Calculations: Significant Digits

In this section we will deal with the issue of significant digits in numbers that have been calculated from measured numbers. The general principle to remember is the precision of a calculated number is limited by the precision of the numbers from which it was calculated. Expressing that precision is done by simply rounding off the calculated number to the proper place. The method for determining where that place is depends on whether the calculation is addition, subtraction, multiplication, division, or a combination of them.

General Guidelines

When you round off a number, it is generally for one of two reasons: convenience or precision. When you round off for convenience, you can do it any way you want. When you round off for precision you have to look at the precision of the numbers that went into the calculation, and make the precision of the answer match the precision of the least precise measurement.

As in the experiment for determining density, the numbers used in calculations are often from measurements. Measurements by their nature have limited precision. Any numbers obtained from the calculations using measurements will also be limited to the same degree of precision as the measurements that went into the calculation. Roughly speaking, any calculated value is no more precise than the least precise number that goes into the calculation.

We will deal with two different kinds of situations. One is for addition and subtraction. The other is for multiplication and division.

Rounding Off After Addition and Subtraction

When adding and subtracting, we round off the answer to the same number of decimal places that are in the number with the fewest decimal places. Notice that I said decimal places, not significant digits. Always do the math and then round off - never do the rounding first!

0.123 + 0.421012 = 0.544

With decimal fractions this can be pretty straight forward, as in the first case here (0.123 + 0.421012). Notice that the "012" part is lost in the final answer. There are no corresponding numbers for them to be added to. We don't know what comes after the 0.123.

0.123 + 0.000012 = 0.123

Sometimes this can be a little bit troublesome as in the next case (0.123 + 0.000012) where the entire second number is lost because the imprecision and the uncertainty of the 0.123 is larger than the number to be added. Consequently, it doesn't get represented.

4100 + 304 = 4400

In this last case we will presume those zeros not to be significant. When 4,100 is added to 304, what we get becomes 4,400. The four in the ones place is not shown. We are not really sure what it is being added to. If we put down 4,404 as an answer, that would imply that we knew that those zeros in the 4,100 were significant digits. If we do know that, then the answer is 4,404. Since we don't know, the last four gets lost in the uncertainty.

Rounding Off After Multiplication and Division

When rounding off the results of multiplication or division you use the least number of significant digits rather than the least number of decimal places.

1.0/3.0 = 0.33

In the first of these, 1.0 and 3.0 both have two significant digits, so the value calculated from 1.0 divided by 3.0 is rounded off to two digits.

1.00000/3.0 = 0.33

In the second of these examples, going across, 1.00000 is very precise with six digits; but 3.0 only has two digits, so the answer is rounded off to two digits.

1/8 = 0.1

Now, this one you may not like because 1/8 is not equal to 0.1. It is supposed to be 0.125, and it is if you have exactly 1/8. But if the one and the eight are both measurements made with only one significant digit of precision, then the answer must also be rounded off to one digit. Now, if a person happened to know that there was more precision, then they should write down additional digits. (By the way, one significant digit is not very precise for any measurement.) If you want your results to come out to three digits of precision, then your measurements have to be three digits of precision also.

1.000/8.00000

In the next case, 1.000 has four digits and 8.00000 has six digits, so the answer is rounded off to four digits. Notice here, even though the last digit is a zero, I put it there because it shows the degree of precision to which the original measurements were made. Calculators by the way won't do that for you. They will drop trailing zeros. Consequently you may have to put them in yourself when it is appropriate.

2.0*3.00 =6.0

The same technique is used for multiplication as well as division. Some more examples are shown, 2.0 x 3.00 is equal to 6.0. Two digits because the 2.0 has two digits. The zero is there because it is the second significant digit and it shows the degree of precision to which the number is known.

4.321*1.2 = 5.2

Next, 4.321*1.2. The answer will be rounded off to two digits because the 1.2 only has two digits.

Let me comment about why we can usually use the least number of significant digits as the key to where to round off the answer. Remember, we are rounding off to indicate the degree of precision or uncertainty in a calculated value. Precision is the part of the number you do know and uncertainty is the part you don't know. You round off at the dividing line between them.

2/467 = 0.00428266

Let's take this calculation as our starting point.

2/467 = 0.00428266 2/468 = 0.0042735

The denominator is a three digit number. We are not completely certain about that last digit. It says 7 but it could be an 8. If it were, the calculated value would be different. That difference shows up in the third significant digit, matching the position of the last digit in these three digit numbers in the denominator.

2/467 = 0.00428266 3/467 = 0.00642398

The numerator is a one digit number. It says 2 but it might possibly be a 3. If it were, the calculated answer would again be different. In this case the difference shows up in the first digit, matching the position of the last digit in these one digit numbers.

Depending on the numbers used, the match-up of variation and significant digits will not always work so smoothly, but it will be close. So we will round off calculations to the least number of significant digits rather than calculate out the various possibilities, unless of course you want to.

Quick Quiz on Rounding Off

Perform each of the following calculations and round off the answer to the appropriate number of significant digits. The answers follow at the bottom of the page.

Question 1:

13 / 72 =

a. 0.18055

b. 0.1806

c. 0.181

d. 0.18

e. 0.2

Question 2

6.7 x 7.6 =

a. 50.920

b. 50.92

c. 50.9

d. 51

e. 50

Question 3

1.30 x 6.789 =

a. 8.8257

b. 8.826

c. 8.83

d. 8.82

e. 8.8

f. 9

Question 4

0.003 x 6.789 =

a. 0.020367

b. 0.02037

c. 0.0204

d. 0.020

e. 0.02

Question 5

2.0407 + 0.02 =

a. 2.0607

b. 2.061

c. 2.06

d. 2.1

e. 2

Question 6

4.03 - 4.021 =

a. 0.0090

b. 0.009

c. 0.01

Answers

Question 1

13 / 72 =

d. 0.18

Question 2

6.7 x 7.6 =

d. 51

Question 3

1.30 x 6.789 =

c. 8.83

Question 4

0.003 x 6.789 =

e. 0.02

Question 5

2.0407 + 0.02 =

c. 2.06

Question 6

4.03 - 4.021 =

c. 0.01

More practice with significant digits in calculations can be found by completing Practice Problem 6 in your workbook.