1.6.3 Accuracy of Calculating Devices

Some of our important functions are mathematical models of reality that are defined by formulas — usually formulas that give an exact value for every input number but which are at best approximations to the reality being modeled. Reality aside, it may be important to know something about exact values — and our ability to know these values may be limited by the accuracy of our computer.

Example 1

If we ask the pop-up calculator for a value of `log_10 text[(]3 text[)]`, it responds with `0.47712125471966244`. How many of these digits are significant?

Solution

The leading zero in the answer is not significant. In fact, leading zeros are never significant. Whether you do or do not put a zero in front of a decimal fraction is a matter of style and clarity. This answer has 17 significant digits (count them), but we have to trust the programmer of the software that the 17^th digit is correct — or check it against some more accurate source. (That's easily done with a computer algebra system by just repeating the calculation with a larger number of digits. In case you're interested, we have checked that last “`4`”, which would be “`3730`” if we went out to 20 digits, so the rounding to “`4`” in the 17^th place is correct.) Note that this answer is not the exact value of `log_10 text[(]3 text[)]`, because exact representation of that number as a decimal expansion would require infinitely many digits.

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.

Checkpoint 2

Contents for Chapter 1