1. Introduction to Significant Figures & Scientific Notation

2. Scientific Method Logical approach to solving problems by observing collecting data formulating a hypotheses testing Hypotheses Formulating theories

3. Significant Figures Scientist use significant figures to determine how precise a measurement is Significant digits in a measurement include all of the known digits plus one estimated digit

4. For example… Look at the ruler below Each line is 0.1cm You can read that the arrow is on 13.3 cm However, using significant figures, you must estimate the next digit That would give you 13.30 cm

5. Let’s try this one Look at the ruler below What can you read before you estimate? 12.8 cm Now estimate the next digit… 12.85 cm

6. The same rules apply with all instruments The same rules apply Read to the last digit that you know Estimate the final digit

7. Let’s try graduated cylinders Look at the graduated cylinder below What can you read with confidence? 56 ml Now estimate the last digit 56.0 ml

8. One more graduated cylinder Look at the cylinder below… What is the measurement? 53.5 ml

9. Rules for Significant figures Rule #1 All non zero digits are ALWAYS significant How many significant digits are in the following numbers? 274274 25.63225.632 8.9878.987 3 Significant Figures3 Significant Figures 5 Significant Digits5 Significant Digits 4 Significant Figures4 Significant Figures

10. Rule #2 All zeros between significant digits are ALWAYS significant How many significant digits are in the following numbers? 504 60002 9.077 3 Significant Figures 5 Significant Digits 4 Significant Figures

11. Rule #3 All FINAL zeros to the right of the decimal ARE significant How many significant digits are in the following numbers? 32.0 19.000 105.0020 3 Significant Figures 5 Significant Digits 7 Significant Figures

12. Rule #4 All zeros that act as place holders are NOT significant Another way to say this is: zeros are only significant if they are between significant digits OR are the very final thing at the end of a decimal

13. For example 0.0002 6.02 x 10 23 100.000 150000 800 1 Significant Digit How many significant digits are in the following numbers? 1 Significant Digit 3 Significant Digits 6 Significant Digits 2 Significant Digits

14. Rule #5 All counting numbers and constants have an infinite number of significant digits For example: 1 hour = 60 minutes 12 inches = 1 foot 24 hours = 1 day

15. How many significant digits are in the following numbers? 0.0073 100.020 2500 7.90 x 10 -3 670.0 0.00001 18.84 2 Significant Digits 6 Significant Digits 2 Significant Digits 3 Significant Digits 4 Significant Digits 1 Significant Digit 4 Significant Digits

16. Rules Rounding Significant Digits Rule #1 If the digit to the immediate right of the last significant digit is less than 5, do not round up the last significant digit. For example, let’s say you have the number 43.82 and you want 3 significant digits The last number that you want is the 8 – 43.82 The number to the right of the 8 is a 2 Therefore, you would not round up & the number would be 43.8

17. Rounding Rule #2 If the digit to the immediate right of the last significant digit is greater that a 5, you round up the last significant figure Let’s say you have the number 234.87 and you want 4 significant digits 234.87 – The last number you want is the 8 and the number to the right is a 7 Therefore, you would round up & get 234.9

18. Rounding Rule #3 If the number to the immediate right of the last significant is a 5, and that 5 is followed by a non zero digit, round up 78.657(you want 3 significant digits) The number you want is the 6 The 6 is followed by a 5 and the 5 is followed by a non zero number Therefore, you round up 78.7

19. Rounding Rule #4 If the number to the immediate right of the last significant is a 5, and that 5 is followed by a zero, you look at the last significant digit and make it even. 2.5350(want 3 significant digits) The number to the right of the digit you want is a 5 followed by a 0 Therefore you want the final digit to be even 2.54

20. Say you have this number 2.5250 (want 3 significant digits) The number to the right of the digit you want is a 5 followed by a 0 Therefore you want the final digit to be even and it already is 2.52

21. Let’s try these examples… 200.99(want 3 SF) 18.22(want 2 SF) 135.50(want 3 SF) 0.00299(want 1 SF) 98.59(want 2 SF) 201 18 136 0.003 99

22. Scientific Notation Scientific notation is used to express very large or very small numbers It consists of a number between 1 & 10 followed by x 10 to an exponent The exponent can be determined by the number of decimal places you have to move to get only 1 number in front of the decimal

