Significance Arithmetic - Multiplication and Division Using Significance Arithmetic

Multiplication and Division Using Significance Arithmetic

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
• 2 × 0.8 = 2

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; more accurate would be 64 ±8).