Significant Digits

A concept that may be unfamiliar to you is the concept of significant digits (also commonly referred to as significant figures). While the concept itself is relatively straightforward - it relates to the precision of a measurement - students frequently have to work hard to apply the rules consistently. Be sure to read this section carefully, do all the practice problems, and ask your instructor for help if you are having a difficult time with it. You will be expected to know how to identify significant digits and apply the concepts to your work throughout this course; it's better to spend a bit more time now to make sure you understand than to skim it and struggle the rest of the term.

Precision & Accuracy

Accuracy and precision are concepts that you are likely already familiar with. Precision and accuracy are both important in chemistry. They are related, but they refer to different aspects of a measured or calculated value. Accuracy refers to how correct a value is. Precision refers to how many digits are used in the number.

For example, consider the value of pi (the ratio of a circle's diameter and circumference). Writing "3.141592" is both an accurate and precise way of writing pi, as it is close to the real value, and it includes a lot of digits. Writing "3.1" is still accurate (though a little less) but not nearly as precise, having fewer digits.

Writing something like "7.81564" would be a precise value for pi, but not a very accurate one. It has lots of digits, but isn't very close to the real value. Writing "7.8" is neither accurate nor precise.

In day to day living the importance of precision and accuracy varies considerably with the situation. If you are describing a window in terms of how it relates to its room, then you might say it is four feet by three feet in size. Not very precise but it gets the message across. On the other hand, if you were giving someone the dimensions to cut a piece of glass so that you could replace a broken window, then saying 4 ft x 3 ft would not be nearly precise enough. If it is cut 1/16th of an inch too large, it won't fit in the frame. If it is cut too small, it won't fill the frame and will either fall through or not seal properly.

Note that whether we deal with general dimensions or with replacing the glass, the precision necessary for an adequate description was dictated by the situation.

Also note that in each case, the numbers used to describe the size were rounded off. Any measured value (except for counting) cannot be described exactly. Measured values are rounded off in a way that depends on how they are measured. Where the number is rounded off is its precision. Whether it is measured correctly is its accuracy.

Significant Digits in Numbers You Read

Counted numbers are exact. For example, if I were to count the number of eggs in my refrigerator, I could get an exact number - say, 5 eggs. Assuming that I didn't miscount - that I made an accurate count - then you know that I have exactly 5 eggs. Measured numbers, on the other hand, are never exact; there is always some component of precision and estimation involved. The amount of precision is determined by the device used to make the measurement; we'll discuss how to make measurements with the correct precision in a later section. Scientists use the concept of significant digits to express the precision of a measured number. When you are reading a measured number, it is important to be able to determine how many significant digits are in the number so that you can retain the proper precision when you perform calculations.

Well, then, how do you determine if a digit is significant or not? Since significant digits represent known values, there is a set of rules to help you figure it out.

• Significant digits represent known values. (We'll use "sd" as shorthand for "significant digits.")

• Every non-zero digit is considered significant.

1223 has 4 sd

12,417 has 5 sd

12,182,373 has 8 sd

• Zeros can be significant digits when they represent a known value rather than an unknown value.

• Between non-zero digits they are always significant (or between non-zero and “known-zero” digits).

2001 4 sd

0.403 3 sd

40.0 3 sd

• At the trailing end of decimal fractions they are always significant.

0.120 3 sd

34.20 4 sd

17.0 3 sd

• A zero with a decimal after it, even if no other digits follow, is significant. (The decimal is indicating that it is a known value.)

1750. 4 sd

• Occasionally trailing end of non-decimal numbers when the zeros are known to be zero values. These are impossible to detect without appropriate clues; if no additional information is given, assume that they are placeholders and not significant digits.

140 2 sd; possibly 3 sd if specified

7200 2 sd; possibly 4 sd if specified

• Placeholders show size without actually indicating a known value. (ph = placeholders)

12 hundred 1200 2 sd 2 ph

12 thousand 12,000 2 sd 3 ph

12 million 12,000,000 2 sd 6 ph

12 millionths 0.000012 2 sd 4 ph

Let's see some examples of these rules.

418 g 3.82 mL

The first two are pretty straight forward: 418 grams contains 3 significant digits; 3.82 milliliters contains 3 significant digits also. Notice that the position of the decimal point is not a factor in determining how many significant digits there are in a number.

4.002 kg

In the next one, 4.002, the zeros are significant digits. They're not there to hold the decimal point; they are there to show that zeros were measured. The 4 was measured; the zeros were measured; then a 2 was measured in the last place. All of those digits were measured.

741.80 g

The next one, 741.80 grams, contains 5 significant digits. The zero is a significant digit. It is not there to hold a decimal point. The zero is there to show precision. The only reason for writing down that zero is to show that it was measured.

0.003 m 3 mm

The leading zeros in the next measurement (0.003 m) are holding the decimal point to show size. They are not significant digits. Nothing was actually measured until the 3 showed up. If this value were measured in millimeters, the value would be 3 mm. Notice how the zeros disappear? They are no longer needed, they were for size rather than for precision. You could apply a similar check by changing the number to scientific notation: 3 x 10^-3 mm. The zeros disappear since they were only needed for size.

74000 g

The next value (74000 g) returns us to the dilemma of the zeros. Certainly the 7 and the 4 are significant digits. Some or all of the zeros might be, depending on how precisely the measurement was made. There are at least two and perhaps as many as five significant digits. Since there is doubt, just say there are two. This dilemma could have been avoided in several ways by the person who recorded this measurement. I could have used kilograms instead of grams, then the place-holding zeros would not have been needed. (74 kg). Or I could have used scientific notation (7.4*10³ g).

Quick Quiz on Significant Digits

Now try your hand at the examples below:

70.0 kg

4.32 kg

0.0033 g

4100 g

40.007 g

0.28 mL

0.010 mL

Answers:

70.0 kg 3

4.32 kg 3

0.0033 g 2

4100 g 2

40.007 g 5

0.28 mL 2

0.010 mL 2

Practice Problem 5 in your workbook can help you get more experience with significant digits.