point |
| $latex 3.00\times 10^8$ | 3 | All digits of the coefficient count |
Type any number into the counter below. Blue digits count, grey digits are place holders, and dashed orange digits are the ambiguous ones.
Significant figure counter and rounder
Type a number. E-notation such as 1.20e3 works too. Blue digits count, grey digits only hold the place, and dashed orange digits are ambiguous.
How to round
To round to n significant figures, look at the digit right after the last digit you keep.
- If it is 5 or more, increase the last kept digit by 1.
- If it is 4 or less, leave the last kept digit as it is.
Replace any dropped digits before the decimal point with zeros. Look only at that one digit: 2.4496 rounded to 2 figures is 2.4, because the next digit is 4. Do not round in steps (2.4496 to 2.450 to 2.45 to 2.5 would give a wrong answer). The tool above rounds exactly this way.
Significant figures in calculations
Adding and subtracting: the answer keeps as many decimal places as the number with the fewest decimal places. The last decimal place is the least precise one.
Multiplying and dividing: the answer keeps as many significant figures as the factor with the fewest significant figures.
Several steps: keep one or two extra digits in the middle of the calculation and round only at the end.
Calculate and round by the rules
Enter two measured numbers and choose an operation. The tool does the exact arithmetic first, then rounds with the correct rule.
Worked examples
Example 1: counting
How many significant figures are in 0.004050? The three leading zeros do not count. Then 4, 0, 5, 0: the inner zero is captive between nonzero digits and the last zero is trailing after a decimal point. Total: 4.
Example 2: rounding
Round 2,847.5 to 3 significant figures. The first three digits are 2, 8, 4 and the next digit is 7, so round up: 2,850. The zero is a place holder, so it is clearer to write $latex 2.85\times 10^3$.
Example 3: multiplying
A rectangle measures 3.2 cm by 2.45 cm. The product is 7.84 cm². The factor 3.2 has 2 significant figures and 2.45 has 3, so keep 2:
$$3.2\times 2.45 = 7.84 \approx 7.8\ \text{cm}^2$$
Example 4: adding
Add 12.52 m, 1.4 m and 0.075 m. The exact sum is 13.995 m. The least precise number is 1.4, with 1 decimal place, so round to 1 decimal:
$$12.52 + 1.4 + 0.075 = 13.995 \approx 14.0\ \text{m}$$
The zero in 14.0 stays: it shows that the tenths place is meaningful.
Example 5: dividing
A runner covers 150.0 m in 12.3 s. 150.0 has 4 figures and 12.3 has 3, so the speed gets 3:
$$\frac{150.0}{12.3} = 12.195\ldots \approx 12.2\ \text{m/s}$$
Example 6: mixing operations
Compute $latex 4.52\times 3.1 + 0.8$. The product is 14.012, and it has 2 significant figures, so its last meaningful place is the ones place. Adding 0.8 gives 14.812, which we round to the ones place: 15. Rounding the product to 14 first and then adding would give 14.8, a precision the product never had.
Significant figures and scientific notation
Scientific notation removes the ambiguity, because every digit of the coefficient is significant. To show exactly how many figures you mean, write a coefficient between 1 and 10 times a power of ten:
- $latex 0.00450 = 4.50\times 10^{-3}$ keeps all 3 figures, including the trailing zero.
- 1,200 can be $latex 1.2\times 10^3$ (2 figures), $latex 1.20\times 10^3$ (3 figures) or $latex 1.200\times 10^3$ (4 figures).
- 602,000 with 3 figures is $latex 6.02\times 10^5$.
It is also the cleanest way to write a rounded answer: 2,847.5 rounded to 3 figures is $latex 2.85\times 10^3$.
| Operation | What to count | Example |
|---|---|---|
| Add or subtract | Decimal places | 12.52 + 1.4 = 13.92, written 13.9 |
| Multiply or divide | Significant figures | 3.2 × 2.45 = 7.84, written 7.8 |
| Powers and roots | Significant figures of the base | $latex 2.5^2 = 6.25$, written 6.3 |
| Counted or defined numbers | Nothing, they are exact | 4 sides × 7.8 cm = 31.2 cm |
Common mistakes
- Counting leading zeros. 0.0306 has 3 figures, not 5.
- Dropping trailing zeros after a decimal point. 2.500 has 4 figures, and writing 2.5 changes the claim.
- Rounding at every step. Round once, at the end.
- Mixing the two rules. Addition looks at decimal places, multiplication looks at significant figures.
- Applying the rule to exact numbers. Converting 3.2 m to centimeters with the exact factor 100 does not reduce the figures.
- Copying every digit from the calculator. The calculator shows 8 or more digits, but your data does not support them.
Practice problems
1. How many significant figures does each number have: 0.0306, 4,000, 2.500, $latex 7.0\times 10^4$?
0.0306 has 3, 4,000 has 1 (ambiguous, since there is no decimal point), 2.500 has 4, and $latex 7.0\times 10^4$ has 2.
2. Round 0.0067345 to 3 significant figures.
Leading zeros do not count, so the figures are 6, 7, 3 and the next digit is 4. Round down: 0.00673.
3. Compute 24.0 × 0.35.
24.0 has 3 figures and 0.35 has 2, so keep 2: $latex 24.0\times 0.35 = 8.4$, so 8.4.
4. Compute 102.5 + 3.27.
The exact sum is 105.77. The fewest decimals is 1 (in 102.5), so 105.8.
5. An object falls for 2.0 s with g = 9.81 m/s². Find the speed, v = gt.
$latex 9.81\times 2.0 = 19.62$. Keep 2 figures: 20 m/s, better written as $latex 2.0\times 10^1$ m/s.
6. A sample has mass 1.50 g and volume 0.0300 L. Find g/L.
$latex 1.50\div 0.0300 = 50$ exactly and both numbers have 3 figures, so the answer is 50.0 g/L.
Frequently asked questions
Do exact numbers count as significant figures?
No. Counted quantities (3 apples) and definitions (1 in = 2.54 cm) are exact, so they never limit the number of figures in the answer.
Why are the zeros in 1,200 ambiguous?
Because a whole number can be written without showing where the measuring stopped. Scientific notation fixes it: $latex 1.2\times 10^3$ means 2 figures and $latex 1.20\times 10^3$ means 3.
Does 5 always round up?
In school and in most measurement work, yes: round half up. Some statistical software and financial systems use "round half to even" to avoid bias, so 2.5 rounds to 2 and 3.5 rounds to 4. Check which rule your course or tool uses.
Is a significant figure the same as a decimal place?
No. Decimal places count digits after the point. Significant figures count meaningful digits from the first nonzero digit. 0.0306 has 4 decimal places and 3 significant figures.
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