Arithmetic
How To Count Significant Figures
Significant figures are the digits in a number that carry real information. They exist because a measurement is only as precise as the instrument that produced it, and reporting extra digits implies a precision that was never measured.
Quick Answer
rounded = round(x, sigFigs - 1 - floor(log10(|x|)))
- x
- The number you are rounding
- sigFigs
- How many significant digits to keep
- floor(log10(|x|))
- The order of magnitude, or power of ten
- rounded
- The result, carrying exactly sigFigs digits
Find the order of magnitude of the number, subtract it from the significant figures you want, and round to that many decimal places. Rounding 12,345 to three significant figures gives 12,300, rounding 3.14159 to four gives 3.142, and rounding 0.0072151 to three gives 0.00722.
What Is How To Count Significant Figures?
A significant figure is a digit that a measurement actually supports. In 12,345 measured to three significant figures, only the 12 and the first 3 carry meaning; the two trailing zeros are placeholders, not information. The rounded answer is written 12,300.
The basic rule for counting is to start at the first non-zero digit and count every digit after it, including zeros that sit between non-zero digits. In 0.0072151 the leading zeros are not significant, so the first significant digit is the 7 and the number carries seven significant figures.
Zeros split into three groups. Leading zeros, as in 0.007, are never significant. Captive zeros, as in 1,003, always are. Trailing zeros, as in 12,300, are significant only if the number is written in a way that says so, such as with a decimal point or scientific notation.
Scientific notation removes the ambiguity entirely. 12,300 to three significant figures is written 1.23 x 10^4, and to five significant figures it is 1.2300 x 10^4. The exponent handles the place value and every digit in the mantissa counts.
The arithmetic of significant figures works differently from ordinary rounding. When adding or subtracting, the answer is limited by the least precise decimal place, not by the count of significant digits. Adding 1.2 and 3.45 gives 4.65 on a calculator but must be reported as 4.7, because 1.2 is only trusted to the tenths.
When multiplying or dividing, the answer carries the same number of significant figures as the factor with the fewest. Multiplying 2.5 by 3.128 gives 7.82 on a calculator but must be reported as 7.8, because 2.5 only carries two significant figures.
Exact numbers do not limit precision. A count of three apples, a defined unit conversion, or a coefficient in an equation is exact and carries unlimited significant figures. Only measured quantities constrain the answer.
Several rules of thumb follow from counting digits rather than estimating error. Rounding first and then operating is usually safer than operating and then rounding, because the intermediate digits were never meaningful in the first place.
Rounding a number up can increase its order of magnitude. Rounding 999 to two significant figures gives 1,000, which now has four digits in it while carrying only two significant figures. Scientific notation is the only honest way to write that result, as 1.0 x 10^3.
The common complaint that significant figures throw away good information is misplaced. The rule is not that the extra digits are wrong, only that they cannot be justified by the measurement. Reporting them transfers your uncertainty onto whoever reads the number, and nobody can then tell how good the measurement was.
Formula
rounded = round(x, sigFigs - 1 - floor(log10(|x|)))
Formula page →Shift the decimal point so the last kept digit becomes the final digit, round, then shift back.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| x | Input | any unit | The number being rounded. |
| n | Significant figures | unitless | How many digits to keep. |
| log10(|x|) | Order of magnitude | unitless | The power of ten the number starts at. |
| rounded | Result | same unit | The number carrying exactly n significant figures. |
result carries the fewest significant figures of any factor
Formula page →Multiplication and division pass the least precise input's precision straight through to the answer.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| min | Fewest | unitless | The limiting input's significant figure count. |
| factors | Inputs | any unit | The measured values entering the calculation. |
| result | Output | same unit | The answer, trimmed to the limiting precision. |
How To Calculate How To Count Significant Figures
- 1
Identify the first non-zero digit
Leading zeros are placeholders and never count. In 0.0072151 the counting starts at the 7, so the number has seven significant figures.
- 2
Read off the order of magnitude
The base ten logarithm, rounded down, tells you the power of ten the number starts at. For 12,345 that is 4; for 0.0072151 it is -3.
- 3
Convert significant figures into decimal places
Subtract the order of magnitude from the significant figures and take one away. Twelve thousand three hundred and forty-five to three significant figures gives a rounding position of -2, meaning two trailing zeros.
- 4
Round, then check the digit count
Round 12,345 to the hundreds place to get 12,300. Count the digits from the first non-zero one to confirm three, and note that the trailing zeros need scientific notation to be unambiguous.
- 5
Carry the result forward without adding digits
Once rounded, the extra digits are gone for good. Re-deriving them by operating on the rounded number simply invents precision that the measurement never had.
