Section Summary
In this section we have offered some cautions about the use of computers and calculators as tools for representing numerical values. Most such devices present every answer with the same number of digits, whether or not those digits are meaningful. It is your responsibility — not your computer's — to keep track of significant digits and to avoid the errors that arise from using either too few or too many digits at various points in your problem-solving processes.
For your reference, we repeat here the “morals” of our examples and activities and the formal definitions they illustrate.
| Moral of Activity 1 Calculation steps never add significant digits, so don't always believe all the digits you see on your computer. (1.6.2) |
| Moral of Example 1 Finite machines (computers and calculators) cannot produce more significant digits than they have been programmed to produce. In particular, they cannot produce an exact numerical answer if that answer requires an infinite decimal expansion. (1.6.3) |
| Moral of Anna's Tale Don't discard digits in an intermediate result. The only time you should round off is at the end of your calculation. (1.6.4) |
| Moral of Activity 2 You should not expect more significant digits in any answer than are in the least accurate input to the calculation. (1.6.4) |
| Moral of Example 2 Watch out for disastrous cancellations. If you can't arrange your work to avoid them, at least be aware that your numbers have fewer significant digits as a result. (1.6.4) |
| Definitions Leading zeros are never significant. The decimal digit in the `10^k` place of an approximation y to a number x is significant if it is not a leading zero, and if
| x - y | < 0.5 × 10 k .
The number `y` is an `n` significant digit approximation to `x` if, after discarding leading zeros, the first `n` of the digits of `y` (reading from left to right) are significant in the sense just defined. |
