Significant figures
Significant figures, also referred to as significant digits or sig figs, are specific digits within a number written in positional notation that carry both reliability and necessity in conveying a particular quantity.
When presenting the outcome of a measurement (such as length, pressure, volume, or mass), if the number of digits exceeds what the measurement instrument can resolve, only the number of digits within the resolution's capability are dependable and therefore considered significant.
In this example, the last digit (8, contributing 0.8 mm) is likewise considered significant despite its uncertainty.
In this case, the actual volume might be 2.94 L or possibly 3.02 L, so all three digits are considered significant.
To avoid conveying a misleading level of precision, numbers are often rounded.
The rounding error (in this example, 0.00025 kg = 0.25 g) approximates the numerical resolution or precision.
Numbers can also be rounded for simplicity, not necessarily to indicate measurement precision, such as for the sake of expediency in news broadcasts.
[3] The significance of trailing zeros in a number not containing a decimal point can be ambiguous.
However, these are not universally used and would only be effective if the reader is familiar with the convention: As the conventions above are not in general use, the following more widely recognized options are available for indicating the significance of number with trailing zeros: Rounding to significant figures is a more general-purpose technique than rounding to n digits, since it handles numbers of different scales in a uniform way.
This reflects the fact that the significance of the error is the same in both cases, relative to the size of the quantity being measured.
This is done because greater precision is immaterial, and usually it is not possible to settle a debt of less than the smallest currency unit.
The following table shows the results for various total precision at two rounding ways (N/A stands for Not Applicable).
The representation of a non-zero number x to a precision of p significant digits has a numerical value that is given by the formula:[citation needed] which may need to be written with a specific marking as detailed above to specify the number of significant trailing zeros.
[1] The implied uncertainty is ± the half of the minimum scale at the last significant figure position.
A mathematical or physical constant (e.g., π in the formula for the area of a circle with radius r as πr2) has no effect on the determination of the significant figures in the result of a calculation with it if its known digits are equal to or more than the significant figures in the measured quantities used in the calculation.
An exact number such as ½ in the formula for the kinetic energy of a mass m with velocity v as ½mv2 has no bearing on the significant figures in the calculated kinetic energy since its number of significant figures is infinite (0.500000...).
For unit conversion, the implied uncertainty of the result can be unsatisfactorily higher than that in the previous unit if this rounding guideline is followed; For example, 8 inch has the implied uncertainty of ± 0.5 inch = ± 1.27 cm.
Another exception of applying the above rounding guideline is to multiply a number by an integer, such as 1.234 × 9.
[citation needed] However, greater accuracy will often be obtained if some non-significant digits are maintained in intermediate results which are used in subsequent calculations.
[citation needed] The base-10 logarithm of a normalized number (i.e., a × 10b with 1 ≤ a < 10 and b as an integer), is rounded such that its decimal part (called mantissa) has as many significant figures as the significant figures in the normalized number.
When performing multiple stage calculations, do not round intermediate stage calculation results; keep as many digits as is practical (at least one more digit than the rounding rule allows per stage) until the end of all the calculations to avoid cumulative rounding errors while tracking or recording the significant figures in each intermediate result.
Then, round the final result, for example, to the fewest number of significant figures (for multiplication or division) or leftmost last significant digit position (for addition or subtraction) among the inputs in the final calculation.
However assuming a normal good quality ruler, it should be possible to estimate tenths between the nearest two marks to achieve an extra decimal place of accuracy.
When estimating the proportion of individuals carrying some particular characteristic in a population, from a random sample of that population, the number of significant figures should not exceed the maximum precision allowed by that sample size.
Hoping to reflect the way in which the term "accuracy" is actually used in the scientific community, there is a recent standard, ISO 5725, which keeps the same definition of precision but defines the term "trueness" as the closeness of a given measurement to its true value and uses the term "accuracy" as the combination of trueness and precision.
In either case, the number of significant figures roughly corresponds to precision, not to accuracy or the newer concept of trueness.
The number of correct significant figures is closely related to the notion of relative error (which has the advantage of being a more accurate measure of precision, and is independent of the radix, also known as the base, of the number system used).
Electronic calculators supporting a dedicated significant figures display mode are relatively rare.
Among the calculators to support related features are the Commodore M55 Mathematician (1976)[17] and the S61 Statistician (1976),[18] which support two display modes, where DISP+n will give n significant digits in total, while DISP+.+n will give n decimal places.
[20][21] The SwissMicros DM42-based community-developed calculators WP 43C (2019)[22] / C43 (2022) / C47 (2023) support a significant figures display mode as well.
