UNIT 3: Numbers Large and Small

1. UNIT 3: Numbers Large and Small • Vocabulary • accuracy • precision • error • percent error • scientific notation • significant figures • powers of ten • rounding numbers • graphic representation of data

2. Accuracy refers to how close a measured value is to an accepted value. • Precision refers to how close a series of measurements are to one another. • Both Good Precision Poor Precision Good Precision • and Good Accuracy but Good Accuracy Poor Accuracy x x x x x x x x x x x x x x x x

3. Error is defined as the difference between and experimental value and an accepted value. • a- most precise • b- most accurate

4. The error equation is: error = experimental value – accepted value. • Percent errorexpresses error as a percentage of the accepted value. • When you calculate percent error, ignore any plus or minus signs because only the size of the error counts.

5. Student A Trial 1 Percent Error: • |-0.05| x 100 = 3.14% • 1.59 • Student B Trial 1 Percent Error: • |-0.19| x 100 = 11.9% • 1.59 • Student C Trial 1 Percent Error: • |+0.11| x 100 = 6.92% • 1.59

6. Scientific Notation • WHY? • We deal with very large and small numbers • FORM - 3.62 x 108 • decimal number • between 1-10 • Starting with a number greater then 10 • 47,602 → 4.7602 x 104 • Number less than 1 • .00671 → 6.71 x 10-3 • 8.5 x 107 → 85,000,000 • 4.6 x 10-5 → .000046

7. The number of places moved equals the value of the exponent. The exponent is positive when the decimal moves to the left and negative when the decimal moves to the right. 800 = 8 × 102 0.0000343 = 3.43 × 10-5 How many “sig figs” are in the numbers listed above?

8. 0.00682 • place holders known exactly estimated • three significant figures

9. Significant Figures - each of the digits of a number that are used to express it to the required degree of accuracy Example Question: How many significant figures are in the following numbers?4.321 g 4 SF 306 s 3 SF 1209 m 4 SF 0.000017 L 2 SF 001235 nm 4 SF 907.0 km 4 SF2.4050 x 10E-4 kg 5 SF300,100,000 g 4 SF 0’s between non 0’s are significant digits 0’s at beginning are never significant 0’s at end of number are always significant IF there’s a decimal point

10. Exact numbers do not affect the number of significant numbers in the answer • Example: 100 m = 1 km • 100 cg = 1 g • 28 Shamrocks

11. Rules for Significant Figures • Rule 1: Nonzero numbers are always significant. • Rule 2: Zeros between nonzero numbers are always significant. • Rule 3: All final zeros to the right of the decimal are significant. • Rule 4: Placeholder zeros are not significant. To remove placeholder zeros, rewrite the number in scientific notation. • Rule 5: Counting numbers and defined constants have an infinite number of significant figures.

12. The Atlantic-Pacific Rule: • "If a decimal point is Present, ignore zeros on the Pacific (left) side. If the decimal point is Absent, ignore zeros on the Atlantic (right) side. Everything else is significant." • If you're not in the Americas, you may prefer the following less colorful way to say the same thing: • 1. Ignore leading zeros. • 2. Ignore trailing zeros, unless they come after a decimal point. • 3. Everything else is significant.

13. Example Question: How many significant figures are in the following numbers? • a. 0.000010 L • b. 9507.0 km • c. 8.400900 x 10E-8 kg • d. 700,103,000 g • Hint: If a decimal point is included, count the zeros. If there is no decimal point, the zeros do not count. Do not start counting until the first nonzero digit is reached as viewed from left to right.

14. Multiplying • (7.86 x 10-8) (4.29 x 10-2) = 33.719 x 10-9 • 7.86 4.29 • EE EE • maybe +/- +/- • 8 2 • 7.86 -08 4.29 -02 = 3.3719 x 10-9 • (7.2 x 1012) (6.01 x 10-21) = 4.3272 x 10-8 • 7.2 12 6.01 -21 = 4.3272 -08 • (calculator result) (correct answer)

15. Multiplication and Division • To multiply, multiply the coefficients, then ADD the exponents. • To divide, divide the coefficients, then SUBTRACT the exponent of the divisor from the exponent of the dividend.

16. Example Problems: • a. (3 x 107 km) x (3 x 107 km) • b. (2 x 10-4 mm) x (2 x 10-4 mm) • c. (90 x 1014 kg) ÷ (9 x 1012 L) • d. (12 x 10-4 m ) ÷ (3 x 10-4 s) • Answers • a. 9 x 1014 km2 • b. 4 x 10-8 mm2 • c. 1 x 103 kg/L • d. 4 m/s

17. Division • 1.29 x 102 = 1.9139 x 105 • 6.74 x 10-4

18. Addition and Subtraction Involving Measured Values Exponents must be the same. Rewrite values with the same exponent. Add or subtract coefficients. Example Questions (keep answers in scientific notation): a. 5.10 x 1020 + 4.11 x 1021 b. 6.20 x 108 - 3.0 x 106 c. 2.303 x 105 - 2.30 x 103 d. 1.20 x 10-4 + 4.7 x 10-5 e. 6.20 x 10-6 + 5.30 x 10-5 f. 8.200 x 102 - 2.0 x 10-1 Answersa. 4.62 x 1021b. 6.17 x 108c. 2.280 x 105d. 1.67 x 10-4e. 5.92 x 10-5f. 8.198 x 102

19. Rules for rounding • Rule 1: If the digit to the right of the last significant figure is less than 5, do not change the last significant figure. • Rule 2: If the digit to the right of the last significant figure is greater than 5, round up to the last significant figure.

20. Round each number to five significant figures. Write your answers in scientific notation. • a. 0.000249950 • b. 907.0759 • c. 24,501,759 • d. 300,100,500 • a. 2.4995 x 10-4 • b. 9.0708 x 102 • c. 2.4502 x 107 • d. 3.0010 x 108

21. Rounding Numbers Calculators are not aware of significant figures. Answers should not have more significant figures than the original data with the fewest figures, and should be rounded.

22. Addition and Subtraction Round numbers so all numbers have the same number of digits to the right of the decimal. Multiplication and Division Round the answer to the same number of significant figures as the original measurement with the fewest significant figures. 3.43 cm + 5.2 cm = 18 cm2 6.210 L + 3 L = 9 L

23. Example Questions: • Complete the following calculations. Round off your answers as needed. • a. 52.6 g + 309.1 g + 77.214 g • b. 927.37 mL - 231.458 mL • c. 245.01 km x 2.1 km • d. 529.31 m ÷ 0.9000 s • Answers • 438.1 g • 695.91 mL • 510 km2 • 588.1 m/s

24. Graphic Representation of Data • A graphis a visual display of data that makes trends easier to see than in a table.

25. A circle graph, or pie chart, has wedges that visually represent percentages of a fixed whole.

26. Bar graphs are often used to show how a quantity varies across categories.

27. On line graphs, independent variables are plotted on the x-axis and dependent variables are plotted on the y-axis.

28. If a line through the points is straight, the relationship is linear and can be analyzed further by examining the slope.

29. Interpolation is reading and estimating values falling between points on the graph. • Extrapolation is estimating values outside the points by extending the line.

30. This graph shows important ozone measurements and helps the viewer visualize a trend from two different time periods.

31. Uncertainty in Measurements • Why is there uncertainty? • Due to nature of the measuring devise.

32. Precision and Accuracy • Often, precision is limited by the tools available: ____________ • Significant figures include all known digits plus one uncertain digit: _____________