1.6.4 Loss of Significance

The information we get from our calculator or computer may be corrupted—in many different ways—by the sequence of operations we choose to perform. Sometimes this loss of significance is preventable, sometimes not. We have already given an example of preventable loss of significance in our tale of Anna and the exam problem.

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.

Activity 2

1. Use Maple Tool 4 to find an approximation to π / 2 to 12 digits. How many digits in your answer do you believe are correct?
2. Write down a five-significant-digit approximation to π / 2.
3. Use the approximations π ≈ 3.14 and 2 ≈ 1.414 to calculate a decimal approximation to π / 2. How many digits of the answer are significant?

Comment on Activity 2

Moral of Activity 2: You should not expect more significant digits in any answer than are in the least accurate input to the calculation.

Example 2

Now we explore a loss of significance that is not preventable. In Maple Tool 5 we calculate a 10-significant-digit value of π, 3.141592654, which is correctly rounded to nine decimal places. We also calculate 355/113 and get 3.141592920 — almost the same number. What happens if we subtract the smaller number (π) from the larger (355/113)? If we were doing it by hand, the calculation would look like this:

It looks like our answer has only three significant digits! And Maple Tool 5 confirms this by reporting 0.266 10^-6 as the answer from the subtraction. Subtraction of nearly equal numbers can be a significance killer. In particular, subtraction of two 10-significant-digit numbers that agree in the first seven digits produces an answer that has only three significant digits.

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.

Anna's subtraction of the nearly equal numbers 0.525 and 0.524 resulted in a cancellation that was disastrous for her: From numbers with only three significant digits, she ended up with an answer that had only one significant digit. However, if she had prevented the preventable part of the problem — loss from twelve down to three significant digits — it wouldn't have mattered much that she had an unpreventable loss of two digits.

There is another (small) significance issue in Example 2 — not the point of that example, but nevertheless worth noting. The 10-significant-digit answer for 355/113 was 3.141592920, which means the "0" at the end is both significant and necessary to indicate that the answer has 10 significant digits rather than 9. This is in contrast to Activity 1 (preceding page), in which the zeros at the end of 240,000,000 were all insignificant.

Checkpoint 3