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Measurements and Calculations Guide: Types, Notation, and Precision

This guide provides an overview of different types of measurements, scientific notation, and the rules for adding, subtracting, multiplying, and dividing numbers in scientific notation. It also explains the concept of accuracy and precision in scientific measurements, as well as the rules for determining significant figures and rounding numbers.

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Measurements and Calculations Guide: Types, Notation, and Precision

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  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

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