Chapter 1
Relationships





1.6 What's Significant about a Digit?

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 the pop-up calculator to find an approximation to π / 2 to 16 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 2Comment 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. Suppose we write down a 10-significant-digit value of π, `3.141592654`, which is correctly rounded to nine decimal places. We also calculate `355//113` to 10 significant digits 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:

`3.141592920`
`-3.141592654`
`0.000000266`

It looks like our answer has only three significant digits! And our computer algebra system confirms this by reporting `0.266 times 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. [If you try this with the pop-up calculator, you will be subtracting 16-significant-digit numbers, and only 9 digits of the answer will be correct, even though 16 digits will be reported.]

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 `240text[,]000text[,]000` were all insignificant.

Checkpoint 3Checkpoint 3

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