Chapter 3: Numbers in the Real World
Lecture notes
Math 1030 Section C
Section C.1: Significant Digits
Significant digits
The digits in a number that represents actual measurements and therefore have meaning are called significant 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 a
pound.
• 2 × 102 students has 1 significant digit and implies a measurement to the nearest hundred students.
• 2.00 × 102 students has 3 significant digits and implies exactly 200 students.
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Chapter 3: Numbers in the Real World
Lecture notes
Math 1030 Section C
Ex.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: Rounding
Significant digits
The basic process of rounding numbers takes two steps:
• Step 1: Decide which decimal place (for example, tens, ones, tenths or hundredths) is the smallest
that 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.
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Chapter 3: Numbers in the Real World
Lecture notes
Math 1030 Section C
Ex.4 Rounding with significant digits.
For each of the following operations, give your answer with the specified number of significant digits:
(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 Errors
Types of Error: Random and Systematic
Types of error: random error and systematic error
There are two types of error:
Significant digits:
• Random errors occur because of random and inherently unpredictable events in the measurement
process. We can minimize the effect of random errors by making many measurements and averaging them.
• Systematic errors occur when there is a problem in the measurement system that affect all measurements in the same way, such as making them all too low or too high by the same amount. If we
discover a systematic error, we can go back and adjust the affected measurements.
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Chapter 3: Numbers in the Real World
Lecture notes
Math 1030 Section C
Ex.5
Suppose you work in a pediatric office and use a digital scale to weigh babies. If you have ever worked
with babies, you know that they usually aren’t very happy about being put on a scale. Their thrashing
and crying tends to shake the scale making the readout jump around. You could equally well record the
baby’s weight as anything between 14.5 and 15.0 pounds. The shaking of the scale introduces a random
error. If you measure the baby’s weight ten times, your measurements will probably be too high in some
case and to low in other cases. When you average the measurements you are likely to get a value that
better represents the true weight.
Now, suppose you have weighed babies all day. At the end of the day, you notice that the scale reads
1.2 pounds when there is nothing on it. In that case, every measurement you made was too high by 1.2
pounds. Therefore, we have a systematic error. Now that you know about this systematic error, you can
go 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, or
the global average temperature, has changed with time. Consider two difficulties (among many others) in
trying 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 warmer
than surrounding rural areas because heat released by human activity.
Discuss whether each of these two difficulties produces random or systematic errors, and consider the
implications of these errors.
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Chapter 3: Numbers in the Real World
Lecture notes
Math 1030 Section C
Ex.7 The Census.
The constitution of the United States mandates a Census of the population every 10 years. The United
State Census Bureau conducts the census by distributing house hold surveys through the mail and through
personal visits. Suggest several sources of both random and systematic error in the census.
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Chapter 3: Numbers in the Real World
Lecture notes
Math 1030 Section C
Size of Errors: Absolute versus Relative
Sizes of error: absolute error and relative error
There 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:
absolute error
measured value − true value
=
.
true value
true value
The absolute and the relative error are positive when the measured value is greater than the true value and
negative 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.
relative error =
Ex.8
• Suppose you go to a store and ask 6 pounds of hamburger. However, because the store’s scale is
poorly calibrated, you actually get 4 pounds.
• Suppose you buy a car which the owner’s manual says weighs 3132 pounds, but you find that it
really weighs 3130 pounds.
Compute the absolute and relative error and discuss why you are disappointed in the first case but you
don’y care too much in the second case.
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Chapter 3: Numbers in the Real World
Lecture notes
Math 1030 Section C
Ex.9
Find the absolute and relative error in each case.
(1) Your true weight is 125 pounds, but a scale says you weigh 130 pounds.
(2) The government claims that a program costs $49.0 billion, but an audit shows that the true cost is
$50.0 billion.
Describing Results: Accuracy and Precision
Accuracy and precision
Once a measurement is reported, we should evaluate it to see whether it is believable in light of any potential errors. In particular, we should consider two key ideas about any reported value: its accuracy and its
precision. The term are often use interchangeably in English, but mathematically they are different.
• 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.
Ex.10 Accuracy.
If a census says that the population of your home town is 72, 453, but the true population is 96, 000, then
the census report is not very accurate. In contrast, if a company projects sales of $7.30 billion and true sales
turn out to be $7.32 billion, we would say that the projection is quite accurate.
Ex.11 Precision.
A distance given as 2.345 kilometers is more precise than a distance given as 2.3 kilometers because the first
number gives detail to the nearest 0.001 kilometer and the second number gives detail only to the nearest
0.1 kilometer.
Similarly, an income of $45, 678.90 has greater precision than an income of $46, 000 because the first income
is precise to the nearest penny and the second income is precise only to the nearest thousand dollars.
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Chapter 3: Numbers in the Real World
Lecture notes
Math 1030 Section C
Ex.12 Accuracy and Precision in your Weight.
Suppose your true weight is 102.4 pounds. The scale at the doctor’s office, which can be read only to the
nearest quarter pound, says that you weigh 102 14 puonds. The scale at the gym, which gives a digital
readout to the nearest 0.1 pound says that you weigh 100.7. Which scale is more precise? Which scale is
more accurate?
Summary: Dealing with Errors
Summary
• Errors can occur in many ways, but generally can be classified as one of two basic types: random
errors and systematic errors.
• Whatever the source of an error, its size can be described in two different ways: as an absolute error
or as a relative error.
• Once a measurement is reported, we can evaluate it in terms of its accuracy and its precision.
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Chapter 3: Numbers in the Real World
Lecture notes
Math 1030 Section C
Section C.3: Combining Measured Numbers
Combining measured numbers
In scientific or statistical work, researchers conduct careful analyses to determine how to combine numbers
properly. We can use two simple rounding rules:
• Rounding rule for addition and subtraction: Rounding your answer to the same precision as the
least precise number in the problem.
• Rounding rule for multiplication or division: Rounding your answer to the same number of significant digits as the measurement with the fewest significant digits.
Note: to avoid errors, you should do the rounding only after completing all the operations, not during
intermediate steps.
Ex.13
Suppose that you live in a city with a population of 300, 000. One day, your best friend move to your city
to share an apartment with you. What is the population of your city now?
Ex.14
(1) A book written in 1962 states that the oldest Mayan ruins are 2000 years old. How old are they now?
(2) The government in a town of 82, 000 people plans to spend $41.5 million this year. Assuming all
this money must come from taxes, what average amount must the city collect from each resident?
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