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Significant Figures and Scientific Notation

Significant Figures and Scientific Notation. Measurement in the scientific method. The key to a good experiment is being able to make good measurements and record our data in the proper way.

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Significant Figures and Scientific Notation

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  1. Significant Figures and Scientific Notation

  2. Measurement in the scientific method • The key to a good experiment is being able to make good measurements and record our data in the proper way. • If we do not make good measurements, we will have incorrect data. Incorrect data results in bad results and wrong conclusions.

  3. What Makes a Good Measurement? • Accuracy – • how close your measurement is to the correct value • Precision – • how close one measurement is to all other measurements in the experiment

  4. Error • ALL MEASUREMENTS HAVE ERROR • So what is error? • Error is a measure of how far off you are from correct • Error depends on a number of different factors, but the main source of error is the tools we use to make the measurements • The accuracy of a measurement is dependent upon the tools we use to make the measurement.

  5. Error • When recording our measurement, we have to make a guess…the guess shows our error…the smaller the guess, the more accurate our measurement… • How long is the box? I know the measurement is at least 5 cm… 0 1 2 3 4 5 6 7 8 cm But since there are not marks between the centimeters, that is all we know for sure… So we guess…5.8cm, the last digit shows our guess Since each person guesses different, the last digit shows our error

  6. Error • To Get less error, we use a tool with smaller guesses… 0 1 2 3 4 5 6 7 8 What is the measurement with a more accurate tool? 5.91 cm…the 1 is a guess

  7. Minimizing Error • There are many ways to minimize your error in measurement, but the main ways are… • Using more accurate tools – since your measurement can only be as accurate as your tool, if your tool is more accurate, your measurement will be more accurate. • Avoiding parallax – • parallax is the apparent shift in location due to the position of an observer… • We avoid parallax by…(volunteers needed for demonstration)

  8. Using Measurements • Since our measurements all have error, when we use them in calculations, we have to carry the error through… • How do we do this you ask? • Significant Figures…

  9. Significant Figures • Significant figures are scientists way of showing accuracy in measurements and in calculations • JUST BECAUSE IT IS ON YOUR CALCULATOR SCREEN DOES NOT MAKE IT SIGNIFICANT!

  10. Rules for Identifying sig. figs. In a Measurement • All non-zero digits are significant 1, 2, 3, 4, 5, 6, 7, 8, 9 • Leading zeros are place holders and not significant 0.0000000000000000002 • Trailing zeros are only significant if they are to the right of the decimal 1000000  zeros are not significant 1.00000  zeros are significant • Zeros between two significant figures, or between a significant digit and the decimal are significant 101  zero is signigicant 10.0  all zeros are significant 10000.  all zeros are significant

  11. Significant Figures • So, is there an easy way to figure this out without memorizing the rules… • YES!

  12. Sig Fig Tool We will use our great nation to identify the sig figs in a number… On the left of the US is the Pacific and on the right is the Atlantic P A

  13. Sig Fig Tool If we write our number in the middle of the country we can find the number of sig figs by starting on the correct side of the country… If the decimal is Present, we start on the Pacific side If the decimal is Absent, we start on the Atlantic side We then count from the first NON zero till we run out of digits… P A 0.05600

  14. Sig Fig Tool Examples P 105200 A 4 This number has _____ sig figs

  15. Sig Fig Tool Examples P 307.10 A 5 This number has _____ sig figs

  16. Sig Fig Tool Examples P 0.00009 A 1 This number has _____ sig figs

  17. Sig Fig Tool Examples P 105200. A 6 This number has _____ sig figs

  18. Calculations with significant figures • Since our measurements have error, when we use them in calculations, they will cause our answers to have error. • Our answer cannot be more accurate than our least accurate measurement. • This means that we have to round our answers to the proper accuracy…

  19. Rounding Rounding is the process of deleting extra digits from a calculated number. 1. If the first digit to be dropped is less than 5, that digit and all the digits that follow it are simply dropped. 1.673 rounded to three significant figures becomes 1.67 2.If the first digit to be dropped is greater than or equal to 5, the excess digits are all dropped and the last significant figure is rounded up. 62.873 rounded to three significant figures becomes 62.9

