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Significant figures

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	Topics Significant figures Topic Home The significant figures (also called significant digits and abbreviated sig figs) of a number are those digitsNumerical digit In mathematics and computer science, a numerical digit is a symbol used in numerals, to represent numbers, in positional num... that carry meaning contributing to its accuracy. Generally this includes all digits except: leading and trailing zeros where they serve merely as placeholders to indicate the scale of the number spurious digits introduced, for example, by calculations carried out to greater accuracy than that of the original data, or measurements reported to a greater precision than the equipment supports. The concept of significant figures is often used in connection with roundingRounding Rounding is the process of reducing the number of significant digits in a number.... . 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 arithmeticSignificance arithmetic Significance arithmetic is a set of heavily-flawed rules for approximating the propagation of error in scientific or statist... . Identifying significant digits The rules for identifying significant digits when writing or interpreting numbers are as follows: All non-zero digits are considered significant. Example: 123.45 has five significant figures: 1, 2, 3, 4 and 5. Zeros appearing anywhere between two non-zero digits are significant. Example: 101.12 has five significant figures: 1, 0, 1, 1 and 2. Leading zeros are not significant. For example, 0.00012 has two significant figures: 1 and 2. Trailing zeros in a number containing a decimal point are significant. For example, 12.2300 has six significant figures: 1, 2, 2, 3, 0 and 0. The number 0.00122300 still has only six significant figures (the zeros before the 1 are not significant). In addition, 120.00 has five significant figures. This convention clarifies the precision of such numbers; for example, if a result accurate to four decimal places is given as 12.23 then it might be understood that only two decimal places of accuracy are available. Stating the result as 12.2300 makes clear that it is accurate to four decimal places. The significance of trailing zeros in a number not containing a decimal point can be ambiguous. For example, it may not always be clear if a number like 1300 is accurate to the nearest unit (and just happens coincidentally to be an exact multiple of a hundred) or if it is only shown to the nearest hundred due to rounding or uncertainty. Various conventions exist to address this issue: A bar may be placed over the last significant digit; any trailing zeros following this are insignificant. For example, has three significant figures (and hence indicates that the number is accurate to the nearest ten). The last significant figure of a number may be underlined; for example, "20000" has two significant figures. A decimal point may be placed after the number; for example "100." indicates specifically that three significant figures are meant. However, these conventions are not universally used, and it is often necessary to determine from context whether such trailing zeros are intended to be significant. If all else fails, the level of rounding can be specified explicitly. The abbreviation s.f. is sometimes used, for example "20 000 to 2 s.f." or "20 000 (2 sf)". Alternatively, the uncertainty can be stated separately and explicitly, as in 20 000 ± 1%, so that significant-figures rules do not apply. A number with all zero digits (e.g. 0.000) has no significant digits, because the uncertainty is larger than the actual measurement. Scientific notation Generally, the same rules apply to numbers expressed in scientific notationScientific notation Scientific notation is a notation for writing numbers that is often used by scientists and mathematicians to make it easier ... . 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. Rounding To round to n significant figures: Start with the leftmost non-zero digit (e.g. the '1' in 1200, or the '2' in 0.0256). Keep n digits. Replace the rest with zeros. Round up by one if appropriate. For example, if rounding 0.039 to 1 significant figure, the result would be 0.04. There are several different rules for handling borderline cases — see roundingRounding Rounding is the process of reducing the number of significant digits in a number.... for more details. Arithmetic For multiplication and division, the result should have as many significant figures as the measured number with the smallest number of 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. Importance Superfluous precision If a sprinter is measured to have completed a 100 m race in 11.71 seconds, what is the sprinter's average speed? By dividing the distance by the time using a calculator, we get a speed of 8.53970965 m/s. 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. Comprehension Numbers are often rounded off to make them easier to read. It's easier for someone to compare (say) 18% to 36% than to compare 18.148% to 35.922%. Similarly, when reviewing a budget, a series of figures like: Division A: $185 000 Division B: $ 45 000 Division C: $ 67 000 is easier to understand and compare than a series like: Division A: $184 982 Division B: $ 44 689 Division C: $ 67 422 To 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 See also Arithmetic precision Additional reading ASTM E29-06b, Standard Practice for Using Significant Digits in Test Data to Determine Conformance with Specifications