Copyright © 2011 Pearson Education, Inc. Numbers in the Real World

Copyright © 2011 Pearson Education, Inc. Slide 3-3 Unit 3B Putting Numbers in Perspective

3-B Copyright © 2011 Pearson Education, Inc. Slide 3-4 Scientific Notation Scientific notation is a format in which a number is expressed as a number between 1 and 10 multiplied by a power of 10. Examples: 6,700,000,000 in scientific notation is 6.7 x is 2.0 x 10 –15

3-B Scientific Notation (cont.) Write in Scientific Notation: 1) 3,320,000,000 2) Write in normal notation: 1) 4.12 x 10 –2 1) x 10 7 Copyright © 2011 Pearson Education, Inc. Slide x x 10 – ,250,00

3-B Scientific Notation on a Calculator Examples: 6,700,000,000 in scientific notation is 6.7 x 10 9 on a calculator it may be expressed as 6.7E is 2.0 x 10 –15 on a calculator it may be expressed as 2E–15 Copyright © 2011 Pearson Education, Inc. Slide 3-6

3-B Entering Scientific Notation on a Calculator To enter a number in Scientific Notation on your calculator, you will use the EE key. Enter 3.25 x 10 3 times 4.14 x EE 3 x 4.14 EE 6 Answer or E10 or x Copyright © 2011 Pearson Education, Inc. Slide 3-7

3-B Copyright © 2011 Pearson Education, Inc. Slide 3-8 Perspective through estimation An order of magnitude estimate specifies a broad range of values. Example: Is the total annual ice cream spending in the United States measured in thousands of dollars, millions of dollars, or billions of dollars? Giving Meaning to Numbers

3-B Copyright © 2011 Pearson Education, Inc. Slide 3-9 Perspective through comparisons Giving Meaning to Numbers

3-B Orders of Magnitude – You try it 1) How many times larger is the energy released by burning 1 kg of coal than the energy needed for 1 hr. of running? Copyright © 2011 Pearson Education, Inc. Slide ) How many times larger is the energy released by the sun annually than the energy used in an average home annually?

3-B Copyright © 2011 Pearson Education, Inc. Slide 3-11 Giving Meaning to Numbers Perspective through scaling Verbally: “1 cm = 1 km” Graphically : As a ratio: 1 cm = 1 km means a scale ratio of 1 to 100,000

3-B Scaling (cont.) 1) If the distance from Fond du Lac to Milwaukee is on the map is 208 cm, how many kilometers is it actually? How many miles? 1) If the distance from Omro to Ladoga is 47 km, how far is it on the map? Copyright © 2011 Pearson Education, Inc. Slide 3-12 Given a scale of “1cm = 500m” 104km, 64.6 mi 94cm

3-B Group Work Assignment p , 17, 20, 32, 54 Copyright © 2011 Pearson Education, Inc. Slide 3-13

3-B Copyright © 2011 Pearson Education, Inc. Slide 3-14 Unit 3C Dealing with Uncertainty

3-B Copyright © 2011 Pearson Education, Inc. Slide 3-15 Significant Digits Not significant unless stated otherwise Zeros to the right of the last nonzero digit but before the decimal point as in (40,000 or 210) Never significantZeros to the left of the first nonzero digit (as in or ) Always significantZeros between nonzero digits (as in 4002 or 3.06) or other significant zeros (such as the first zero in 30.0) Always significantZeros that follow a nonzero digit and lie to the right of the decimal point (as in 4.20 or 3.00) Always significantNonzero digits SignificanceType of Digit

3-B Copyright © 2011 Pearson Education, Inc. Slide 3-16 Random errors occur because of random and inherently unpredictable events in the measurement process. Systematic errors occur when there is a problem in the measurement system that affects all measurements in the same way, such as making them all too low or too high by the same amount. Types of Error

3-B Errors (cont.) Acme Inc. has a machine that cuts tubes to length as a part for their famous rocket skates. The machine cuts the tubes to 45cm ± 2mm. 1) What is the longest tube the machine can cut that is still within spec? the shortest? 2) If the machine cut 100 tubes, and you laid them end-to-end, what would be the maximum total length? The mininum? Copyright © 2011 Pearson Education, Inc. Slide cm, 44.8cm 4520cm, 4480cm

3-B Copyright © 2011 Pearson Education, Inc. Slide 3-18 The absolute error describes how far a measured (or claimed) value lies from the true value. absolute error = measured value – true value The relative error compares the size of the error to the true value. Size of Errors

3-B Copyright © 2011 Pearson Education, Inc. Slide 3-19 absolute error = measured value – true value Example: A projected budget surplus of $17 billion turns out to be $25 billion at the end of the fiscal year. = $25 billion – $17 billion = $8 billion = $8 billion / $17 billion ≈ = 47.1% relative error Absolute vs. Relative Error

3-B Copyright © 2011 Pearson Education, Inc. Slide 3-20 Describing Results Accuracy describes how closely a measurement approximates a true value. An accurate measurement is very close to the true value. Precision describes the amount of detail in a measurement.

3-B Copyright © 2011 Pearson Education, Inc. Slide 3-21 Combining Measured Numbers Rounding rule for addition or subtraction: Round the answer to the same precision as the least precise number in the problem. Rounding rule for multiplication or division: Round the answer to the same number of significant digits as the measurement with the fewest significant digits. To avoid errors, round only after completing all the operations, not during the intermediate steps.

3-B Group Work Assignment p odd, odd Copyright © 2011 Pearson Education, Inc. Slide 3-22

3-B Copyright © 2011 Pearson Education, Inc. Slide 3-23 Unit 3D Index Numbers: The CPI and Beyond

3-B Copyright © 2011 Pearson Education, Inc. Slide 3-24 Index Numbers An index number provides a simple way to compare measurements made at different times or in different places. The value at one particular time (or place) must be chosen as the reference value.

3-B Copyright © 2011 Pearson Education, Inc. Slide 3-25 Example: Suppose the current price of gasoline is $3.20 per gallon. Using the 1975 price of 56.7¢ as the reference value, find the price index number for gasoline today. The current price is 564.4% of the 1975 price. Making Comparisons with Index Numbers

3-B Copyright © 2011 Pearson Education, Inc. Slide 3-26 Any reference value can be used to calculate an index number, but changing the reference value results in a different index number. Index Numbers: Changing the Reference Value Example:

3-B Copyright © 2011 Pearson Education, Inc. Slide 3-27 The Consumer Price Index The Consumer Price Index (CPI) is based on an average of prices for a sample of more than 60,000 goods, services, and housing costs. The rate of inflation from one year to the next is usually defined as the relative change in the Consumer Price Index.

3-B Copyright © 2011 Pearson Education, Inc. Slide 3-28 Consumer Price Index

3-B Copyright © 2011 Pearson Education, Inc. Slide 3-29 Adjusting Prices for Inflation Given a price in dollars for year X ($ X ), the equivalent price in dollars for year Y ($ Y ) is where X and Y represent years.

3-B Group Work Assignment p odd Copyright © 2011 Pearson Education, Inc. Slide 3-30