Significance arithmetic

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.

Contents

1 Multiplication and division using significance arithmetic 2 Addition and subtraction using significance arithmetic 3 The even-odd rule, also known as bankers' rounding 4 External links 5 References

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. Some examples using significant figures rules:

• 1 + 1.1 = 2
  • Because 1 has only one significant digit, the answer must as well.

• 1.0 + 1.1 = 2.1
  • 1.0, 1.1, and 2.1 each have two significant digits.

• 100 + 110 = 200
  • 100 has one significant digit, while 110 has two. The answer must have one significant digit.

• 1.0×10² + 111 = 210
  • 1.0×10² has two significant digits, while 111 has three.

• 123.25 + 46.0 + 86.26 = 256
  • 123.25 has five significant digits, 46.0 has three, and 86.26 has four.

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, when dealing with finite data, such as money, to always round up or down if the digit is equal to exactly five would skew data in one direction or the other. (The normal rule to round up on 5 is based upon the real underlying value 'measured' being a little bit more than 5. This is not true for descrete data, like money.)

Thus, when using the significant figures system and rounding in such a 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.
• If 2.5 had to be rounded to one significant figure, it would become 2.

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

External links

• The Decimal Arithmetic FAQ

References

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