1 2.3 Measured Numbers and Significant Figures Chapter 2 Measurements Copyright © 2008 by Pearson Education, Inc. Publishing as Benjamin Cummings

2 Measured Numbers A measuring tool Is used to determine a quantity such as height or the mass of an object. Provides numbers for a measurement called measured numbers. Copyright © 2008 by Pearson Education, Inc. Publishing as Benjamin Cummings

3. l l.... l l.... l 4.. cm The markings on the meter stick at the end of the orange line are read as The first digit 2 plus the second digit 2.7 The last digit is obtained by estimating. The end of the line might be estimated between 2.7–2.8 as half-way (0.5) or a little more (0.6), which gives a reported length of 2.75 cm or 2.76 cm. Reading a Meter Stick

4 Known + Estimated Digits In the length reported as 2.76 cm, The digits 2 and 7 are certain (known) The final digit 6 was estimated (uncertain) All three digits (2.76) are significant including the estimated digit

5 Learning Check. l l.... l l.... l 10.. cm What is the length of the red line? 1) 9.0 cm 2) 9.03 cm 3) 9.04 cm

6 Solution. l l.... l l.... l 10.. cm The length of the red line could be reported as 2) 9.03 cm or 3) 9.04 cm The estimated digit may be slightly different. Both readings are acceptable.

7. l l.... l l.... l 5.. cm For this measurement, the first and second known digits are 4.5. Because the line ends on a mark, the estimated digit in the hundredths place is 0. This measurement is reported as 4.50 cm. Zero as a Measured Number

8 Significant Figures in Measured Numbers Significant figures Obtained from a measurement include all of the known digits plus the estimated digit. Reported in a measurement depend on the measuring tool.

9 Significant Figures Copyright © 2008 by Pearson Education, Inc. Publishing as Benjamin Cummings

10 All non-zero numbers in a measured number are significant. MeasurementNumber of Significant Figures cm4 5.6 ft lb m5 Counting Significant Figures

11 Sandwiched zeros Occur between nonzero numbers. Are significant. MeasurementNumber of Significant Figures 50.8 mm min lb m 5 Sandwiched Zeros

12 Trailing zeros Follow non-zero numbers in numbers without decimal points. Are usually place holders. Are not significant. MeasurementNumber of Significant Figures cm kg mL g 5 Trailing Zeros

13 Leading zeros Precede non-zero digits in a decimal number. Are not significant. Measurement Number of Significant Figures mm oz lb mL 3 Leading Zeros

14 Significant Figures in Scientific Notation In scientific notation All digits including zeros in the coefficient are significant. Scientific NotationNumber of Significant Figures 8 x 10 4 m1 8.0 x 10 4 m x 10 4 m3

15 State the number of significant figures in each of the following measurements: A m B L C g D m Learning Check

16 State the number of significant figures in each of the following measurements: A m2 B L4 C g1 D m3 Solution

17 A. Which answer(s) contain 3 significant figures? 1) ) ) 4.76 x 10 3 B. All the zeros are significant in 1) ) ) x 10 3 C. The number of significant figures in 5.80 x 10 2 is 1) one3) two3) three Learning Check

18 A. Which answer(s) contain 3 significant figures? 2) ) 4.76 x 10 3 B. All the zeros are significant in 2) ) x 10 3 C. The number of significant figures in 5.80 x 10 2 is 3) three Solution

19 In which set(s) do both numbers contain the same number of significant figures? 1) 22.0 and ) and 40 3) and Learning Check

20 Solution In which set(s) do both numbers contain the same number of significant figures? 3) and Both numbers contain two (2) significant figures.

21 Examples of Exact Numbers An exact number is obtained When objects are counted Counting objects 2 soccer balls 4 pizzas From numbers in a defined relationship. Defined relationships 1 foot = 12 inches 1 meter = 100 cm

22 Exact Numbers Copyright © 2008 by Pearson Education, Inc. Publishing as Benjamin Cummings

23 Learning Check A. Exact numbers are obtained by 1. using a measuring tool 2. counting 3. definition B. Measured numbers are obtained by 1. using a measuring tool 2. counting 3. definition

24 Solution A. Exact numbers are obtained by 2. counting 3. definition B. Measured numbers are obtained by 1. using a measuring tool

25 Learning Check Classify each of the following as (1) exact or (2) measured numbers. A.__Gold melts at 1064°C. B.__1 yard = 3 feet C.__The diameter of a red blood cell is 6 x cm. D.__There are 6 hats on the shelf. E.__A can of soda contains 355 mL of soda.

26 Classify each of the following as (1) exact or (2) measured numbers. A. 2 A measuring tool is required. B. 1 This is a defined relationship. C. 2 A measuring tool is used to determine length. D. 1 The number of hats is obtained by counting. E. 2 The volume of soda is measured. Solution