Examples
Example 1: 12,345 rounded to three significant figures
- Number
- 12345
- Significant figures
- 3
| Step | Calculation | Result |
|---|---|---|
| Magnitude | abs(12345) | 12345 |
| Order of magnitude | floor(log(12345) + 0.0000000001) | 4 |
| Rounding position | 3 - 1 - 4 | -2 |
| Rounded value | round(12345, -2) | 12300 |
| Size of the rounding change | abs(12345 - 12300) | 45 |
Result: 12,345 to three significant figures is 12,300. The rounding position is -2, meaning two trailing zeros, and the rounding change moves the value by 45 in total.
Example 2: 3.14159 rounded to four significant figures
- Number
- 3.14159
- Significant figures
- 4
| Step | Calculation | Result |
|---|---|---|
| Magnitude | abs(3.14159) | 3.14159 |
| Order of magnitude | floor(log(3.14159) + 0.0000000001) | 0 |
| Rounding position | 4 - 1 - 0 | 3 |
| Rounded value | round(3.14159, 3) | 3.142 |
| Size of the rounding change | abs(3.14159 - 3.142) | 0.00041 |
Result: 3.14159 to four significant figures is 3.142, which is the usual textbook rounding of pi. The rounding position is three decimal places and the value moves by 0.00041.
Example 3: 0.0072151 rounded to three significant figures
- Number
- 0.0072151
- Significant figures
- 3
| Step | Calculation | Result |
|---|---|---|
| Magnitude | abs(0.0072151) | 0.0072151 |
| Order of magnitude | floor(log(0.0072151) + 0.0000000001) | -3 |
| Rounding position | 3 - 1 + 3 | 5 |
| Rounded value | round(0.0072151, 5) | 0.00722 |
| Size of the rounding change | abs(0.0072151 - 0.00722) | 0.0000049 |
Result: 0.0072151 to three significant figures is 0.00722. The first significant digit is the 7, the rounding position is five decimal places, and the value moves by 0.0000049.
Calculator
Rounded value
12,300
- Order of magnitude
- 4
- Size of the rounding change
- 45
- Rounding position, negative means trailing zeros
- -2
Values update as you type. This calculator covers the single scenario its formula assumes — see Common Mistakes for what it leaves out.
Prefer a full-width tool? Open the How To Count Significant Figures calculator page.
Common Mistakes
Counting leading zeros as significant
Zeros before the first non-zero digit only fix the place value. 0.0072151 has seven significant figures, not nine, and counting the leading zeros invents precision.
Counting every digit in a trailing-zero number as significant
12,300 written without further qualification is normally read as three significant figures, not five. If you really mean five, write 12,300. or 1.2300 x 10^4.
Adding significant-figure rules to a sum
Addition is limited by decimal places, not by digit count. 1.2 plus 3.45 is 4.7, not 4.65, because the 1.2 stops at the tenths.
Rounding before multiplying
Rounding each factor first, then multiplying, compounds the error. Multiply at full precision and round the single final answer to the limiting factor's count.
Letting an exact number limit the answer
Coefficients, defined conversion factors and counted items are exact. Three of something does not reduce a precision that was set by a measured length.
Writing the rounded answer without scientific notation
Rounding 999 to two significant figures gives 1,000, which reads as four digits. Only 1.0 x 10^3 says unambiguously that two figures are meant.
FAQ
How many significant figures does 12,300 have?
Normally three, unless the source says otherwise. A trailing decimal point, as in 12,300., or scientific notation, as in 1.2300 x 10^4, tells the reader you mean five.
What is 3.14159 to four significant figures?
It is 3.142. The order of magnitude is zero, so four significant figures means four digits after rounding at three decimal places.
Are zeros between non-zero digits significant?
Always. The zeros in 1,003 are captive and count, which is why 1,003 carries four significant figures.
Do significant figures apply to multiplication only?
No. Multiplication and division are limited by the fewest significant figures among the inputs, while addition and subtraction are limited by the least precise decimal place.
How do I round to a number of significant figures in a spreadsheet?
Use ROUND with a decimal-place count of significant figures minus one minus the integer part of the base ten logarithm of the number. For 12,345 to three figures that is ROUND(12345, -2).
What happens when rounding increases the order of magnitude?
The digit count changes even though the significant figure count is fixed. Rounding 999 to two figures gives 1,000, and you should re-express it as 1.0 x 10^3.
References
- [1]Wikipedia, Significant figures — https://en.wikipedia.org/wiki/Significant_figures
- [2]Wikipedia, Order of magnitude — https://en.wikipedia.org/wiki/Order_of_magnitude
- [3]Wikipedia, Rounding — https://en.wikipedia.org/wiki/Rounding