Significant Figures

Significant FiguresSignificant Figures refers to two things:

1) How many digits you should record when taking a measurement

2) How many digits should appear in your answer after you perform a calculation

1. Taking MeasurementsWhen recording a measurement, you should record all the

certain digits and one uncertain digit. For example consider the red line below.

The line is somewhere between 6.6 and 6.7 cm on the ruler. This is certain. Therefore the measurement would be recorded as 6.63 cm, 6.64 cm, or 6.65 cm (any one is ok). The numbers in yellow are certain. The last digit in green is uncertain and has to be estimated (hopefully it looks the same to you as it does to me, i.e. the red line ends a little less than halfway between 6.6 and 6.7).

5 6 7

Significant Figures

Now consider the alcohol thermometer to the right. If you look closely you should be able to see the top level of the alcohol is somewhere between 48 and 49. This we know for sure. Now we need to estimate one more digit. To me, it looks like the level is about 2/3 of the way between 48 and 49. Therefore, any of the following would be correct for reporting the temperature: 48.6 oC, 48.7 oC, or 48.8 oC. Once again, yellow digits are certain, green is estimated.

Significant FiguresYour turn #1:

Record the length of the red line below. The ruler is marked in cm. (click for answer)

2 3 4

Answer: 2.3 cm

Note that the 2 is certain and the 3 is an estimated digit. If you put 2.4 cm, that is ok too. To me it looks like 2.2 would definitely be too low and 2.5 would definitely be too high.

Significant FiguresA graduated cylinder, shown on the right, is used for measuring volumes of liquids. Notice that the top of the liquid is curved and not flat. This curvature is called the meniscus. (click to animate)

meniscusThe proper way to read a graduated cylinder is to record the value at the bottom of the meniscus.

Your turn #2:

Record the volume of the liquid in the graduated cylinder to the right. (click for answer)

Answer: 64 mL

6 is certain. 4 is estimated. 63 mL also ok.

Significant FiguresAnother device used for measuring volumes of liquids is a buret (shown on the right). Unlike a graduated cylinder, which measures the actual volume of the liquid in the cylinder, burets are used to measure how much liquid was used for an experiment (i.e. an initial volume is read, some of the liquid is drained out of the bottom, and then the final volume is read; the volume used is then the difference between the initial and final.) The details of exactly how to use a buret is not important now. All you need to know to answer the next question is that the graduations on a buret, as shown in the picture, are read from top to bottom. Your turn #3:

Record the level of the liquid in the buret which is shown. (click for answer)

Answer: 20.15 mL

20.1 is certain. 5 is estimated. 20.14 or 20.16 mL also ok.

Significant FiguresAt this point it is worth making a special note about electronic balances. Electronic balances are calibrated so that they automatically provide you with the correct number of significant figures. Therefore, if you were measuring the mass of something and the electronic balance read 89.2863 g, you would record exactly what the balance says, 89.2863 g.

As mentioned in the first slide, significant figures relates to two things, how to take measurements and how to perform calculations based upon these measurements. Now that we know how to take a measurement, let’s learn how to incorporate these measurements into a calculation…

Significant Figures2. Performing Calculations

When performing calculations, there are certain guidelines which determine how much we need to round off our answer. For example, consider the following calculation:

24.5 x 1.713 = 41.9685 (this is the answer given on a calculator)

Significant figure rules will tell us how much, if at all, we need to round off this answer. To answer this question we need to a) recognize how many significant figures are in a number and b) learn how to perform calculations with that number.

The rules for recognizing how many significant figures are in a number can be found on the next slide.

Significant Figuresa. Recognizing how many significant figures in a number (click to animate)

For example, 34.56 has four significant figures because all nonzero digits are significant.Another example would be that 0.0457 has three significant figures (sig figs in yellow). The leading zeros are not significant.

Another example would be that 0.000405 has three significant figures. Of course the 4 and 5 are significant, but also the zero that is “captive” between the 4 and 5.

So, 5,500 would have only two significant figures. The trailing zeros on the right are not significant. 5,500.0 would have five sig figs, however, because of the presence of the decimal point.

