The concept of significant figures is often used in connection with rounding. Rounding to n significant figures is a more general-purpose technique than rounding to n decimal places, since it handles numbers of different scales in a uniform way. Computer representations of floating point numbers typically use a form of rounding to significant figures, but with binary numbers.
The term "significant figures" can also refer to a crude form of error representation based around significant figure rounding; for this use, see Significance arithmetic.
A number with all zero digits (e.g. 0.000) has no significant digits, because the uncertainty is larger than the actual measurement.
Generally, the same rules apply to numbers expressed in scientific notation. However, in the normalized form of that notation, placeholder leading and trailing digits do not occur, so all digits are significant. For example, 0.00012 (two significant figures) becomes 1.2×10−4, and 0.000122300 (six significant figures) becomes 1.22300×10−4. In particular, the potential ambiguity about the significance of trailing zeros is eliminated. For example, 1300 (four significant figures) is written as 1.300×103, while 1300 (three significant figures) is written as 1.30×103.
To round to n significant figures:
For addition and subtraction, the result should have as many decimal places as the measured number with the smallest number of decimal places.
The most straightforward way to indicate the precision of this result (or any result) is to state the uncertainty separately and explicitly, for example as 8.540±0.085 m/s or equivalently 8.540(85) m/s. This is particularly appropriate when the uncertainty itself is important and precisely known. In this case, it is safe and indeed advantageous to provide more digits than would be called for by the significant-figures rules.
If the degree of precision in the answer is not important, it is again safe to express trailing digits that are not known exactly, for example 8.5397 m/s.
If, however, we are forced to apply significant-figures rules, expressing the result as 8.53970965 m/s would seem to imply that the speed is known to the nearest 10 nm/s or thereabouts, which would improperly overstate the precision of the measurement. Reporting the result using three significant figures (8.54 m/s) might be interpreted as implying that the speed is somewhere between 8.535 and 8.545 m/s. This again overstates the accuracy, but not nearly so badly. Reporting the result using two significant figures (8.5 m/s) would introduce considerable roundoff error and degrade the precision of the result.
Division A: $185 000
Division B: $ 45 000
Division C: $ 67 000is easier to understand and compare than a series like:
Division A: $184 982
Division B: $ 44 689
Division C: $ 67 422To reduce ambiguity, such data are sometimes represented to the nearest order of magnitude, like:
Revenue (in thousands of dollars):
Division A: 185
Division B: 45
Division C: 67
Practicing scientists commonly express uncertain quantities in the form 1.23±0.06 or equivalently 1.23(6). The benefit is that the nominal value of the quantity is expressed by one numeral (1.23) while the uncertainty is expressed by a separate numeral (0.06). Expressing these two things explicitly and separately is more sensible than trying to encode both the nominal value and the uncertainty into a single numeral, where the uncertaintly range is constrained to being a power of ten.
"Significant figures" primarily refers to a type of rounding, and is arguably appropriate when roundoff of the final answer is the dominant contribution to the uncertainty. However, there are many important situations where roundoff of the final answer is not the dominant contribution to the uncertainty. Indeed, in experimental research (especially metrology), only in a very badly designed experiment would such roundoff error be dominant, because roundoff errors are so easily reduced. Furthermore, even when roundoff error is dominant, it is preferable to indicate this explicitly, as in 1.24(½) or equivalently 1.24(⁄).
Secondarily, "significant figures" may refer to a crude scheme for significance arithmetic, but as discussed in the significance arithmetic article and elsewhere, there is generally not any rigorous way to express the uncertainty using significant figures.
In computer science and numerical analysis, good practice demands the use of guard digits. This is incompatible with any notion of significant figures. For a discussion, see Acton.
Good examples of how real scientists express uncertain quantities can be found in the NIST compendium of physical constants. None of the values there conform to any "significant figures" rules.
Procedures for how to properly represent uncertainty, and the rationale for these procedures, can be found in the references.
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