23. Large Numbers If the number you start with is greater than 1, the exponent will be positive Write the number 39923 in scientific notation First move the decimal until 1 number is in front – 3.9923 Now at x 10 – 3.9923 x 10 Now count the number of decimal places that you moved (4) Since the number you started with was greater than 1, the exponent will be positive 3.9923 x 10 4

24. Small Numbers If the number you start with is less than 1, the exponent will be negative Write the number 0.0052 in scientific notation First move the decimal until 1 number is in front – 5.2 Now at x 10 – 5.2 x 10 Now count the number of decimal places that you moved (3) Since the number you started with was less than 1, the exponent will be negative 5.2 x 10 -3

25. Scientific Notation Examples 99.343 4000.1 0.000375 0.0234 94577.1 9.9343 x 10 1 4.0001 x 10 3 3.75 x 10 -4 2.34 x 10 -2 9.45771 x 10 4 Place the following numbers in scientific notation:

26. Going from Scientific Notation to Ordinary Notation You start with the number and move the decimal the same number of spaces as the exponent. If the exponent is positive, the number will be greater than 1 If the exponent is negative, the number will be less than 1

27. Going to Ordinary Notation Examples 3 x 10 6 6.26x 10 9 5 x 10 -4 8.45 x 10 -7 2.25 x 10 3 3000000 6260000000 0.0005 0.000000845 2250 Place the following numbers in ordinary notation:

28. Significant Digits Calculations

29. Significant Digits in Calculations Now you know how to determine the number of significant digits in a number How do you decide what to do when adding, subtracting, multiplying, or dividing?

30. Rules for Addition and Subtraction When you add or subtract measurements, your answer must have the same number of decimal places as the one with the fewest For example: 20.4 = 104.722 1.322 83

31. Addition & Subtraction Continued Because you are adding, you need to look at the number of decimal places 20.4 + 1.322 + 83 = 104.722 (1) (3) (0) Since you are adding, your answer must have the same number of decimal places as the one with the fewest The fewest number of decimal places is 0 Therefore, you answer must be rounded to have 0 decimal places Your answer becomes 105

32. Addition & Subtraction Problems 1.23056 + 67.809 = 23.67 – 500 = 40.08 + 32.064 = 22.9898 + 35.453 = 95.00 – 75.00 = 69.03956  69.040 - 476.33  -500 72.1440  72.14 58.4428  58.443 20  20.00

33. Rules for Multiplication & Division When you multiply and divide numbers you look at the TOTAL number of significant digits NOT just decimal places For example: 67.50 x 2.54 = 171.45

34. Multiplication & Division Because you are multiplying, you need to look at the total number of significant digits not just decimal places 67.50 x 2.54 = 171.45 (4) (3) Since you are multiplying, your answer must have the same number of significant digits as the one with the fewest The fewest number of significant digits is 3 Therefore, you answer must be rounded to have 3 significant digits Your answer becomes 171

35. Multiplication & Division Problems 890.15 x 12.3 = 88.132 / 22.500 = (48.12)(2.95) = 58.30 / 16.48 = 307.15 / 10.08 = 10948.845  1.09 x 10 4 3.916977  3.9170 141.954  142 3.5376  3.538 30.47123  30.47

36. More Significant Digit Problems 18.36 g / 14.20 cm 3 105.40 °C –23.20 °C 324.5 mi / 5.5 hr 21.8 °C + 204.2 °C 460 m / 5 sec = 1.293 g/cm 3 = 82.20 °C = 59 mi / hr = 226.0 °C = 90 or 9 x 10 1 m/sec

37. 1. How many significant digits are in each of the following? a. 12.5 b..00230 c..01000 d. 100.025 e. 100.000 3 3 4 66 2.Round each of the following to three sig figs. a. 125.365 b..3002536458c. 455.5 d. 278.96e. 9.96543 125. 300456 279 9.97 3.Change each of the following to scientific notation. a. 100.00 b..0020 c. 1000000 d..02500 e. 70 1.0000 x 10 2 2.0 x 10 -3 1 x 10 6 2.500 x 10 -2 7 x 10 1 4.Add the following using sig figs. a. 110.1b. 78.59681 250.326 10. 360.4 89

38. 5. Subtract each of the following using sig figs. a. 125.63b. 56.056 25.364 25.4 100.2730.7 6.Multiply each of the following using sig figs. a. 200.00 x 30.0b. 25.11 x 5.0 6.00 x 10 3 130