Significant Figuresa. Recognizing how many significant figures in a number (click to animate)

Remember:

Leading zeros are never significant.

Captive zeros are always significant.

Trailing zeros are sometimes significant (when decimal point is present).

Remember this when you practice the problems on the next page.

Significant FiguresYour turn #4:

Indicate how many significant figures are in each of the following numbers. If you need to recall the rules, go here. (click to reveal answers one at a time)

a. 406

b. 54.700

c. 0.000060

d. 5.003

e. 4.30 x 105

f. 60

g. 60.

Answers

3 sig figs (406)

5 sig figs (54.700)

2 sig figs (0.000060)4 sig figs (5.003)

3 sig figs (4.30 x 105)

1 sig fig (60)

2 sig figs (60.)

Now that we know how to recognize how many significant figures in a number, there’s one more thing we need to cover before we do calculations. This has to do with exact numbers…

With scientific notation, only consider the coefficient (4.30 in this case) and not the power of 10 (105 in this case) when evaluating sig figs.

Significant Figures…exact numbers are numbers that are not obtained by measurement.Exact numbers can be numbers which are counted.

Or, exact numbers can be numbers which are given by definition.

Either way, exact numbers are considered to be an infinite number of significant figures and they never limit the number of sig figs in an answer (you’ll see what that refers to on the next slide).

Now let’s apply sig fig rules to calculations…

Significant Figuresb. Rules for significant figures in calculations

Rule 1: For multiplication and division: Your answer should contain the same number of significant figures as the number that had the least number of significant figures for the numbers used in the calculation.

For example:

4.57 (3 sig figs)x 0.0024 (2 sig figs) 0.010968 (unrounded answer from calculator)

0.011 (2 sig figs)

This is the final answer; answer only has 2 sig figs because 2 sig figs was the least number of sig figs for the numbers used in the calculation. (Notice that the final digit was rounded up from 0 to 1 because the digit removed, 9, was equal to or greater than 5. If the digit removed was less then 5, the preceding digit, in this case 0, would have stayed the same.)

Significant Figuresb. Rules for significant figures in calculations

Rule 2: For addition and subtraction: Your answer should contain the same number of decimal places as the number that had the least number of decimal places for the numbers used in the calculation.

For example:

7.57 (2 decimal places)+ 0.0524 (4 decimal places) 7.6224 (unrounded answer from calculator)

7.62 (2 decimal places)

This is the final answer; answer only has 2 decimal places because 2 decimal places was the least number of decimal places for the numbers used in the calculation.

Significant FiguresRemember, for multiplication and division, the answer relates to the least number of significant figures. For addition and subtraction, the answer relates to the least number of decimal places.

Your turn #5:

Try this interactive significant figure activity. (Requires internet connection and Shockwave plugin. If it doesn’t work, just proceed to the next page.)

45.7 + 62.48

34.567=

108.18

34.567

Addition rules say this intermediate result should be 108.1 (45.7 had only one decimal place). But we don’t round yet. We just remember that the number really only has four sig figs (in yellow). Then we do the final division and report the answer with the least number of sig figs (four in this case).

= 3.130

Finally, a note about rounding. In a series of calculations don’t round

off until the final result. You should, however, keep a mental note of how many significant figures an intermediate result has so you can apply it to subsequent calculations; but you don’t actually round until the end. For example:

Significant FiguresYour turn #6:

Compute each of the following and report each answer to the correct number of significant figures. (click for answer one at a time; sig figs in yellow)a. 0.0040 x 3.23

= ?0.0040 x 3.23 = 0.0129

round to 2 sig figs because0.0040 only has 2 sig figs, less than the 3 sig figs in 3.23

0.013

b. 503/65.4 = ?

we need to apply multiplication and division rule

503/65.4= 7.6911round to 3 sig figs becauseboth 503 and 65.4 are considered to be the least number of sig figs (they both have three)

7.69

we need to apply multiplication and division rule

c. 800 - 47.25 = ?

800 - 47.25 = 752.75round to no decimal placesbecause 800 has no decimal places (the least) compared to 47.25 which has two decimal places

753

we need to apply addition and subtraction rule

no decimal places

two decimal places

The End