# How many sig figs are in a number?

Counts the significant figures (sig figs) in a number, rounds a number to any number of them, and adds, subtracts, multiplies, or divides measured numbers under the sig fig rules.

- Page: https://www.acalculator.org/math/significant-figures-calculator
- JSON spec: https://www.acalculator.org/math/significant-figures-calculator.json
- Version: bd80484c3719

## Default answer

Example with the default inputs (What to do Count, Number 0.004050): 0.004050 has 4 significant figures.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| mode | What to do | Count the significant figures, round to a number of them, or do arithmetic with them. |
| x | Number | A measured number as written, such as 0.00450, 1200, 1200. or 1.20e3; trailing zeros matter. |
| n | Significant figures | How many significant figures to round to, from 1 to 30. |
| op | Operation | Add, subtract, multiply, or divide the first number by the second. |
| y | Second number | The second measured number, written with the digits that were measured. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| result | Answer | The rounded number or the result, written so its significant figures are clear: 2.50 × 10^3 where 2500 would not show them. |
| sigFigs | Significant figures | How many significant figures the number (or the answer) has. |
| decimals | Decimal places | How many decimal places the number (or the answer) keeps; 0 when its last significant digit is in the ones place or higher. |
| scientific | Scientific notation | The number (or the answer) as a × 10^k, with every significant figure in a. |
| significant | Significant digits | The digits that count, in order, from the first non-zero digit. |
| exact | Unrounded result | The exact result before rounding; ≈ marks a value cut to 15 significant figures. |
| rule | Rule used | Which significant-figure rule sets the precision of the answer. |
| note | Note | Shown when a whole number has trailing zeros that are not counted. |
| summary | In short | The answer as one sentence. |

## Method

Count: every non-zero digit, zeros between them, and trailing zeros after a decimal point. × and ÷: keep the fewest significant figures of the numbers. + and −: round to the last significant place of the less precise number. Halves round away from zero.

## Assumptions

- Leading zeros never count. Trailing zeros count only when the number has a decimal point (1200 has 2 significant figures, 1200. has 4, 1.200 × 10^3 has 4).
- Every number is a measurement; the rules do not treat exact counts or defined constants differently.
- Rounding works on the exact value; a value exactly halfway rounds away from zero (2.25 to 2 figures is 2.3), not to the even digit that some chemistry texts use.
- The answer is written so that typing it back gives the same significant figures: 2.50 × 10^3 where 2500 would read as 2 figures.

## Worked examples

1. mode = count, x = 0.004050 gives sigFigs = 4, decimals = 6, scientific = 4.050 × 10^-3, significant = 4050. Source: Rules from OpenStax Chemistry 2e, section 1.5: https://openstax.org/books/chemistry-2e/pages/1-5-measurement-uncertainty-accuracy-and-precision.
2. mode = count, x = 1200 gives sigFigs = 2, scientific = 1.2 × 10^3, note = The trailing zeros in 1200 are not significant, because it has no decimal point..
3. mode = count, x = 1200. gives sigFigs = 4, decimals = 0.
4. mode = round, x = 0.0034567, n = 3 gives result = 0.00346, decimals = 5, scientific = 3.46 × 10^-3.
5. mode = round, x = 31.57, n = 2 gives result = 32.
6. mode = round, x = 8.1649, n = 3 gives result = 8.16.
7. mode = round, x = 0.051065, n = 4 gives result = 0.05107.
8. mode = round, x = 2468, n = 2 gives result = 2500, scientific = 2.5 × 10^3.
9. mode = round, x = 2496, n = 3 gives result = 2.50 × 10^3.
10. mode = round, x = 2499.7, n = 4 gives result = 2.500 × 10^3.
11. mode = arithmetic, x = 3.24, op = *, y = 2.5 gives result = 8.1, sigFigs = 2, exact = 8.1, rule = Multiplication: the answer keeps as many significant figures as the number with the fewest, 2..
12. mode = arithmetic, x = 1.0023, op = +, y = 4.383 gives result = 5.385, sigFigs = 4, decimals = 3, exact = 5.3853.
13. mode = arithmetic, x = 486, op = -, y = 421.23 gives result = 65, sigFigs = 2, exact = 64.77.
14. mode = arithmetic, x = 0.6238, op = *, y = 6.6 gives result = 4.1, sigFigs = 2, exact = 4.11708.
15. mode = arithmetic, x = 421.23, op = /, y = 486 gives result = 0.867, sigFigs = 3.
16. mode = arithmetic, x = 10, op = /, y = 3.0 gives result = 3, sigFigs = 1, exact = ≈ 3.33333333333333.
17. mode = arithmetic, x = 5.0, op = -, y = 5.0 gives result = 0.0, decimals = 1.

## FAQ

### How do I count significant figures?

Count every non-zero digit, every zero between non-zero digits, and every zero at the end of a number that has a decimal point. Leading zeros never count. 0.004050 has 4 significant figures (4, 0, 5, 0), and 1.20 × 10³ has 3.

### Are trailing zeros significant?

After a decimal point, yes: 3.10 has 3 significant figures. In a whole number with no decimal point, such as 1200, trailing zeros are ambiguous, and this calculator does not count them (1200 has 2). To show that they were measured, write 1200. with a decimal point or use scientific notation, 1.200 × 10³.

### How do significant figures work when multiplying or dividing?

The answer keeps as many significant figures as the number with the fewest. 0.6238 × 6.6 = 4.11708, and 6.6 has 2 significant figures, so the answer is 4.1.

### How do significant figures work when adding or subtracting?

Round the answer to the last decimal place that both numbers know. 1.0023 + 4.383 = 5.3853; 4.383 stops at the thousandths place, so the answer is 5.385. 486 − 421.23 = 64.77 rounds to 65, because 486 stops at the ones place.

### How do I round to a number of significant figures?

Find the first non-zero digit, count that many digits to the right, and look at the next digit: 5 or more rounds up, less than 5 rounds down. 0.0034567 to 3 significant figures is 0.00346.

### Why is the answer sometimes in scientific notation?

Because the plain number would hide the significant figures. 2496 rounded to 3 significant figures is 2500, but 2500 reads as 2 significant figures, so the calculator writes 2.50 × 10³. You can copy any answer back into the calculator and it counts the same number of figures.

### What happens when the dropped digit is exactly 5?

The calculator rounds halves up (away from zero): 0.051065 to 4 figures is 0.05107. Some chemistry texts round a final 5 to the even digit instead and get 0.05106, so check which rule your class uses.

## Sources

- OpenStax, Chemistry 2e, section 1.5 Measurement Uncertainty, Accuracy, and Precision (significant figures, rounding, and calculations): https://openstax.org/books/chemistry-2e/pages/1-5-measurement-uncertainty-accuracy-and-precision
- Wikipedia, Significant figures (identifying significant figures, rounding, and arithmetic): https://en.wikipedia.org/wiki/Significant_figures