Measurements and Calculations Guide: Types, Notation, and Precision

1. Chapter 2 Measurements and Calculations

2. Types of measurement • Quantitative- use numbers to describe • Qualitative- use description without numbers • 4 feet • extra large • Hot • 100ºF

3. Scientific Notation • A decimal point is in standard position if it is behind the first non-zero digit. • Let X be any number and let N be that number with the decimal point moved to standard position. Then: • If 0 < X < 1 then X = N x 10negative number • If 1 < X < 10 then X = N x 100 • If X > 10 then X = N x 10positive number

4. Some examples • 0.00087 becomes 8.7 x 10¯4 • 9.8 becomes 9.8 x 100 (the 100 is seldom written) • 23,000,000 becomes 2.3 x 107

5. Adding and Subtracting • All exponents MUST BE THE SAME before you can add and subtract numbers in scientific notation. • The actual addition or subtraction will take place with the numerical portion, NOT the exponent.

6. Adding and Subtracting • Example: 1.00 x 103 + 1.00 x 102 • A good rule to follow is to express all numbers in the problem in the highest power of ten. • Convert 1.00 x 102 to 0.10 x 103, then add:                                 1.00 x 103       +        0.10 x 103       =        1.10 x 103

7. Multiplication and Division • Multiplication: Multiply the decimal portions and add the exponential portions. • Example #1:      (3.05 x 106) x (4.55 x 10¯10) • Here is the rearranged problem: (3.05 x 4.55) x (106+ (-10)) • You now have 13 x 10¯4 = 1.3 x 10¯3

8. Multiplication and Division • Division: Divide the decimal portions and subtract the exponential portions. • Example:      (3.05 x 106) ÷ (4.55 x 10¯10) • Here is the rearranged problem: (3.05 ÷ 4.55) x (106- (-10)) • You now have 0.7 x 1016 = 7.0 x 1015

9. Scientists prefer • Quantitative- easy check • Easy to agree upon, no personal bias • The measuring instrument limits how good the measurement is

10. How good are the measurements? • Scientists use two word to describe how good the measurements are • Accuracy- how close the measurement is to the actual value • Precision- how well can the measurement be repeated

11. Differences • Accuracy can be true of an individual measurement or the average of several • Precision requires several measurements before anything can be said about it • examples

12. Let’s use a golf anaolgy

13. Accurate? No Precise? Yes

14. Accurate? Yes Precise? Yes

15. Precise? No Accurate? Maybe?

16. Accurate? Yes Precise? We cant say!

17. In terms of measurement • Three students measure the room to be 10.2 m, 10.3 m and 10.4 m across. • Were they precise? • Were they accurate?

18. 1 2 3 4 5 Significant figures (sig figs) • How many numbers mean anything • When we measure something, we can (and do) always estimate between the smallest marks.

19. Significant figures (sig figs) • The better marks the better we can estimate. • Scientist always understand that the last number measured is actually an estimate 1 2 3 4 5

20. Sig Figs • What is the smallest mark on the ruler that measures 142.15 cm? • 142 cm? • 140 cm? • Here there’s a problem does the zero count or not? • They needed a set of rules to decide which zeroes count. • All other numbers do count

21. Which zeros count? • Those at the end of a number before the decimal point don’t count • 12400 • If the number is smaller than one, zeroes before the first number don’t count • 0.045

22. Which zeros count? • Zeros between other sig figs do. • 1002 • zeroes at the end of a number after the decimal point do count • 45.8300 • If they are holding places, they don’t. • If they are measured (or estimated) they do

23. Sig Figs • Only measurements have sig figs. • Counted numbers are exact • A dozen is exactly 12 • A a piece of paper is measured 11 inches tall. • Being able to locate, and count significant figures is an important skill.

24. Sig figs. • How many sig figs in the following measurements? • 458 g • 4085 g • 4850 g • 0.0485 g • 0.004085 g • 40.004085 g

25. Sig Figs. • 405.0 g • 4050 g • 0.450 g • 4050.05 g • 0.0500060 g • Next we learn the rules for calculations

26. More Sig Figs

27. Problems • 50 is only 1 significant figure • if it really has two, how can I write it? • A zero at the end only counts after the decimal place • Scientific notation • 5.0 x 101 • now the zero counts.

