Significance arithmetic - Definition and Overview

Significance arithmetic is a collection of rules-of-thumb which attempt to indicate the propagation of error in a scientific experiment or in statistics when perfect accuracy is not attainable or not required.

The rules are derived on an assumption that the number of significant figures in the operands to an operation is a useful guide to the error bounds of the number.

Multiplication and division using significance arithmetic

When multiplying and dividing numbers together, the product or quotient is rounded to the number of significant figures of that of the factor with the least. For instance, using significant figures rules:

• 8 × 8 = 60
• 8 × 8.0 = 60
• 8.0 × 8.0 = 64
• 8.02 × 8.02 = 64.3

In the above, all numbers are assumed to be measurements (therefore potentially inexact). For example: the answer yielded from 8 × 8 is actually 64, but because 8 is treated as a measurement, it only has one significant figure, and so the answer must be rounded to 60. If we are particularly unlucky in the measurement, this still might be incorrect; if each "8" is actually nearly 8.5, the result could be over 70.

Exact numbers are treated as having a limitless number of significant figures. A trivial example of such a number would be the quotient used in taking the mean, or a defined conversion factor.

When squaring or taking the square root of a value, the number of significant figures decreases by one using some systems of significant digits.

Addition and subtraction using significance arithmetic

When you add or subtract significant figures, limit to, and round your answer to the least number of decimal places in any of the numbers that make up the problem. For instance, using significant figures rules:

• 1 + 1.1 = 2

(The answer in significant figures is 2, because 1 has no decimal place, so the answer can have no decimal place)

• 1.0 + 1.1 = 2.1

(The answer in significant figures is 2.1, since 1.0 and 1.1 both have one decimal place, so the answer must have one decimal place also)

• 100 + 110 = 200

(The answer in significant figures is 200, since 100 and 110 have no decimal places, so the answer cannot have any decimal places either)

• 1.0 × 10² + 111 = 210

(The answer in significant figures is 210; since 1.0×10² has no ones digit or decimal places, the answer cannot have either)

• 123.25 + 46.0 + 86.257 = 255.5

(The answer in significant figures is 255.5, because 46.0 only has one decimal place, so the answer can only have one decimal place)

The even-odd rule, also known as bankers' rounding

As with all rounding procedures, if the number directly to the right of the digit to be rounded to is less than five, the digit stays the same; if more than five, the digit is rounded up. However, to always round up or down if the digit is equal to exactly five would skew data in one direction or the other. Thus, when using the significant figures system and rounding in such situation, the even-odd rule is used: round in whichever direction would make the last digit of the final product even. For example:

• If 3.5 had to be rounded to one significant figure, it would become 4, since four is even
• If 2.5 had to be rounded to one significant figure, it would become 2, since two is even

In this way, the even-odd rule avoids skewing data either upwards or downwards.

References

See the decimal arithmetic FAQ (http://www2.hursley.ibm.com/decimal/decifaq4.html#signif) for a discussion of why significance arithmetic results in grossly inaccurate error bounds. The classic paper on this is: Computation with Approximate Numbers, Daniel B. Delury, The Mathematics Teacher 51, pp521-530, November 1958.