Significant Figures… Bluefield High School

1. Significant Figures… Bluefield High School

2. What is a significant digit? Significant digits is a set of internationally accepted rules for measurement. The rule of thumb is to record all digits that are certain, plus one that is estimated. The rules of significant digits apply to all reporting and calculating unless you are dealing with defined values (those by definition. For example …60 minutes = 1 hour).

3. The Rules of Significant Digits… • 1. Every nonzero digit is significant. 24.7 has three significant digits • 2. Zeroes between nonzero digits are significant. 2.07m and 109m each have three significant digits. • 3. Zeroes in front of all nonzero digits are not significant. They are merely place holders. The measurements of 0.00037m and 0.46m each have two significant digits. • 4. Zeroes at the end of the number and to the right of the decimal point are significant. The measurement 43.00m, 1.010m and 9.500m all have four significant digits. • 5. Zeroes at the end of a measurement and to the left of the decimal point are unclear. You may consider all such zeroes to be not significant. To avoid confusion, express the number in scientific notation.

4. When to use Significant figures • When a measurement is recorded only those digits that are dependable are written down.

5. When to use Significant digits… • If you measured the width of a paper with your ruler you might record 21.7cm. To a mathematician 21.70, or 21.700 is the same.

6. But, to a scientist 21.7cm and 21.70cm is NOT the same • 21.700 cm to a scientist means the measurement is accurate to within one thousandth of a cm. • significant figures video

7. How many sig digits? • 7 • 40 • 0.5 • 0.00003 • 7 x 105 • 7,000,000 • 1 • 1 • 1 • 1 • 1 • 1

8. How many sig digits here? • 1.2 • 2100 • 56.76 • 4.00 • 0.0792 • 7,083,000,000 • 2 • 2 • 4 • 3 • 3 • 4

9. How many sig digits here? • 3401 • 2100 • 2100.0 • 5.00 • 0.00412 • 8,000,050,000 • 4 • 2 • 5 • 3 • 3 • 6

10. Counting Significant Figures Number of Significant Figures 38.15 cm 4 5.6 ft 2 65.6 lb ___ 122.55 m ___ All non-zero digits in a measured number are (significant or not significant).

11. Leading Zeros Number of Significant Figures 0.008 mm 1 0.0156 oz 3 0.0042 lb ____ 0.000262 mL____ Leading zeros in decimal numbers are (significant or not significant).

12. Sandwiched Zeros Number of Significant Figures 50.8 mm 3 2001 min 4 0.702 lb ____ 0.00405 m ____ Zeros between nonzero numbers are (significant or not significant).

13. Trailing Zeros Number of Significant Figures25,000 in. 2 200 yr 1 48,600 gal 3 25,005,000 g ____ Are trailing zeros, serving as place holders in numbers without decimals, significant or not significant?

14. Learning Check A. Which answer contains 3 significant figures? 1) 0.4760 2) 0.00476 3) 4760 B. All the zeros are significant in 1) 0.00307 2) 25.300 3) 2.050 x 103 C. 534,675 rounded to 3 significant figures is 1) 535 2) 535,000 3) 5.35 x 105

15. Solution A. Which answers contain 3 significant figures? 2) 0.00476 3) 4760 B. All the zeros are significant in 2) 25.300 3) 2.050 x 103 C. 534,675 rounded to 3 significant figures is 2) 535,000 3) 5.35 x 105

16. Learning Check In which set(s) do both numbers contain the samenumber of significant figures? 1) 22.0 and 22.00 2) 400.0 and 40 3) 0.000015 and 150,000

17. Solution … 3) 0.000015 and 150,000

18. Learning Check State the number of significant figures in each of the following: A. 0.030 m 1 2 3 B. 4.050 L 2 3 4 C. 0.0008 g 1 2 4 D. 3.00 m 1 2 3 E. 2,080,000 bees 3 5 7

19. Solution A. 0.030 m 2 B. 4.050 L 4 C. 0.0008 g 1 D. 3.00 m 3 E. 2,080,000 bees 3

20. Significant Numbers in Calculations • A calculated answer cannot be more precise than the measuring tool. • A calculated answer must match the least precise measurement. • Significant figures are needed for final answers from 1) adding or subtracting 2) multiplying or dividing

