Significance arithmetic is a set of rules (sometimes called significant figure rules) for approximating the propagation of error in scientific or statistical calculations. These rules can be used to find the appropriate number of significant figures in the result of a calculation. This is important lest the result of a calculation be written with too many or too few significant figures and implicitly show an uncertainty that is either overconfident or underconfident. Understanding these rules requires a good understanding of the concept of significant and insignificant figures.

The rules are an approximation based on statistical rules for dealing with probability distributions. See the article on propagation of uncertainty for these more advanced and precise rules. Significance arithmetic rules rely on the assumption that the number of significant figures in the operands gives accurate information about the uncertainty of the operands and hence the uncertainty of the result. For an alternative see interval arithmetic.

Multiplication and division using significance arithmetic

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When multiplying or dividing numbers, the result is rounded to the number of significant figures in the factor with the least significant figures. Here, the quantity of significant figures in each of the factors is important—not the position of the significant figures. For instance, using significance arithmetic rules:

• 8 × 8 = 6 × 10¹
• 8 × 8.0 = 6 × 10¹
• 8.0 × 8.0 = 64
• 8.02 × 8.02 = 64.3
• 8 / 2.0 = 4
• 8.6 /2.0012 = 4.3

If, in the above, the numbers are assumed to be measurements (and therefore probably inexact) then "8" above represents an inexact measurement with only one significant digit. Therefore, the result of "8 × 8" is rounded to a result with only one significant digit, i.e., "6 × 10¹" instead of the unrounded "64" that one might expect. In many cases, the rounded result is less accurate than the non-rounded result; a measurement of "8" has an actual underlying quantity between 7.5 and 8.5. The true square would be in the range between 56.25 and 72.25. So 6 × 10¹ is the best one can give, as other possible answers give a false sense of accuracy. Further, the 6 × 10¹ is itself confusing (as it might be considered to imply 60 ±5, which is over-optimistic).

Exact numbers are treated as having an unlimited number of significant figures. Examples of such numbers include integer counts (e.g., the number of oranges in a bag, the divisor used in calculating a mean), legal conversion factors such as monetary conversion rates (e.g., there are 2.20371 Dutch guilders to the Euro), or constants that have a value by definition (e.g., one inch = 25.4 mm). Physical constants such as Avogadro's number have a limited number of significant digits, because these constants are only known to us by measurement.

Addition and subtraction using significance arithmetic

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When adding or subtracting using significant figures rules, results are rounded to the position of the least significant digit in the most uncertain of the numbers being summed (or subtracted). That is, the result is rounded to the last digit that is significant in each of the numbers being summed. Here the position of the significant figures is important, but the quantity of significant figures is irrelevant. Some examples using these rules:

• 1 + 1.1 = 2
  • 1 is significant up to the ones place, 1.1 is significant up to the tenths place. Of the two, the least accurate is the ones place. The answer cannot have any significant figures past the ones place.

• 1.0 + 1.1 = 2.1
  • 1.0, 1.1 are significant up to the tenths place. So will the answer.

• 100 + 110 = 210
  • 100, 110 are significant up to the ones place, even though these digits are zeroes. So will the answer.

• 1×10² + 1.1×10² = 2×10²
  • 100 is significant up to the hundreds place, while 110 is up to the tens place. Of the two, the least accurate is the hundreds place. The answer should not have significant digits past the hundreds place.

• 1.0×10² + 111 = 2.1×10²
  • 1.0×10² is significant up to the tens place while 111 has numbers up until the ones place. The answer will have no significant figures past the tens place.

• 123.25 + 46.0 + 86.26 = 255.5
  • 123.25 and 86.26 are significant until the hundredths place while 46.0 is only significant until the tenths place. The answer will be significant up until the tenths place.

Rounding rules

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Because significance arithmetic involves rounding, it is useful to understand a specific rounding rule that it is wise to use when doing scientific calculations: the round-to-even rule (also called banker's rounding). It is especially useful when dealing with large data sets or doing calculations on large data sets.

This rule helps to eliminate the upwards skewing of data when using traditional rounding rules. Whereas traditional rounding always rounds up when the following digit is 5, Banker's sometimes rounds down to eliminate this upwards bias.

See the article on rounding for more information on rounding rules and a detailed explanation of the round-to-even rule.

See also

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• Propagation of uncertainty

External links

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• The Decimal Arithmetic FAQ
• Advanced methods for handling error propagation and some explanations of the shortcomings of significance arithmetic and significant figures.

References

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• Daniel B. Delury. "Computation with Approximate Numbers". The Mathematics Teacher, v51, pp521-530. November 1958.

Elementary arithmetic | Statistics