Significant Figures Calculator
- This tool counts, computes, and edits significant figures (sig. figs.) of quantities submitted in different formats. The tool accepts positive and negative values and expresses the results in conventional and scientific notation.
- To use the tool, please enter one quantity per line, ending each line by pressing the
- First time users may want to try the example provided by pressing the Try This Example button.
The number of significant figures attributed to a quantity is related to the notion of relative error (Burden & Faires, 2010; Norminton, 2007; Harder, 2010). Let the number p* be an approximation of p. Then p* is said to approximate p to t significant figures if t is the largest nonnegative integer for which the relative error (rel_err) is
All kind of guidelines for handling significant figures can be derived from (1). For instance, note that no significant figures attribution is possible if p, p*, or both are zero or if p = p*.
Accordingly, no significant figures attribution is possible for the following cases: zero as a value, exact numbers, conversion factors, and non-measured constants. However, all known digits of a measured constant are significant.
If the relative error of a quantity is known or can be estimated, the number of significant figures t can be computed. Solving (1) for $t,
where $t is the number of significant figures, $rel_err is the relative error, and floor() is a function that rounds a number down to the nearest integer, if necessary. This function is found in most programming languages.
- Most students are taught to memorize the so-called rules for counting significant figures. They often do not know or understand from where these rules come from. They blindly implement these because a teacher or textbook told them to do so, when in fact said "rules" can be derived from the previous equation (1) and from the notion of relative errors. Simply put, the number of significant figures of a quantity are computed when the relative error of said quantity is known or can be estimated. See, for instance, Burden & Faires book, pp 20, or previous (1) and (2) equations.
Although giving a precise definition for the correct number of sig. figs. is quite subtle (Higham, 2002), the following guidelines can be easily derived from the theory of relative errors.
- Non-zero digits and zeros within non-zero digits are significant.
- Leading zeroes (zeros before the first non-zero digit) are not significant.
- Trailing zeros in a number (integer or not) can be significant, depending on the measurement or reporting precision.
Note: The ambiguity of trailing zeros in an integer can be avoided by expressing said integer in scientific notation. A somehow arbitrary convention consists in adding a decimal point after the last trailing zero. Another convention consists in overlining the trailing zero deemed as significant. Both conventions make all previous zeros significant. If neither of these marks are given, it is assumed that none of the trailing zeros are significant.
- All digits of a number given in scientific notation are significant.
- All known digits of a measured constant are significant.
- No significant figures are attributed to non-measured constants, exact numbers, and conversion factors.
- No significant figures are attributed to a number where all of its digits are zero; e.g., 0, 0.0, 0.00...
- For addition and subtraction, the result should have the same number of decimal places as the quantity with the fewest decimal places.
- For multiplication and division, the result should have the same number of significant figures as the quantity with the fewest number of significant figures.
- Although there are many significant figures calculators out there, many do not pass what can be called "the test of zeros"; i.e., to grasp the quality of said calculators, we may enter one or more zeros as a quantity (0, 0.0, 0.00...) and check the result. According to the theory of relative errors, no significant figures attribution should be possible. Expressing these in scientific notation does not add any artificial significance to them as we still cannot compute relative errors for them.
- Another test that unveils the quality of these tools is "the test of formats". A significant figures tool is considered robust against formatting if the tool returns correct results for a variety of input formats. Many tools out there fail to pass this test as these accept input in one or a few format. For instance, try several tools by entering something like -00040808000, -2,000., 9.9995, 1.56e+3, 2.33x10^-3, and so forth, and check the results. If a tool lets you edit the reported number of significant figures, instruct it to return results rounded to different numbers of sig. figs. and, again, check the results.
- Let us now address the question of how many significant figures are in "0.", i.e., in a zero followed by a decimal point. Some authors claim that this zero has one significant figure. Their argument here is that the convention of trailing zeros ending with a decimal place applies. This implies that said zero is trailing itself. This is an invalid argument as we can also imply that said zero is a leading zero, leading itself. Again, no relative errors can be computed for self-trailing or self-leading zeros. We must conclude that "0." has no significant figures. The same goes for "0", "0.0", "0.00", and so forth.
- Not all significant figures are meaninful figures. For instance, specialized temperature sensors can handle temperatures to a tenth of a degree at the most. For typical temperature sensors, half a degree increments are more likely to be within the temperature sensor's instrumental error range. This is why temperature readings from analog displays are commonly reported as integers, or to no more than one decimal place.
- Reporting temperature values to more decimal places than those measured by a temperature sensor (e.g., 76.004 degree Fahrenheit), are not very meaningful. The same goes for barometric pressures. In the mmHg scale, pressure is commonly, though not always, reported to two decimal places. In other words, not all significant figures computed with a computer program are meaningful figures.
- In general, analog displays should be read to the nearest marked gradation or half-way between if possible. With digital displays the procedure is simpler: use exactly the displayed reading. If the uncertainty of a value is not given, the least count is assumed. In general the reading uncertainty for an analog or digital device is defined as
- Users (students, teachers, researchers) that need to express results to a given number of sig. figs..
- How many significang figures are in
- Express the numbers given in the previous exercise to three significant figures.
- 25 degrees Celsius has two significant figures. When converted to kelvins using 273 or 273.15, the result is 298, to three significant figures. Why?
- 25.0 degrees Celsius has three significant figures. When converted using 273, the result is 298, to three significant figures, but when converted using 273.15, the result is 298.2, to four significant figures. Why?
- Burden, R. L. and Faires, J. D. (2010). Numerical Analysis. 9th Edition. Chapter 1, p. 20. Brooks/Cole, Cengage Learning.
- Harder, D. W. (2010). Numerical Analysis for Engineering. University of Waterloo, Canada.
- Higham, N. (2002). Accuracy and Stability of Numerical Algorithms. (PDF) (2nd ed.). SIAM. pp. 3-5. Retrieved on May 19, 2021.
- Norminton, T. (2007). Errors. Math3806. Carleton, Canada.
- Wikipedia (2021). Significant figures Retrieved on May 19, 2021.
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