28. Adding and subtracting with sig figs • The last sig fig in a measurement is an estimate. • Your answer when you add or subtract can not be better than your worst estimate. • have to round it to the least place of the measurement in the problem

29. 27.93 + 6.4 27.93 27.93 + 6.4 6.4 For example • First line up the decimal places Then do the adding Find the estimated numbers in the problem 34.33 This answer must be rounded to the tenths place

30. Rounding rules • look at the number behind the one you’re rounding. • If it is 0 to 4 don’t change it • If it is 5 to 9 make it one bigger • round 45.462 to four sig figs • to three sig figs • to two sig figs • to one sig fig

31. Practice • 4.8 + 6.8765 • 520 + 94.98 • 0.0045 + 2.113 • 6.0 x 102 - 3.8 x 103 • 5.4 - 3.28 • 6.7 - .542 • 500 -126 • 6.0 x 10-2 - 3.8 x 10-3

32. Multiplication and Division • Rule is simpler • Same number of sig figs in the answer as the least in the question • 3.6 x 653 • 2350.8 • 3.6 has 2 s.f. 653 has 3 s.f. • answer can only have 2 s.f. • 2400

33. Multiplication and Division • Same rules for division • practice • 4.5 / 6.245 • 4.5 x 6.245 • 9.8764 x .043 • 3.876 / 1983 • 16547 / 714

34. The Metric System An easy way to measure

35. Measuring • The numbers are only half of a measurement • It is 10 long • 10 what. • Numbers without units are meaningless. • How many feet in a yard • A mile • A rod

36. The Metric System • Easier to use because it is a decimal system • Every conversion is by some power of 10. • A metric unit has two parts • A prefix and a base unit. • prefix tells you how many times to divide or multiply by 10.

37. Base Units • Length - meter more than a yard - m • Mass - grams - a bout a raisin - g • Time - second - s • Temperature - Kelvin or ºCelsius K or C • Energy - Joules- J • Volume - Liter - half f a two liter bottle- L • Amount of substance - mole - mol

38. kilo- mega- M k 106 103 deci- BASE UNIT d --- 100 10-1 centi- c 10-2 milli- m 10-3 micro-  10-6 nano- n 10-9 pico- p 10-12 SI Prefix Conversions Prefix Symbol Factor move left move right

39. Prefixes • kilo k 1000 times • deci d 1/10 • centi c 1/100 • milli m 1/1000 • kilometer - about 0.6 miles • centimeter - less than half an inch • millimeter - the width of a paper clip wire

40. Dimensional Analysis • The “Factor-Label” Method • Units, or “labels” are canceled, or “factored” out

41. Dimensional Analysis • Steps: 1. Identify starting & ending units. 2. Line up conversion factors so units cancel. 3. Multiply all top numbers & divide by each bottom number. 4. Check units & answer.

42. Dimensional Analysis • Lining up conversion factors: = 1 1 in = 2.54 cm 2.54 cm 2.54 cm 1 = 1 in = 2.54 cm 1 in 1 in

43. qt mL  Dimensional Analysis • How many milliliters are in 1.00 quart of milk? 1.00 qt 1 L 1.057 qt 1000 mL 1 L = 946 mL

44. lb cm3 Dimensional Analysis • You have 1.5 pounds of gold. Find its volume in cm3 if the density of gold is 19.3 g/cm3. 1000 g 1 kg 1 cm3 19.3 g 1.5 lb 1 kg 2.2 lb = 35 cm3

45. in3 L Dimensional Analysis • How many liters of water would fill a container that measures 75.0 in3? (2.54 cm)3 (1 in)3 75.0 in3 1 L 1000 cm3 = 1.23 L

46. cm in Dimensional Analysis 5) Your European hairdresser wants to cut your hair 8.0 cm shorter. How many inches will he be cutting off? 8.0 cm 1 in 2.54 cm = 3.2 in

47. cm yd Dimensional Analysis 6) Taft football needs 550 cm for a 1st down. How many yards is this? 1 ft 12 in 1 yd 3 ft 550 cm 1 in 2.54 cm = 6.0 yd

48. cm pieces Dimensional Analysis 7) A piece of wire is 1.3 m long. How many 1.5-cm pieces can be cut from this wire? 1.3 m 100 cm 1 m 1 piece 1.5 cm = 86 pieces

49. Volume • calculated by multiplying L x W x H • Liter the volume of a cube 1 dm (10 cm) on a side • so 1 L = 10 cm x 10 cm x 10 cm • 1 L = 1000 cm3 • 1/1000 L = 1 cm3 • 1 mL = 1 cm3

50. Volume • 1 L about 1/4 of a gallon - a quart • 1 mL is about 20 drops of water or 1 sugar cube