  20. Round The following • 423.78 to three significant figures • 424 • B. 0.000123 to two significant figures • 0.00012 • C. 22.550 to four significant figures • 22.55 • D. 129.6 to three significant figures • 130. (must have decimal) • E. 0.365 to one significant figure • 0.4 • F. 7.206 to three significant figures • 7.21

  21. Calculations with significant figures • When we add or subtract, our error only makes a small difference. So, when adding or subtracting we base our rounding on the number of decimal places. • Rule for Adding and Subtracting – • the answer must have the same number of decimal places as the measurement used in the calculation that has the fewest decimal places

  22. Example 1 35.0 cm + 2.98 cm – 7 cm = ? 30.98 cm This is what your calculator gives you… However, as we just discussed, the answer cannot be more accurate than your least accurate measurement… The least accurate measurement is 7 cm… So by the adding rule, our answer must be rounded to zero decimal places, or the ones place Which gives us the answer of 31 cm

  23. Calculations with significant figures • When we multiply or divide, our error makes a large difference. So, when multiplying or dividing numbers, we round based on significant figures. • Rule for Multiplying and Dividing – • the answer must have the same number of significant figures as the measurement used in the calculation that has the fewest significant figures

  24. Example 2 3.0 x 89.54 ÷ 0.000000001 = ? 268620000000 We have to round to proper sig figs… So we get 300000000000 Or in scientific notation (which we will discuss next) 3 x 1011

  25. Example 3 • What if we have both add/sub and mult/div in the same problem? (2.4 m + 5 m) ÷ (1.889 s – 3.9 s) = ? Order of operations means we do the addition and subtraction first… (7.4 m) ÷ (-2.011s) We have to round these before we go on to the division… 7 m ÷ -2.0 s Now divide -3.480855296 m/s Now Round -3 m/s

  26. Scientific Notation • Use to express very large or small numbers • 60221415000000000000000 or 6.02x1023 • 0.000000000000000000435 or 4.35x10-19

  27. Scientific Notation • Scientific (Exponential) notation is a system in which a number is expressed as a product of a number between 1 and 9 multiplied by 10 raised to a power (exponent).

  28. Scientific Notation • C x 10n • C= the coefficient= only the significant figures are used • n= the exponent (power)= location of the decimal point

  29. Scientific Notation 1. The value of the exponent is determined by counting the number of places the original decimal point must be moved to give the coefficient. Remember that the coefficient must be a number from 1 to 10.

  30. Scientific Notation 2. If the original number (in standard notation)is greater than 1, the exponent is a positive number. 60221415000000000000000 or 6.02x1023 3. If the original number is less than 1, the exponent is a negative number. 0.000000000000000000435 or 4.35x10-19

  31. Example 1 • Write 628,000 in scientific notation a. Determine the number of sig figs __ 3 b. Write the coefficient ____ 6.28 c.Determine the number of places to move the decimal point __ 628000  larger than 1 = positive exponent 5 Answer = 6.28 x 105

  32. Example 2 • Write 0.00260 in scientific notation a. Determine the number of sig figs __ 3 b. Write the coefficient ____ 2.60 c. Determine the number of places to move the decimal point __ 0.00260  smaller than 1 = negative exponent 3 Answer = 2.60 x 10 -3

  33. Write the Following in Scientific Notation • 2305.7 • 2.3057 x 103 • 3,000,000 • 3 x 106 • 300. • 3.00 x 102 • 0.00010 • 1.0 x 10-4 • 402.0 • 4.020 x 102 • 0.1005 • 1.005 x 10-1 • 3.57 • 3.57 or 3.57 x 100

  34. Scientific Notation • To convert from scientific notation back to standard (ordinary) notation: • Simply move the decimal point the number of places indicated by the exponent. • Positive exponent= make number larger • Move decimal to the right  • Negative exponent= make number smaller • Move decimal to the left 

  35. Write the Following in Standard Notation a. 3.4 x 103 3400 b. 8.10 x 10-2 0.0810 c. 5.600 x 103 5600. d. 6.5 x 10-4 0.00065 e. 5 x 106 5,000,000

  36. The overbar • You can make a zero in the middle of a number have significance by adding an overbar above it. • 3,300 has 3 sig figs (or 3.30 x 103) • 2,000,000,000 has 4 sig figs (or 2.000 x 109)

  37. THE END Presentation created by: Mr. Kern Information gathered from years of scientific research and data collection Assignment provided by : BHS Chemistry Department

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