21. Adding and Subtracting The answer has the same number of decimal places as the measurement with the fewest decimal places. 25.2 one decimal place + 1.34two decimal places 26.54 answer 26.5 one decimal place

22. Learning Check In each calculation, round the answer to the correct number of significant figures. A. 235.05 + 19.6 + 2.1 = 1) 256.75 2) 256.8 3) 257 B. 58.925 - 18.2 = 1) 40.725 2) 40.73 3) 40.7

23. Solution A. 235.05 + 19.6 + 2.1 = 2) 256.8 B. 58.925 - 18.2 = 3) 40.7

24. Multiplying and Dividing Round (or add zeros) to the calculated answer until you have the same number of significant figures as the measurement with the fewest significant figures.

25. Learning Check A. 2.19 X 4.2 = 1) 9 2) 9.2 3) 9.198 B. 4.311 ÷ 0.07 = 1)61.582) 62 3) 60 C. 2.54 X 0.0028 = 0.0105 X 0.060 1) 11.3 2) 11 3) 0.041

26. Solution A. 2.19 x 4.2 = 2) 9.2 B. 4.311÷ 0.07 = 3) 60 C.2.54x 0.0028 = 2) 11 0.0105 x 0.060 Continuous calculator operation = 2.54 x 0.0028  0.0105  0.060

27. 3 sig figs round to 3 sig figs 2 sig figs round to 2 sig figs Significant Figures Multiplication or Division The number of significant figures in the result is set by the original number that has the smallest number of significant figures 4.51 x 3.6666 = 16.536366 = 16.5 6.8 ÷ 112.04 = 0.0606926 = 0.061 1.8

28. Accuracy – how close a measurement is to the true value Precision – how close a set of measurements are to each other accurate & precise precise but not accurate not accurate & not precise 1.8

29. Scientific Notation A short-hand way of writing large numbers without writing all of the zeros.

30. The Distance From the Sun to the Earth 93,000,000

31. Step 1 • Move decimal left • Leave only one number in front of decimal 93,000,000 = 9.3000000

32. Step 2 • Write number without zeros 93,000,000 = 9.3

33. 7 93,000,000 = 9.3 x 10 Step 3 • Count how many places you moved decimal • Make that your power of ten

34. The power of ten is 7 because the decimal moved 7 places. 7 93,000,000 = 9.3 x 10

35. 93,000,000 --- Standard Form • 9.3 x 107 --- Scientific Notation

36. 9.85 x 107 -----> 6.41 x 1010 -----> 2.79 x 108 -----> 4.2 x 106 -----> Practice Problem Write in scientific notation. Decide the power of ten. • 98,500,000 = 9.85 x 10? • 64,100,000,000 = 6.41 x 10? • 279,000,000 = 2.79 x 10? • 4,200,000 = 4.2 x 10?

37. More Practice Problems On these, decide where the decimal will be moved. • 734,000,000 = ______ x 108 • 870,000,000,000 = ______x 1011 • 90,000,000,000 = _____ x 1010 Answers 3) 9 x 1010 • 7.34 x 108 2) 8.7 x 1011

38. Complete Practice Problems Write in scientific notation. • 50,000 • 7,200,000 • 802,000,000,000 Answers 1) 5 x 104 2) 7.2 x 106 3) 8.02 x 1011

39. 3.40000 --- move the decimal ---> Scientific Notation to Standard Form Move the decimal to the right • 3.4 x 105 in scientific notation • 340,000 in standard form

40. 6.27 x 106 9.01 x 104 tutorial 6,270,000 90,100 Write in Standard Form Move the decimal to the right.

41. 1000 mL 1L L2 1.63 L x = 1630 mL mL 1L 1.63 L x = 0.001630 1000 mL Dimensional Analysis Method of Solving Problems • Determine which unit conversion factor(s) are needed • Carry units through calculation • If all units cancel except for the desired unit(s), then the problem was solved correctly. How many mL are in 1.63 L? 1 L = 1000 mL 1.9

42. 60 min m x x x 343 60 s 1 mi s 1 hour = 767 1 min 1609 m mi hour The speed of sound in air is about 343 m/s. What is this speed in miles per hour? meters to miles seconds to hours 1 mi = 1609 m 1 min = 60 s 1 hour = 60 min 1.9

43. Please do the following questions… • Page 349 in your text… • #’s 2,3,4,5,6,8 and 9