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- All leading zeros;
- Trailing zeros when they are merely placeholders to indicate the scale of the number (exact rules are explained at identifying significant figures); and
- Spurious digits introduced, for example, by calculations carried out to greater precision than that of the original data, or measurements reported to a greater precision than the equipment supports.
Numbers are often rounded to avoid reporting insignificant figures. For example, it would create false precision to express a measurement as 12.34500 kg (which has seven significant figures) if the scales only measured to the nearest gram and gave a reading of 12.345 kg (which has five significant figures) . Numbers can also be rounded merely for simplicity rather than to indicate a given precision of measurement, for example to make them faster to pronounce in news broadcasts.
Arithmetic precision can also be defined with reference to a fixed number of decimal places (the number of digits following the decimal point). This second definition is useful in applications where the number of digits in the fractional part has particular importance, but it does not follow the rules of significance arithmetic.
Identifying significant figures
Specifically, the rules for identifying significant figures when writing or interpreting numbers are as follows:
- All non-zero digits are considered significant. For example, 91 has two significant figures (9 and 1), while 123.45 has five significant figures (1, 2, 3, 4 and 5).
- Zeros appearing anywhere between two non-zero digits are significant. Example: 101.1203 has seven significant figures: 1, 0, 1, 1, 2, 0 and 3.
- Leading zeros are not significant. For example, 0.00052 has two significant figures: 5 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 0.000122300 still has only six significant figures (the zeros before the 1 are not significant). In addition, 120.00 has five significant figures since it has three trailing zeros. This convention clarifies the precision of such numbers; for example, if a measurement precise to four decimal places (0.0001) is given as 12.23 then it might be understood that only two decimal places of precision are available. Stating the result as 12.2300 makes clear that it is precise to four decimal places (in this case, six significant figures).
- 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 precise 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 figure; any trailing zeros following this are insignificant. For example, 1300 has three significant figures (and hence indicates that the number is precise to the nearest ten).
- The last significant figure of a number may be underlined; for example, "2000" 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 with a plus-minus sign, as in 20 000 ± 1%, so that significant-figures rules do not apply. This also allows specifying a precision in-between powers of ten (or whatever the base power of the numbering system is).
In most cases, 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 0.00122300 (six significant figures) becomes 1.22300×10−3. In particular, the potential ambiguity about the significance of trailing zeros is eliminated. For example, 1300 to four significant figures is written as 1.300×103, while 1300 to two significant figures is written as 1.3×103.
The part of the representation that contains the significant figures (as opposed to the base or the exponent) is known as the significand or mantissa.
Alternatively: 1. All non-zero digits are significant 2. In a number without a decimal point, only zeros between non-zero digits are significant. 3. In a number with a decimal point, all zeros to the right of the first non-zero digits are significant.
Rounding and decimal places
The basic concept of significant figures is often used in connection with rounding. Rounding to 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. For example, the population of a city might only be known to the nearest thousand and be stated as 52,000, while the population of a country might only be known to the nearest million and be stated as 52,000,000. The former might be in error by hundreds, and the latter might be in error by hundreds of thousands, but both have two significant figures (5 and 2). This reflects the fact that the significance of the error (its likely size relative to the size of the quantity being measured) is the same in both cases.
To round to n significant figures:
- If the first non-significant figure is a 5 followed by other non-zero digits, round up the last significant figure (away from zero). For example, 1.2459 as the result of a calculation or measurement that only allows for 3 significant figures should be written 1.25.
- If the first non-significant figure is a 5 not followed by any other digits or followed only by zeros, rounding requires a tie-breaking rule. For example, to round 1.25 to 2 significant figures:
- Replace non-significant figures in front of the decimal by zeros.
In financial calculations, a number is often rounded to a given number of places (for example, to two places after the decimal separator for many world currencies). Rounding to a fixed number of decimal places in this way is an orthographic convention that does not maintain significance, and may either lose information or create false precision.
As an illustration, the decimal quantity 12.345 can be expressed with various numbers of significant digits or decimal places. If insufficient precision is available then the number is rounded in some manner to fit the available precision. The following table shows the results for various total precisions and decimal places rounded to the nearest value using the round-to-even method.
The representation of a positive number x to a precision of p significant digits has a numerical value that is given by the formula:
For negative numbers, the formula can be used on the absolute value; for zero, no transformation is necessary. Note that the result may need to be written with one of the above conventions explained in the section "Identifying significant figures" to indicate the actual number of significant digits if the result includes for example trailing significant zeros.
