Chapter 3: Numbers in the Real World Lecture notes Math 1030 Section CSection C.1: Significant DigitsSignificant digitsThe digits in a number that represents actual measurements and therefore have meaning are called signifi-cant digits.Significant digits:• Nonzero digits.• Zeros that follow a nonzero digit and lie to the right of the decimal point:• Zeros between nonzero digits or other significant zeros:Not significant digits:• Zeros to the left of the first nonzero digit:• Zeros to the right of the last nonzero digit but before the decimal point:Ex.1• 132 pounds has 3 significant digits and implies a measurement to the nearest pound.• 132.00 pounds has 5 significant digits and implies a measurement to the nearest hundredth of apound.• 2 × 102students has 1 significant digit and implies a measurement to the nearest hundred students.• 2.00 × 102students has 3 significant digits and implies exactly 200 students.1Chapter 3: Numbers in the Real World Lecture notes Math 1030 Section CEx.2 Counting significant digits.State the number of significant digits and the implied meaning of the following numbers:(1) a time of 11.90 seconds;(2) a length of 0.000067 meter;(3) a population reported as 240, 000;(4) a population reported as 2.40 × 105.Section C.2: RoundingSignificant digitsThe basic process of rounding numbers takes two steps:• Step 1: Decide which decimal place (for example, tens, ones, tenths or hundredths) is the smallestthat should be kept.• Step 2: Look at the number in the nearest place to the right (for example, if rounding the tenths,look at hundredths). If the value in the next place is less than 5 round down, if it is 5 or greater than 5,round up.Ex.3• 382.2593 rounded to the nearest thousandth is 382.259.• 382.2593 rounded to the nearest hundredth is 382.26.• 382.2593 rounded to the nearest tenth is 382.3.• 382.2593 rounded to the nearest one is 382.• 382.2593 rounded to the nearest ten is 380.• 382.2593 rounded to the nearest hundred is 400.2Chapter 3: Numbers in the Real World Lecture notes Math 1030 Section CEx.4 Rounding with significant digits.For each of the following operations, give your answer with the specified number of significant dig-its:(1) 7.7 mm × 9.92 mm; give your answer with 2 significant digits;(2) 240, 000 × 72, 106; give your answer with 4 significant digits.Section C.3: Understanding ErrorsTypes of Error: Random and SystematicTypes of error: random error and systematic errorThere are two types of error:Significant digits:• Random errors occur because of random and inherently unpredictable events in the measurementprocess. We can minimize the effect of random errors by making many measurements and averag-ing them.• Systematic errors occur when there is a problem in the measurement system that affect all measure-ments in the same way, such as making them all too low or too high by the same amount. If wediscover a systematic error, we can go back and adjust the affected measurements.3Chapter 3: Numbers in the Real World Lecture notes Math 1030 Section CEx.5Suppose you work in a pediatric office and use a digital scale to weigh babies. If you have ever workedwith babies, you know that they usually aren’t very happy about being put on a scale. Their thrashingand crying tends to shake the scale making the readout jump around. You could equally well record thebaby’s weight as anything between 14.5 and 15.0 pounds. The shaking of the scale introduces a randomerror. If you measure the baby’s weight ten times, your measurements will probably be too high in somecase and to low in other cases. When you average the measurements you are likely to get a value thatbetter represents the true weight.Now, suppose you have weighed babies all day. At the end of the day, you notice that the scale reads1.2 pounds when there is nothing on it. In that case, every measurement you made was too high by 1.2pounds. Therefore, we have a systematic error. Now that you know about this systematic error, you cango back and adjust the affected measurements.Ex.6 Errors in global warming data.Scientists studying global warming need to know how the average temperature of the entire Earth, orthe global average temperature, has changed with time. Consider two difficulties (among many others) intrying to interpret historical temperature data from the early 20th century:(1) temperatures were measured with simple thermometers and the data were recorded by hand;(2) most temperature measurements were recorded in or near urban areas, which tend to be warmerthan surrounding rural areas because heat released by human activity.Discuss whether each of these two difficulties produces random or systematic errors, and consider theimplications of these errors.4Chapter 3: Numbers in the Real World Lecture notes Math 1030 Section CEx.7 The Census.The constitution of the United States mandates a Census of the population every 10 years. The UnitedState Census Bureau conducts the census by distributing house hold surveys through the mail and throughpersonal visits. Suggest several sources of both random and systematic error in the census.5Chapter 3: Numbers in the Real World Lecture notes Math 1030 Section CSize of Errors: Absolute versus RelativeSizes of error: absolute error and relative errorThere are two types of error:• The absolute error describes how far a measured 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:relative error =absolute errortrue value=measured value − true valuetrue value.The absolute and the relative error are positive when the measured value is greater than the true value andnegative when the measured value is less than the true value.Note that the above formula gives the relative error as a fraction which can be converted to a percentage.Ex.8• Suppose you go to a store and ask 6 pounds of hamburger. However, because the store’s scale ispoorly calibrated, you actually get 4 pounds.• Suppose you buy a car which the owner’s manual says weighs 3132 pounds, but you find that itreally weighs 3130 pounds.Compute the absolute and relative error and discuss why you are disappointed in the first case but youdon’y care too much in the second case.6Chapter 3: Numbers in the Real World Lecture notes Math 1030 Section CEx.9Find the absolute and relative error in each case.(1) Your true weight is 125 pounds, but a scale says you