Chapter 3: Numbers in the Real World Lecture notes Math 1030 Section BSection B.1: Writing Large and Small NumbersLarge and small numbersWorking with large and small numbers is much easier when we write them in a special format calledscientific notation.Scientific notationScientific notation is a format in which a number is expressed as a number between 1 and 10 multiplied by apower of 10.Ex.1one billion = 109(ten to the ninth power)6 billion = 6 × 109420 = 4.2 × 1020.5 = 5 × 10−1Ex.2 Numbers in Scientific Notation.Rewrite each of the following statement using scientific notation.(1) The U.S. federal debt is about $9, 100, 000, 000, 000.(2) The diameter of a hydrogen nucleus is about 0.000000000000001 meter.Approximations with Scientific Notati onApproximations with scientific notationWe can use scientific notation to approximate answers without a calculator.Ex.3Approximate 5795 × 326.1Chapter 3: Numbers in the Real World Lecture notes Math 1030 Section BEx.4 Checking Answers with Approximations.You and a friend are doing a rough calculation of how much garbage New York City residents produceevery day. You estimate that, on average, each of the 8 million residents produces 1.8 pounds or 0.0009ton of garbage each day. Thus the total amount of garbage is8, 000, 000 person × 0.0009 ton.Your friend quickly presses the calculator buttons and tells you that the answer is 225 tons. Without usingyour calculator, determine whether this answer is reasonable.2Chapter 3: Numbers in the Real World Lecture notes Math 1030 Section BSection B.2: Giving Meaning to NumbersGiving meaning to numbersNow that we have a method for writing large and small numbers, we can put numbers in perspective. Wewill study three techniques to put the number in perspec tive : through estimation, through comparisonsand through scaling.Perspective through Estimat ionDefinition of order of magnitude estimateAn order of magnitude estimate specifies only a broad range of v alues, such as ”in the ten thousands“ or ”inthe millions“.Ex.5We might say that the population of the United State s is ”on the order of 300 million“, by which we meanit is nearer to 300 million then to, say, 200 million or 400 million.Ex.6When astronomers say that the number of stars in a ga laxy is ”of order 100 billion“, they mean it couldbe anywhere within about a factor of 10 of this numbers, that is, between about 10 billion and 1 tr illion.This is a much wider range than the range implied when we say ”on the order of 300 million“ for thepopulation of the United States, but it is appropriate to the context.3Chapter 3: Numbers in the Real World Lecture notes Math 1030 Section BEx.7 Order of Magnitude of Ice cream Spending.Make an order of magnitude estimate of total annual spending on ice cream in the United Sa tes.4Chapter 3: Numbers in the Real World Lecture notes Math 1030 Section BPerspective through ComparisonsEx.8Consider $100 billion, which is more or less the wealth of the world’s richest individuals. It’s easy to saya number like 100 billion, but how big is it?Ex.9 U.S. versus World Energy Consumption.Compare the U.S. population to the world population and U.S . energy consumption to world energyconsumption. What does this tell you about energy usage by Americans?5Chapter 3: Numbers in the Real World Lecture notes Math 1030 Section BPerspective through ScalingWays of expressing scalesThere a re three ways of exp ressing scales:• Verbally: A scale can be de scribed in words such as ”One ce ntimeter represents one kilometer“ or,more simply, as ”1 cm = 1 km“.• Geographically: A marked miniruler can show the scale visually.• As a ratio: We can state the ratio of a scaled size to an actual size. For example, there are 100, 000 ina kilometer. Thus, a scale where 1 centimeter represents 1 kilometer can be described as a scale ratioof 1 to 100, 000.Ex.10 Scale Ratio.A city map states, ” One inch represents one mile“. What is the scale ratio for this map?6Chapter 3: Numbers in the Real World Lecture notes Math 1030 Section BEx.11 Earth and Sun.The distance from the Earth to the Sun is about 150 million kilometers. The diameter of the Sun is about1.4 million kilometers and the diameter of the Earth is about 12, 760 kilometers. Put this numbers inperspective by using a scale model of the scalar system with a 1 to a 10 billion scale.7Chapter 3: Numbers in the Real World Lecture notes Math 1030 Section BEx.12 Distances to the Stars.The distance from the E arth to the nearest stars besides the Sun is about 4.3 light-years. On the 1 to 10billion scale of the last example, how far are these sta rs from the Earth?Note: A light-year is the distance that the light can travel in one year:1 light-year = 9.5 × 1012km.8Chapter 3: Numbers in the Real World Lecture notes Math 1030 Section BEx.13 Timeline.Human civilization, at le ast since the time of ancient Egypt, is on the order of 5000 years old. The ageof the Earth is on the order of 5 billion years. Suppose we use the length of a football field, or about 100meters, as a timeline to represent the age of the Earth. If you put the birth of the Earth at the start of thetimeline, how far from the line’s end would human civilization