A common guide often used when performing calculations by hand is as follows.
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 (for example, 100.0 + 1.111 = 101.1).
In a base 10 logarithm of a normalized number, the result should be rounded to the number of significant figures in the normalized number. For example, log10(3.000×104) = log10(104) + log10(3.000) ≈ 4 + 0.47712125472, should be rounded to 4.4771.
When taking antilogarithms, the resulting number should have as many significant figures as the mantissa in the logarithm.
When performing a calculation, do not follow these guidelines for intermediate results; keep as many digits as is practical (at least 1 more than implied by the precision of the final result) until the end of calculation to avoid cumulative rounding errors.
When using a ruler, initially use the smallest mark as the first estimated digit. For example, if a ruler's smallest mark is cm, and 4.5 cm is read, it is 4.5 (±0.1 cm) or 4.4 – 4.6 cm.
It is possible that the overall length of a ruler may not be accurate to the degree of the smallest mark and the marks may be imperfectly spaced within each unit. 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. Failing to do this adds the error in reading the ruler to any error in the calibration of the ruler.
When multiplying several quantities, the number of significant figures in the final answer is the number of significant figures in the factor having the least number of significant figures (the least precise). For example, given the two measurements 16.3 cm (±0.1 cm) and 4.5 cm (±0.1 cm), the range is: 16.2*4.4 – 16.4*4.6 (71.28–75.44) cm2 and the average is 73.36; however, only two significant figures (i.e. 73 cm2) can be claimed in the result (the area calculated).[clarification needed]
When adding numbers, the number of decimal places in the result is the smallest of the number decimal places in any term. For example, 123 + 5.35 == 128 and 1.001 + 0.0031 == 1.004.
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 minimum precision allowed by that sample size. The correct number of significant figures is given by the order of magnitude of sample size. This can be found by taking the base 10 logarithm of sample size and rounding to the nearest integer.
For example, in a poll of 120 randomly chosen viewers of a regularly visited web page we find that 10 people disagree with a proposition on that web page. The order of magnitude of our sample size is Log10(120) = 2.0791812460... which rounds to 2. Our estimated proportion of people who disagree with the proposition is therefore 0.083, or 8.3%, with 2 significant figures. This is because in different samples of 120 people from this population our estimate could would vary in units of 1/120 and any additional figures would misrepresent the size of our sample by giving spurious precision. To interpret our estimate of the number of viewers who disagree with the proposition we should then calculate some measure of our confidence in this estimate.
Relationship to accuracy and precision in measurement
In various technical fields, "accuracy" refers to the closeness of a given measurement to its true value; "precision" refers to the stability of that measurement when repeated many times. The number of significant figures roughly corresponds to precision, not accuracy.
Computer representations of floating point numbers typically use a form of rounding to significant figures, but with binary numbers. 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 of the number system used).
- Accuracy and precision
- Benford's Law (First Digit Law)
- Engineering notation
- Error bar
- False precision
- IEEE754 (IEEE floating point standard)
- Interval arithmetic
- Kahan summation algorithm
- Precision (computer science)
- Round-off error
- Chemistry in the Community; Kendall-Hunt:Dubuque, IA 1988
- Giving a precise definition for the number of correct significant digits is surprisingly subtle, see Higham, Nicholas (2002). Accuracy and Stability of Numerical Algorithms (2nd ed.). SIAM. p. 3.
- Myers, R. Thomas; Oldham, Keith B.; Tocci, Salvatore (2000). Chemistry. Austin, Texas: Holt Rinehart Winston. p. 59. ISBN 0-03-052002-9.
- Numerical Mathematics and Computing, by Cheney and Kincaid.
- de Oliveira Sannibale, Virgínio (2001). "Measurements and Significant Figures (Draft)" (PDF). Freshman Physics Laboratory. California Institute of Technology, Physics Mathematics And Astronomy Division. Archived from the original on June 18, 2013.
- Serway, Raymond A. (1990). Physics for Scientists and Engineers: With Modern Physics. Saunders Golden Sunburst Series (3rd ed.). Harcourt School. ISBN 0030313538.
- Subcommittee E11.30 (2013). Standard Practice for Using Significant Digits in Test Data to Determine Conformance with Specifications. ASTM International. doi:10.1520/E0029-08. ASTM E29-13.