1.6.5 How to Tell Whether a Digit Is Significant

The significance of digits in an approximation has to be measured relative to an exact value that is being approximated. Regardless of where the decimal point is located, we ignore 0's to the left of the first nonzero digit. They function only as place holders and do not tell us anything about digits of the number being approximated. Starting from the first nonzero digit on the left, we ask of each digit in turn how well it matches the corresponding digit in the exact number. We will say it matches if the error in the approximation is less than 5 in the next decimal place. For example, if we are checking for a match in the thousands place, the next place, reading from left to right, is the hundreds place. To have a match in the thousands place, the error has to be less than 500.

Example 3

Suppose the exact number being approximated is `1text[,]342text[,]709`, and the following numbers are calculated as proposed approximations:

`1text[,]340text[,]000    1text[,]341text[,]624    1text[,]342text[,]000    1text[,]343text[,]000`

How many digits of each approximation are significant?

Solution

The number `1text[,]340text[,]000` is a three-significant-digit approximation, because the error, `2709`, is less than `5000`. Notice that trailing zeros are sometimes necessary, even when they are not significant.

Similarly, `1text[,]341text[,]624` is also a three-significant-digit approximation to `1text[,]342text[,]709`, for exactly the same reason. The fact that it has other nonzero digits is irrelevant — its error is less than `5000` but not less than `500`.

The number `1text[,]342text[,]000` is yet another three-significant-digit approximation to `1text[,]342text[,]709`, but not a four-significant-digit approximation, because its error, `709`, is not less than `500`.

On the other hand, `1text[,]343text[,]000` is a four-significant-digit approximation, because its error,

`|1text[,]343text[,]000 - 1text[,]342text[,]709| = 291`,

is less than `500`. Thus, from the exact number `1text[,]342text[,]709`, we can report a four-significant-digit approximation (`1text[,]343text[,]000`) by correct rounding in the fourth place from the left.

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.

We will abbreviate the phrase “four significant digits” by 4SD, and similarly for other numbers of SDs. We provide two more examples to illustrate the meaning of SD, and then we ask you to check your understanding in the exercises.

Example 4

Find a 4SD approximation to π / 40 .

Solution

A computer or calculator approximation to π / 40 is `0.078539816`. The fourth significant digit of this approximation is in the `10^(-5)` position, so we need an error of less than `5` in the sixth decimal place. We get this by rounding to five decimal places: `0.07854`. Note that significance is not directly connected with numbers of decimal places.

Example 5

Find a 4SD approximation to 20013 10006 .

Solution

A computer or calculator approximation is `2.00009994...`. Thus a 4SD approximation is `2.000`. It does not make sense to report this answer as `2` or even as `2.0`. The first suggests either an exact answer or a 1SD answer — only the context could make it clear which you intended. The second definitely suggests 2SD. Thus sometimes trailing zeros are not only significant but necessary to convey that significance.

Contents for Chapter 1