PROBLEM 4. What are the absolute and relative errors if a computer that has an eight-bit binary digit string represents 0.2 as 0.11001100 x 2 ^-2 ?
Solution: From Problem 3c, the absolute error is l/l280 or about 0.0008. The relative error is approximately 0.0008/0.2 = 0.004, or 0.4 percent
The use of significant figures is helpful in error analysis. The number of significant figures is defined as the number of digits that can be assumed to be correct, starting at the left with the first nonzero digit, and proceeding to the right. By this definition, 10.62, 0.05713, and 4.600 all have four significant figures. A number such as 4300 is ambiguous. This ambiguity may be resolved by using scientific notation, since we may write the number as 4.3 x 10^3, 4.30 x 10^3, or 4.300 x 10^3 according to whether the number has two, three, or four significant figures, respectively.
When approximate numbers are added or subtracted, it can be shown that the absolute error in the sum or difference could be as large as the sum of the absolute errors of the individual numbers. When approximate numbers are multiplied or divided, it can be shown that the relative error of the result could be as large as the sum of the relative errors of the individual numbers. This means that for sums and differences of approximate numbers, the number of decimal places considered significant can never be greater than the number of decimal places in the least precise addend. For products and quotients, the number of significant figures can never be more than the smallest number of significant figures in the individual factors. Wherever appropriate, numerical results will be given in accordance with these guidelines.