1. The following lesson is one lecture in a series of Chemistry Programs developed by Professor Larry Byrd Department of Chemistry Western Kentucky University

2. Excellent Assistance has been provided by: Dr. Robert Wyatt Ms. Elizabeth Romero Ms. Kathy Barnes

3. PART 4 Significant Figures

4. SIGNIFICANT FIGURES III.Exact Numbers: Simple Counting and Exact Conversion Factors (Definitions) We must remember that not all numbers representing measurements have only a specific number of significant figures. Numbers that occur in Simple Counting Operations, such as, the chickens laid ten eggs, or numbers that are given in definitions (12 inches equal 1 foot, 100 centimeters equals 1 meter, etc.) are called EXACT NUMBERS.

5. SIGNIFICANT FIGURES III.Exact Numbers: Simple Counting and Exact Conversion Factors (Definitions) When using exact numbers in calculations, we will consider these numbers to have an unlimited (infinite) numbers of significant figures. We need to be concerned with significant figures only when dealing with measurements that have required some estimation.

6. SIGNIFICANT FIGURES III.Exact Numbers: Simple Counting and Exact Conversion Factors (Definitions) If we are using Exact Conversion Factors (these are defined values) such as: 36 inches= 1 yard Or 1000 millimeters = 1 liter, these are exact conversions factors that have an unlimited number of significant figures.

7. SIGNIFICANT FIGURES III.Exact Numbers: Simple Counting and Exact Conversion Factors (Definitions) However, if we use 2.54 centimeters= 1.00 inch or 454 grams= 1.00 pound, these are not exact conversion factors and each contains only "3" significant figures.

8. SIGNIFICANT FIGURES IV. Rounding-Off Numbers From Calculations: In chemistry the values that are obtained during measurements are most often used in further calculations. When using hand calculators for these calculations, the results usually contain more significant figures than are justified.

9. SIGNIFICANT FIGURES IV. Rounding-Off Numbers From Calculations: It is understood that the calculated results cannot be more accurate than the “ LEAST ACCURATE “ of the measured quantities. Thus, in most calculations we need to round-off the results to the correct number of significant figures.

10. SIGNIFICANT FIGURES IV. Rounding-Off Numbers From Calculations: The following procedure should be used: (1)First, determine how many significant figures are justified in your answer. (2)Next, starting with the first number to the left-hand side of the given value, count over until you reach the number of required significant figures. The number you are now presently at will be called the “ Last Significant Figure Number “.

11. SIGNIFICANT FIGURES IV. Rounding-Off Numbers From Calculations: The following procedure should be used: (3)This “ Last Significant Figure Number “ will either remain the same value or its value will be increased by one. The following two rules will be used to determine if the last significant figure number is increased by one or if it stays the same value: We will call this important number “The Rounding Number “ What happens to this number will be determined only by the value of the number just to the right of this last significant figure number.

12. SIGNIFICANT FIGURES IV. Rounding-Off Numbers From Calculations: a)If the rounding number is 4 or less, the value of the last significant figure number will remain the same value and all the numbers to the right of the last significant figure number will be dropped. Examples of rounding-off to four significant figures: last significant figure number The "2" is the rounding number and it is dropped 193.42 thus, to 4 significant figures the value is 193.4

13. SIGNIFICANT FIGURES IV. Rounding-Off Numbers From Calculations: Examples of rounding-off to four significant figures: last significant figure number The "4" is the rounding number (The "4" and "9" will be dropped!) 6.30649thus, to 4 significant figures the value is 6.306 a)If the rounding number is 4 or less, the value of the last significant figure number will remain the same value and all the numbers to the right of the last significant figure number will be dropped.

