Numerical Methods in Science

1. Numerical Methods in Science --How many scientists does it take to change a light bulb? --Scientists don’t change light bulbs, that’s what engineers are for.

2. Rounding • Choose where (at which digit) you want to round. • If the NEXT digit is 5 or more, round up; otherwise round down • Rounding does not change the size of the number, just its precision.

3. 27,454,352 Round to the nearest million .00088536 Round to the nearest 100,000th 7432 Round to the nearest ten .0653 Round to the nearest 1000th Examples

4. 27,454,352 Round to the nearest million .00088536 Round to the nearest 100,000th 7432 Round to the nearest ten .0653 Round to the nearest 1000th Examples

5. 27,454,352 Check .00088536 Check 7432 Check .0653 Check Examples

6. 27,454,352 Check .00088536 Check 7432 Check .0653 Check Examples Round down Round down Round down Round up

7. 27,454,352 =27,000,000 (fill in 0’s to keep the same size) .00088536 =.00089 (change the 8 to 9, do not fill in 0’s after a decimal!) 7432 =7430 (fill in 0 to keep the same size) .0653 =.065 (do not fill in 0’s after a decimal!) Examples

8. Round to the nearest: • 1.22 (tenth) • .0004528 (1000th) • 12,900,000 (million) • .00100 (10000th) • 3,045,000,000 (million) • .00003 (100th) • 7 (10)

9. Significant figures • All non-zero digits are significant • Zeros • A) Leading, not significant. • B) Trapped (by SF)--significant • C) Trailing, with a decimal--significant

10. Which digits are SF? • 1.22 • .0004528 • 12,900,000 • .00100 • 3,045,000,000 • .00003 • 5.30 x 10 14

11. Which digits are SF? • 1.22 • .0004528 • 12,900,000 • .00100 • 3,045,000,000 • .00003 • 5.30 x 10 14

12. Adding and subtracting 1.22 + .452 =

13. Adding and subtracting 1.22 + .452 = 1.67 Your calculator says “1.672”, but you don’t know how many thousandths there are in the first number. Round where your knowledge ends

14. Adding and subtracting • 1.22 - .047 • 1290 + 100 • .00034 + .000038 • 5.30 - 2.30 • 153000 - 12

15. Adding and subtracting • 1.22 - .047 = 1.17 • 1290 + 100 = 1400 • .00034 + .000038 = .00038 • 5.30 - 2.30 = 3.00 • 153000 - 12 = 153000

16. Multiplying and dividing Suppose there are 20,000 pairs of Nike Air Pegasus running shoes in the Denver area. Suppose each pair cost $53.47, like mine did. How much did those shoes cost?

17. Multiplying and dividing Suppose there are 20,000 pairs of Nike Air Pegasus running shoes in the Denver area. Suppose each pair cost $53.47, like mine did. How much did those shoes cost? $1 million.

18. Multiplying and dividing Suppose there are 20,000 pairs of Nike Air Pegasus running shoes in the Denver area. Suppose each pair cost $53.47, like mine did. How much did those shoes cost? $1 million. Not $1,069,400

19. Multiplying and dividing • Round to match the precision of the least number of SF in your problem. • The “20,000 pairs” is a round number, 1SF. Don’t use more than 1SF in your answer.

20. Multiplying and dividing • 138422 x .047 • 1390 ÷ 150 • .34 x .038 • 5.30 ÷ 23521 • 3 x 4

21. Multiplying and dividing • 138422 x .047 = 6500 • 1390 ÷ 150 = 9.3 • .34 x .038 = .013 • 5.30 ÷ 23521 = .000225 • 3 x 4 = 10

22. A little bit of algebra • If Density = mass/volume (It does.) then: D=m/v , m=vD, v=m/D and

23. A little bit of algebra • You will have to be able to solve for any variable in a formula. • The steps are: 1) Start with your original formula. D=m/v

24. A little bit of algebra • You will have to be able to solve for any variable in a formula. • The steps are: 2) Multiply both sides by v (the denominator) vD=vm/v

25. A little bit of algebra • You will have to be able to solve for any variable in a formula. • The steps are: 2) Multiply both sides by v (the denominator) vD=vm/v = m V cancels on the right

26. A little bit of algebra • You will have to be able to solve for any variable in a formula. • The steps are: 3) Divide both sides by D m = vD D D

27. A little bit of algebra • You will have to be able to solve for any variable in a formula. • The steps are: 3) Divide both sides by D m = vD =v D D D cancels on the right

28. A little bit of algebra • So: D=m/v m=vD v=m/D

29. In general: Solve by undoing • If something is added, subtract • If something is subtracted, add • If something is multiplied, divide • If something is divided, multiply • If something is raised to a power, take that root Practice, Practice, Practice!

