Significant figures: Difference between revisions
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==Identifying significant digits== |
==Identifying significant digits== |
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1. All non-zero digits are significant. Example: 123.45 has five significant figures: 1,2,3,4 and 5. |
1. All non-zero digits are significant. Example: '123.45' has five significant figures: 1,2,3,4 and 5. |
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2. Zeros appearing in between two non-zero digits are significant. Example: 101.12 has five significant figures: 1,0,1,1,2. |
2. Zeros appearing in between two non-zero digits are significant. Example: '101.12' has five significant figures: 1,0,1,1,2. |
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3. All zeros appearing to the right of an understood decimal point and non-zero digits are significant. Example: 12.2300 has six significant figures: 1,2,2,3,0 and 0. The number 0.00122300 still only has six significant figures (the zeros before the '1' are not significant). |
3. All zeros appearing to the right of an understood decimal point and non-zero digits are significant. Example: '12.2300' has six significant figures: 1,2,2,3,0 and 0. The number '0.00122300' still only has six significant figures (the zeros before the '1' are not significant). |
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4. All zeros appearing in a number without a decimal point and to the right of a non-zero digit are not significant unless indicated by a bar. Example: 1300 has two significant figures: 1 and 3. The zeros are not considered significant because they don't have a bar. However, 1300.0 has five significant figures. |
4. All zeros appearing in a number without a decimal point and to the right of a non-zero digit are not significant unless indicated by a bar. Example: '1300' has two significant figures: 1 and 3. The zeros are not considered significant because they don't have a bar. However, 1300.0 has five significant figures. |
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However, this last convention is not universally used; it is often necessary to determine from context whether trailing zeros without a decimal point are intended to be significant. |
However, this last convention is not universally used; it is often necessary to determine from context whether trailing zeros in a number without a decimal point are intended to be significant. |
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Digits may be important without being 'significant' in this usage. For instance, the zeros in '1300' or '0.005' are not considered significant digits, but are still important as placeholders that establish the number's magnitude. |
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==Rounding conventions== |
==Rounding conventions== |
Revision as of 03:26, 1 August 2007
Rounding to n significant figures is a form of rounding. Significant figures (also called significant digits) can also refer to a crude form of error representation based around significant figure rounding. For this use, see Significance arithmetic.
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.
Identifying significant digits
1. All non-zero digits are significant. Example: '123.45' has five significant figures: 1,2,3,4 and 5.
2. Zeros appearing in between two non-zero digits are significant. Example: '101.12' has five significant figures: 1,0,1,1,2.
3. All zeros appearing to the right of an understood decimal point and non-zero digits are significant. Example: '12.2300' has six significant figures: 1,2,2,3,0 and 0. The number '0.00122300' still only has six significant figures (the zeros before the '1' are not significant).
4. All zeros appearing in a number without a decimal point and to the right of a non-zero digit are not significant unless indicated by a bar. Example: '1300' has two significant figures: 1 and 3. The zeros are not considered significant because they don't have a bar. However, 1300.0 has five significant figures.
However, this last convention is not universally used; it is often necessary to determine from context whether trailing zeros in a number without a decimal point are intended to be significant.
Digits may be important without being 'significant' in this usage. For instance, the zeros in '1300' or '0.005' are not considered significant digits, but are still important as placeholders that establish the number's magnitude.
Rounding conventions
When rounding to n significant digits, there are a few general rules that are followed:[1]
- If the digit immediately to the right of the nth digit is greater than 5, the number is rounded up.
- If the digit immediately to the right of the nth digit is less than 5, the number is rounded down.
- If the digit immediately to the right of the nth digit is 5 and there are non-zero digits after the 5, the number is rounded up.
- If the digit immediately to the right of the nth digit is 5 and there are no subsequent non-zero digits, there are two commonly-used conventions (see rounding for longer discussion). In 'common rounding', such a digit is always rounded up; in 'unbiased rounding' (also known as 'round-to-even'), it is rounded in whichever direction leaves the nth digit even. For instance, under unbiased rounding, 51.5 would be rounded up to 52, but 54.5 would be rounded down to 54.
Examples
Rounding to 2 significant figures:
- 12,300 becomes 12,000
- 0.00123 become 0.0012
- 0.1 becomes 0.10 (the trailing zero indicates that we are rounding to 2 significant figures).
- 0.02084 becomes 0.021
- 19,800 becomes 20,000
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One issue with rounding to n significant figures is that the value of n is not always clear. This occurs when the last significant figure is a zero. For example, in the final example above, when 19,800 is rounded to 20,000, it is not clear from the rounded value what n was used - n could be anything from 1 to 5. The level of rounding can be specified explicitly. The abbreviation s.f. is sometimes used, for example "20,000 to 2 s.f.".[citation needed] More commonly, scientific notation is used to handle the ambiguity.
Importance
The number of significant digits in a stated number is important for several reasons:
- Spurious accuracy:[2] If a sprinter is measured to have completed a 100.0 m race in 11.71 seconds, what is his average speed? By dividing the distance by the time using a calculator, we get a speed of 8.53970965 m/s Obviously, the speed of the sprinter is not known to the nearest 10 nm/s, so it is more sensible to report it to four significant figures (8.540 m/s), because the time is only measured to four significant figures.
- Comprehension - 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
Often, to reduce ambiguity, such data are represented to the nearest order of magnitude, like:
Revenue (in thousands of dollars): Division A: 185 Division B: 45 Division C: 67
Method
- Start with the left-most non-zero digit. E.g. the '1' in 1,000, or the '2' in 0.02.
- 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 rounding for more details.
Zeros appearing between nonzero digits are significant, for example:
- 60.8 has three significant figures
- 39008 has five significant figures
Zeros appearing in front of nonzero digits are not significant, for example:
- 0.093827 has five significant figures
- 0.0008 has one significant figure
- 0.012 has two significant figures
Zeros at the end of a number and to the right of a decimal are significant, for example:
- 35.00 has four significant figures
- 8,000.000000 has ten significant figures
Zeros at the end of a number without a decimal point may or may not be significant, and are therefore ambiguous, for example:
- 1,000 could have between one and four significant figures.
This ambiguity could be resolved by placing a decimal after the number, e.g. writing "1,000." to indicate specifically that four significant figures are meant.[3]
To specify unambiguously how many significant figures are implied, scientific notation can be employed:
- 1×103 or 1e3 has one significant figure, while
- 1.000×103 has four.
See also
References
- ^ Trinklein, Frederick. "1". Modern Physics. Austin, Texas: Holt Rinehart Winston. pp. 26–28. ISBN 0-03-014514-7.
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- ^ Myers, R. Thomas. "2". Chemistry. Austin, Texas: Holt Rinehart Winston. p. 59. ISBN 0-03-052002-9.
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