Significant Figures

Measurement WS 6 Date:

Significant Figures

Making measurements in science most often involves using instruments. In the chemistry lab, for example, you will use rulers to measure length, graduated cylinders to measure liquid volume, and balances to measure mass. Making a measurement usually involves a certain process and the use of an instrument.

Making Measurements

The process of measurement can be illustrated by measuring the length of a wooden block, as shown in the drawing. According to the ruler, the length of the block lies somewhere between 4 and 5 cm. The exact measurement might be 4.3 or 4.4 cm. The second figure is estimated or doubtful.

Significant Figures

All measurements are doubtful to some extent. To indicate the degree of a certainty in a measurement or the number derived from measurements, scientists use significant figures. The significant figures in a quantity are those digits that are known accurately plus one doubtful digit. The block measurement contained one certain digit and one estimated or uncertain, digit. Thus it consisted of two significant figures.

The number of significant figures appropriate for a given measurement depends on the limitations of the measuring instrument. For example, a ruler with a mark for every millimeter could be read to a greater number of significant figures than the one used to measure the block. The estimated digit would be in hundredths, rather than tenths, of a centimeter.

Derived Quantities

Suppose the volume of a block was calculated using measurements of its length, width, and height. The exactness of the calculated volume would depend on the exactness of the measurements of length, width, and height. The number of significant digits in the derived quantity—that is, the volume—should indicate its degree of exactness. By using some simple rules concerning the use of significant figures, the exactness of derived quantities can be indicated.

In addition and subtraction, the result must be only as exact as the least exact measurement. For example, if a ruler measures only to the nearest 0.1cm, then the sum of measurements made with that ruler can be exact only to the nearest 0.1 cm.

Example 1 The length of an object is the sum of three measurements, each made with a different measuring device. The measurements are 27.15 cm, 5.6 cm, and 8.725 cm. What is the sum, expressed in the appropriate number of significant figures?

Solution The answer is 41.5cm. Because the least accurate measurements is doubtful in the tenths place, the sum is also doubtful in the tenths place. Measurement WS 6 Date:

In multiplication and division, the result should have no more significant figures than are present in the least exact quantity used to obtain the result.

Example 2 A piece of lead has a mass of 404.2 g and a volume of 51.6 cm3. Its density equals 404.2 g divided by 51.6 cm3. What is the density, expressed to the appropriate number of significant figures?

Solution The answer is 7.83 g/cm3. The appropriate number of significant figures is that of the less accurate of the two measurements—that is, 51.6 cm3.

When you obtain a result—with the use of a hand calculator, for example—that includes more than the appropriate number of significant figures, you must round it off. For example, if a hand calculator gave the density as 7.833333 g/cm3, you would round off to 7.83 g/cm3 because the digit following the 3 is less than five. If the result had been 7.836333 g/cm3, then you would have to round off to 7.84 g/cm3 because the digit following the 3 is greater than five. If the digit following the 3 had been five, (that is, 7.835333), then the rule is to round up to 7.84.

Zeros in Significant Figures

Zeros are not significant figures when they only serve to mark the decimal place. For example, the zeros in 0.0054 are not significant. The number could be written in exponential notation as 5.4 × 10-3, which clearly shows that it only contains two significant figures. In the number 705000, the last three zeros are probably not significant, but the first one is. In exponential notation the number might be written as 7.05 × 105. If the last three zeros were significant, the exponential form of the number would be written 7.05000 × 105.

Exercises

Indicate the number of significant figures.

1. 453.2 1. _____4______

2. 0.0356 2. _____3______

3. 11.0 3. _____3______

4. 1.75 × 10 4. _____3______

5. 0.0010 5. _____2______

Express the results of the following calculations to the correct number of significant figures.