30. Conversions 1) Start with the measurement given. 2) Multiply it by a fraction called a conversion factor. It has three properties: --The units you start with go on the bottom (You want them to cancel) --The units you want go on the top (You want to end up with them next) --The numbers make the top and the bottom equal (So the fraction is equal to 1, it won't change the value of the measurement) 3) Cancel your units, multiply the numerators, and divide by the denominator 4) Repeat if necessary

31. For example: • 74.32 mm = _______ m

32. For example: • 74.32 mm = _______ m • 74.32 mm Start with the measurement given.

33. For example: • 74.32 mm = _______ m • 74.32 mm x ____________ = Multiply it by a fraction called a conversion factor.

34. For example: • 74.32 mm = _______ m • 74.32 mm x ____________ = mm --The units you start with go on the bottom (You want them to cancel)

35. For example: • 74.32 mm = _______ m • 74.32 mm x ________m___ = mm --The units you want go on the top (You want to end up with them next)

36. For example: • 74.32 mm = _______ m • 74.32 mm x __1 x 10-3 m___ = 1 mm --The numbers make the top and the bottom equal (So the fraction is equal to 1, it won't change the value of the measurement)

37. For example: • 74.32 mm = _______ m • 74.32 mm x __1 x 10-3 m___=7.432x10-2m 1 mm (or .07432m) 3) Cancel your units, multiply the numerators, and divide by the denominator

38. Convert • 1.26 cm = _____m • 5.28 m = ______ mm • .00084 km = _______ mm • 8.00 mm = _______nm

39. Metric System prefixes • Prefix Symbol Meaning • giga G 109 (1 000 000 000) • mega M 106 (1 000 000) • kilo k 103 (1 000) • deka dk 101 (10) • deci d 10-1 (0.1) • centi c 10-2 (0.01) • milli m 10-3 (0.001) • micro m 10-6 (0.000 001) • nano n 10-9 (0.000 000 001)

40. SI System • --the International system • --used by scientists worldwide • --more consistent than the English system • --defines seven standard units • --allows combinations for derived units • (it is no more precise or accurate than any other system)

41. Measurement Unit Symbol • Length meter m • Mass kilogram   kg • Time second s • electric current ampere A • temperature       kelvin K • amount of substance mole mol • luminous intensity candela cd

42. Commonly Used Derived Units • Area • Volume • Velocity • Acceleration • Density • Dynamic viscosity

43. Commonly Used Derived Units • Area =length x width (in m2) • Volume =area x height (in m3) • Velocity =length / time (in m/s) • Acceleration =velocity / time (in m/s2 ) • Density =mass / volume (in kg/m3) • Dynamic viscosity (Just kidding, it’s not common)

44. For a chemist • Mass: gram, kilogram, milligram • Length: centimeter, meter, millimeter, nanometer • Volume: milliliter, liter, cubic meter • Time: second, minute, hour

45. Making measurements • Read the numbers • Count the marks • Estimate one final digit.

46. 7 3 10 15 7 6 50 6 2 9 10 8 4 40 5 1 8 5 9 2 30

47. 1 2 3 4 5 6 1 2 3 4 5 6 10 20 30 40 50 60

48. Scientific Notation • For any real number, A, there is some a and b, such that: • A= a x 10b • a is between 1 and 10 • b is a whole number

49. 27,000,000 .00089 7430 .065 Examples

50. 27000000 = 2.7 x 10 7 .00089 = 8.9 x 10 -4 7430 = 7.43 x 10 3 .065 = 6.5 x 10 -2 Examples