14. SIGNIFICANT FIGURES IV. Rounding-Off Numbers From Calculations: b)If the rounding number is 5 or greater, the value of the last significant figure number will be increased by one and all the numbers to the right of the last significant figure number will be dropped. Examples of rounding-off to four significant figures: Last significant figure number—this 5 will be increased by one. This "5" is the Rounding Number (This "5" and "3" will be dropped!) 9.06553=9.066

15. SIGNIFICANT FIGURES IV. Rounding-Off Numbers From Calculations: c)If the rounding number is 5 or greater, the value of the last significant figure number will be increased by one and all the numbers to the right of the last significant figure number will be dropped. Examples of rounding-off to four significant figures: This number, 0, is the last significant figure number and it will be increased by one The "9" is the Rounding Number ( The 9 and "6" will be dropped! ) 149.096=149.1

16. SIGNIFICANT FIGURES IV. Rounding-Off Numbers From Calculations: Some other examples of rounding-off are as follows: (1) 0.00654 to two significant figures = 0.0065 but it must be written as (2) 316.67 to four significant figures = 316.7 (3) 316.67 to two significant figures = 320 but it must be written as (4) 16.0250 to four significant figures = 16.03 (5) 94,560 to three significant figures = 94,600 but it must be written as

17. SIGNIFICANT FIGURES V. Using Significant Figures in Calculations: When using numbers in calculations that involve addition, subtraction, multiplication, and/or division, we will use two different rules to determine how many significant figures are justified in our final answer. One of the rules applies only to addition and subtraction while the other rule applies only to multiplication and division.

18. SIGNIFICANT FIGURES V. Using Significant Figures in Calculations: 1. Rule for Addition and Subtraction : When you are adding or subtracting numbers with different numbers of significant figures, you must first line them up according to their decimal points and then do the addition or subtraction. The final answer must have the same number of values to the right of its decimal point as the measurement with the “ LEAST Number “ of values to the right of its decimal point.

19. SIGNIFICANT FIGURES V. Using Significant Figures in Calculations: Study the following examples: 147.769The measurement 38.67 (two decimal places!) is the "least accurate" of the values. -38.67Thus, the answer is rounding off to a value having two decimal places. 109.099 Final Answer = 109.10 (a) Subtract 38.67 from 147.769 1. Rule for Addition and Subtraction :

20. SIGNIFICANT FIGURES Study the following examples: (b) Add 31.04, 16.4, 1.96, and 179.768 31.04 The measurement 16.4 (one decimal place!) is the "least accurate" of the values. 16.4 Thus, the answer must be rounded off so that only one decimal place remains. 71.96 179.768 299.168 Final Answer = 299.2 V. Using Significant Figures in Calculations: 1. Rule for Addition and Subtraction :

21. SIGNIFICANT FIGURES V. Using Significant Figures in Calculations: 2. Rule for Multiplication and Division : When you are multiplying and/or dividing Numbers with different number of significant figures, the answer must contain the same number of significant figures as the measurement with the least number of significant figures. The position of the decimal point IS NOT involved in this rule!

22. SIGNIFICANT FIGURES Study the following examples: a. 16.9 = 5.6521739 by calculator. 2.99 However, the answer can only contain 3 significant figures. Final answer = 5.65 b. (0.1511) (14.1) (66.66) = 142.01979 by calculator. Since 14.1 contains only 3 significant figures, the answer can have only have 3 significant figures. Final answer = 142 V. Using Significant Figures in Calculations: 2. Rule for Multiplication and Division :

23. SIGNIFICANT FIGURES Study the following examples: c.= 2475 by calculator. We can “only” have “one” significant figure in our answer. Final answer = 2000 which must be given as = 2 x 10³ Notice, that we had to give the value in Scientific Notation so everyone will know that it was only accurate to one significant figure V. Using Significant Figures in Calculations: 2. Rule for Multiplication and Division :

24. SIGNIFICANT FIGURES Study the following examples: d. = 0.5988023 x 10 = 5.988023 x 10 -8 -9-9 Final answer = 5.99 x 10 ( must have only 3 significant figures ) -9 V. Using Significant Figures in Calculations: 2. Rule for Multiplication and Division :