6. 25.56 cm + 5.755 cm + 9.5 cm 6. ___40.8 cm_____

7. 7.06 × 23 7. ___1.6 x 104_____

8. 0.0012 ÷ 5.0 8. ___2.4 x 10-4_____

Round off the following number to the indicated number of significant figures: 5.40457

9. Four significant figures. 9. ____5.404______Measurement WS 6 Date:

10. Five significant figures. 10. ___5.4046_____ In-class practice

1. Non-zero digits are significant (1,2,3,4,5,6,7,8, and 9). How many significant figures?

.01426 4 33.817 5 1,418,235 7 .7 1 9.6 2 46.1 3

.23 2 .4438714 7 48.72 4

113 3 6114 4 2 1

2. Zeros that fall in-between non-zero digits are significant. How many sig figs?

30.17 4 1004.6 5 3,000.0001 8 1060.5 5 8,000.7 5 8604 4

.32000701 8 1.0001 5 5,703.0 5

3. Zeros to the right of the decimal point, but to the left of a non-zero digit are not significant; they serve only to locate the decimal point. How many sig figs?

.0051 2 .000004 1 .00205 3 .01 1 .03808 4 .000005337 4

.0666 3 .000465 3 .000000099 2

4. This zero is never significant; it helps you to locate the decimal point. How many sig figs?

0.0415 3 0.8001 4 0.0006 3 06

5. Zeros to the right of a non-zero digit and also to the right of the decimal point are significant. How many sig figs?

1.200 4 10.0 3 0.320 3 9.100 4 0.0230 3 0.00450 3

1200.0 5 70.0 3 0.040 2

1.6060 5 1,000.0 5 6,501.003 7

6. Zeros to the left of the decimal point (it is understood) and to the right of a non-zero digit are never significant; They serve only to locate the decimal point. How many sig figs?

3,200 2 400 1 10 1 Measurement WS 6 Date:

3,000,000 1 789,770 5 300,600 4

8,000,850 6 7,000 1 3,000,010 6

Caution!! If you put a decimal point after these zeros, you are saying that all of the zeros are significant digits!! How many sig figs now?

1000. 4 15,000. 5 970,000. 6 0.04 1 5015.00 6 0.005600 4 Measurement WS 6 Date:

Significant Figures and Exponential Notation

1. How many significant figures are in each of the following numbers?

(a) 7.4___2____ (b) 462___3____ (c) 1.40___3____ (d) 16,000___2____

(e) 0.072___2____ (f) 0.0130___3____ (g) 0.0058___2____ (h) 0.1090___4____

2. Write each of the following numbers in proper exponential notation: (a) 557 (a) ___5.57 x 102______(b) 0.064 (b) ___6.4 x 10-2______(c) 4,300 (c) ____4.3 x 103______(d) 382.0 (d) ___3.820 x 102______(e) 11,800,000 (e) ____1.18 x 107______(f) 0.007 (f) ____7 x 10-3______1 sf 3 sfs 3. a) Yolanda found that each of two pine boards had a mass of 3000 grams and measured 4.15 cm by 8.90 cm by 175 cm. Determine the number of significant figures in each of these numbers. 3 sfs 3 sfs

3 sfs 4 sfs b) A rectangular building block that measures 2.00 cm on each of two sides and 10.00 cm on the third side has a mass of 120.0 grams. How many significant figures are in each of these values? 4 sfs

2 sfs 3 sfs 4 sfs c) The length of a long wall is measured in three segments of 4.2 meters, 6.17 meters, and 14.15 meters. How many significant figures are in each of these measurements?

4. How many significant figures should be in the answer to each of the following calculations?

(a) 17.1 (b) 47.826 (a) ____3______+ 0.77 - 9.4 (b) ____3______

(c) 12.4 x 5.3 = (d) 1.27 x 3.1416 = (c) ____2______

(d) ____3______(e) 0.517 = (f) 0.072 = 0.2742 4.36 (e) ____3______

(f) ____2______(g) 5.82 x 760 x 325 = (h) 0.37 x 454 x 5.620 = 723 x 273 22.4 (g) ____2______

(h) ____2______Measurement WS 6 Date: