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CHAPTER 2. REAL NUMBERS, ALGEBRA AND FUNCTIONS
Both quantities we computed represent the same length, but in different units.25
2.3.2
Multiplying Units: Simplest Cases
Shortly after one is secure adding units of length to each other, it is natural to begin to consider
area and volume units built from these. For instance, in abstract geometry, we define the area
of a rectangle to be the product of its length and width, i.e., the product of two adjacent sides’
lengths (and which is called length, the other being the width, is entirely up to the individual,
and varies from text to text). Thus a rectangle with length 3 and width 5 would have area
3 · 5 = 15. But what of the units? If both are in feet, and we assume multiplication is always
commutative (which it is in calculus, until you define some “vector” products in multivariable
calculus, much later), and we have
3ft · 5ft = 15ft · ft = 15(ft)2 = 15ft2 .
Note that there are conventions that arise over the years. For instance, we are to assume “ft” is
one indivisible concept, and so 15ft2 means 15(ft)2 , and certainly not 15f 2 t2 , which is meaningless
in this context. Other conventions arise for convenience in other fields, and usually the meaning
is obvious from the context.
Note also that it is easy to imagine a “foot squared,” or a “square foot.” It is a unit of
area equal to that of a square with dimensions one foot by one foot. Similarly we can discuss a
volume of solid which is a right parallelpiped measuring 3 ft long, 4 ft wide and 5 ft tall, having
volume 3 ft · 4 ft · 5 ft = 60 ft3 , where one cubic foot is the volume of a cube with each edge length
being 1 ft.
Now consider the area of a rectangle which is 4 feet by 3 yards. We have basically three
immediate options for computing this area: we could simply multiply these two quantities; we
could convert feet into yards and have an answer in square yards; or we could convert the yards
into feet and have an answer in square feet. Each answer will represent the same area, but in
different units.
4ft · 3yd =
12ft-yd
4
4ft · 3yd = yd · 3yd = 4yd2
3
4ft · 3yd = 4ft · 9ft = 36ft2 .
Note first that there is a convention when expressing a product of two different units, in
which a short “dash” is used where we might expect to see a “dot.” The typsetters and writers
are careful to be sure it is short enough that it can be distinguised from the longer subtraction
mark (“-” versus “−”), but it takes some getting used to. The first answer would be read aloud
as, “twelve foot-yards,” where the other two would be read, respectively, “four yards square(d)”
(or “four square yards”), and “thirty-six feet squared” or similar.
Now it is rare to see a unit such as a foot-yard, but it is simply the amount of area equal
to a rectangular area which is one foot long and one yard wide, for instance. (Occasionally we
25 There
are other ways one writes feet and inches, such as
1.5 ft = 1.5� = 18�� = 18 in.
In some contexts this is then written 1� 6�� , meaning 1 ft + 6 in. That is not so easy to manipulate arithmetically.
A similar notation is used for angle measure in degrees-minutes-seconds, and with time in hours-minutes-seconds,
but we will rarely mix units like that. For instance, for time we will leave the answer in seconds, or in minutes, or
in hours, etc., but not in a combination of these unless we are curious as to such a form for our answer. Certainly
the arithmetic we set up will not mix these different units of the same type of quantity.
2.3. ARITHMETIC OF UNITS; SIGNIFICANT DIGITS
91
see such mixed units in commerce, where for instance one might buy an area of fabric or carpet
which comes off of a roll with a fixed width, and the customer officially pays by the length of
the cut piece.)
Another place where units are multiplied is in dealing with torques. If an arm of some length
is attached at one end to a pivot about which it can rotate in a circle within a plane, and a
force is applied at the other end of the arm in a direction tangent to the circle, then the torque
experienced at that central pivot is given by the product of the length of the arm. If the force is
in pounds (lbs), and the length of the arm is in feet, then the torque is in ft-lbs (foot-pounds).
If we prefer metric units, the force could be in newtons (n) and the length in meters (m), and
we would have torque in newton-meters, i.e., n-m.
Oddly enough, energy and work can be given in the same units as torque, though they are
completely distinct from the concept of torque. Energy is the ability to do work, and the amount
of energy an object has is defined by the work it can accomplish, so energy and work have the
same units. Work is defined as the product of a force and the displacement (or signed distance)
of the motion in the direction at which the force acts. This will be clarified more later, but for
a simple example we can consider a young man pushing a broken down car along a road, using
perhaps 100 lbs of force as he pushes the car 40 ft in the direction of motion. In this action the
young man contributed (40 ft) · (100 lbs), or 4000 ft-lbs of energy to the car. (Most likely the
energy was dissipated by internal friction of the car, and the flexion of the car-road interface.)
We will see applications of work and energy later in the text.
2.3.3
Division of Units: Rates
While multiplication of units is particularly common in the integral calculus (starting with Chapter 8), rates actually appear more prevalently in the earlier parts of the text, being particularly
important in the study of applied differential calculus (beginning in Chapter 4). However, both
concepts appear throughout the rest of the text.
For our first example of a rate, note that in the United States the speed (rate) of travel in
vehicles is still commonly computed in miles per hour, or mph. In fact, this is mathematically
represented as mi/hr, i.e., miles divided by hours. Indeed, the word “per” (sometimes rather
carelessly read “for every”) always indicates a division. The reason for this is clear if we look at
applications in which we multiply these compound units by other units in obvious applications.
For instance, suppose a car travels at exactly 60 mph for exactly 1.5 hr. Then the total travel
distance is given by
�
�
90 mi · h� r
60 mi
· (1.5 hr) =
= 90 mi.
hr
h� r
This is the well-known elementary school idea that “distance equals rate times time.” Note,
though, how the units resolve correctly when we allow “per” to mean division by the unit which
follows.
While in calculus we usually avoid the elementary school “times” notation, × for multiplication, it is fairly common in the physical sciences, so the above computation might be written26
60 mi
× 1.5 hr = 90 mi.
hr
26 Of course, one has to distinguish the multiplication operation × with the variable x. While this is easy enough
with typesetting, some care has to be taken when writing by hand, particularly for other readers. One device used
by lecturers in the physical sciences is to be sure there is extra space around, and careful vertical centering of,
the operator but not so much with the variable. Also, using a more cursive style with the variable,and straighter
“45◦ ” strokes with the operator is standard in writing these. It is up to the writer to be clear, and not always
assume the reader will understand from the context.
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CHAPTER 2. REAL NUMBERS, ALGEBRA AND FUNCTIONS
The answer is completely correct, but unwieldy for interpretation, as we are led to wonder just
what is a km-sec/hr. Instead we could have (1) converted seconds to hours from the start, (2)
converted km/hr into km/sec from the start, or (3) perform the conversion after we multiplied
our rate and time:
45 km
1 hr
× 90 sec ×
= 1.125 km.
hr
3600 sec
Again we may feel more comfortable reporting the final answer as 1.13 km, 1.1 km or even 1 km,
depending upon our confidence in the original velocity (and how constant it was for the 90 sec).
Note also that if we wished, we could have converted the final answer to express the distance
in some other unit of length. Chemistry and Physics students especially learn this “conversion as
we go” habit, beginning from the relevant formula, as it is often easier than looking ahead to the
end calculation and attempting to convert the original, often disparate units we are given at the
beginning in order to have complicated formulas output a simple answer. In fact, multiple early
conversions require the original data to be stored or written down in their converted form, where
converting the final answer, using the original numbers, can be done at once on the calculator,
utilizing all of its internal precision.
2.3.5
Significant Digits
“Significant digits aren’t the only, and are not the best, way of keeping track of errors
in experimental data, but they are often used in beginning science classes. Don’t get
hung up on the rules for significant digits. They are only a crude tool for showing the
approximate amount of error in a measurement or calculated result. Use the rules
with good judgment, rather than following them rigidly at all times.”
—David Dice, Carlton
(This subsection is heavily footnoted because the exact understanding and best rules for
significant digits are not uniform among physical scientists, but it is useful to be aware of the
different meanings and methods as one encounters the discussions among these scientists.)
In applied problems, it is important to communicate the level of precision to which a quantity
is measured. Two methods for doing so are (1) through tolerances, and (2) by the significant
digits that are listed. We will return to the topic of tolerances in Chapter 3, and introduce them
briefly here, but will concentrate on the topic of significant digits.
A tolerance is an upper bound on the error in measuring a quantity. For instance, one
might give the deisgned gap between the top of a valve stem in a particular combustion engine,
and the valve’s respective cam on the camshaft, as 0.003 in ± 0.0008 in, or (0.003±0.0008) in.
This means that the actual value of the gap’s length would be between 0.003 − 0.0008 in and
0.003 + 0.0008 in, putting the gap in the interval [.0022 in, .0038 in]. In electrical applications,
components are often given tolerances as percentages, but it is the same idea. For instance, a
resistor might be listed as 330 Ohms of resistance, usually written 330Ω, and if it is allowed a
10% tolerance, that is also a tolerance of 33Ω, so the actual value of the resistance should be
within [297 Ω, 363 Ω].
Tolerances can be given in abolute terms, which describes the exact interval by its endpoints
or its center point plus/minus the possible error, or tolerances can be given in relative terms,
usually by percentages, where one often writes, for instance ±5% or claims a 5% tolerance. It
is called a relative measure of error because 5% is considered relative to the size of the target
value. Later we will define relative error to be the actual (absolute) error divided by the target
value, but that is all for later. There is a more compact, if less precise way to communicate the
error, and that is through significant digits.
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2.3. ARITHMETIC OF UNITS; SIGNIFICANT DIGITS
When one writes a measured or approximated quantity using the conventional approach of
significant digits, the basic idea is that the last written digit, as we move left to right, is considered
to be correct to ±1. Furthermore, to see how many significant digits a written quantity conveys,
we count all the digits from the first nonzero digit on the left to the last nonzero digit on the
right.
For instance, suppose we are asked to measure approximately 150 g (150 grams) of some
substance for a chemistry exercise. Further suppose that we are to write down our actual
measurement of the mass we measure. Then the following table represents what we might
report, and what is meant in terms of tolerances. (All quantities are in units of grams.)
Number of Significant Digits
2
3
4
5
6
Significant Digits
150
150.
150.0
150.00
150.000
Tolerances
150 ± 10
150 ± 1
150 ± .1
150 ± .01
150 ± .001
Similarly, if we wrote some measurement as 123.456, we are claiming that the actual value
is 123.456 ± .001. We would say that this represents six significant digits of precision. It has the
disadvantage that the tolerance is always given as some power of 10 (such as 10−3 in this case),
but many of the advantages of concision. For instance, perhaps we really know this number is
correct to within ±.0008, but are satisfied to report it within ±.001. As we will see in Chapter 3,
it is not always easy to get an exact tolerance, and we will often instead estimate the tolerance,
being sure to never “under-estimate,” in the colloquial sense, the possible error.28
There are some conventions for communicating the “significance” of digits. In our original
example, note that 150 is assumed to have only two significant digits, the 1 and the 5. If there is
a decimal written after the 0, it is also considered significant. It becomes confusing if one writes a
number such as 150,000 because it is not clear how many (at least two) digits are significant. For
that reason, the scientific notation is often used, where we write any nonzero number as n× 10m ,
where 1 ≤ n < 10 and m ∈ {0, ±1, ±2, ±3, · · · }. Here n is the it coefficient or mantissa (which
unfortunately has other, very different meanings), and m the exponent or order of magnitude of
the given number. For any nonzero number, it is not hard to show that there is only one way to
express it in scientific notation exactly.29 However we are Thus we can write 150,000 as 1.5 × 105
for two significant digits, and 1.50 × 105 for three, and so on.
Note that we can also write 150,000 as 0,150,000 but the leading zero is not considered
significant. Similarly, if we write 0.123, or 0.0103, or 0.0000000120, we still have three significant
digits, so the “leading zeroes” are not considered significant.
28 In higher mathematics, the term “estimate” is used differently than it is in colloquial English. The colloquial
meaning of “estimate” is instead expressed by professionals in the mathematical sciences as approximate, in both
noun and verb forms. A mathematician using the term “estimate” means he or she is finding bounds for a
quantity, either upper or lower bounds. For instance, if a quantity ξ is known to be less than 10, and a “good
guess” is that it is close to 6, we have the estimate ξ < 10, and the approximation ξ ≈ 6. The estimate is
expressed as a truth that is provable, while the approximation is, in fact, subjective, but its voracity should be
easily judged from the evidence given.
29 In fact this is not quite true, as we can write, for instance, 1 = 0.9999999 · · · , and so 1 × 100 = 9.9 × 10−1 ,
but this become moot in actual empirical sciences where measurements are never exact, and tolerances and thus
significant digits become important. For practical pursposes (involving tolerances), there is only one way to write
a nonzero number in scientific notation. Since we never have infinitely many significant digits for a measured
quantity, we will not find ourselves dealing with expressions like 9.9 × 10−1 or, for that matter, 1.0 × 100 , though
of course we have many times where we have, at least theoretically, the exact number 1.
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CHAPTER 2. REAL NUMBERS, ALGEBRA AND FUNCTIONS
Science students are some loose guidelines for determining how many significant digits a
computational result should contain, but they all have their exceptions so we point out these
guidelines here but with the understanding that they are not rigid.
For instance, adding a number with three significant digits to another with only one significant
digit can result in a number with one, two or three significant digits. Consider the following:
123 + 20, 000 ≈ 20, 000;
123 + 10 ≈ 130;
123 + 1 ≈ 124
The first line, if all things were exact, would yield 20,123, but we do not know our answer to
such precision because the 20,000 term was only known to an accuracy ±10, 000, as only the
2 was communicated to be significant. Similary, when we added 123 to 10, we did not know
the accuracy of the units digit in the 10, so we can not include it in the final answer as being
significant. The final line is more likely to be considered at face value.30
Often, if not usually, a textbook in the physical sciences would simply use = with the understanding that there is a tolerance expressed by the number of significant digits and the final
digit’s placement. These textbooks usually do not use the approximation symbol ≈ (or its vari.
ations such as = or �) in these circumstances. However, this being a mathematics textbook, we
will use the approximation when we finish with our (approximate) answer. In fact we will adopt
the convention that we assume the input data is exact, i.e., solve the problem hypothetically,
and then report the final answer to the number of significant digits actually warranted by the
precision in the data, using ≈ in the final step. We will also not perform numerical, intermediate
computations, but instead allow our calculators to assume the 10–12 digits of accuracy they are
capable of, before truncating our final answer to a reasonable number of significant digits for the
problem.
For instance, suppose we wish to calculate the area of a rectangle with length 6.25 cm and
height 12.67 cm. The usual method for doing so is to allow the calculator to report the product
as if these inputs are exact, and then do our rounding. Multiplication and division differ from
addition and subtraction in that we usually leave our answer with minimum number of significant
digits posessed by the factors. For this computation, we have
2
2
6.25 cm × 12.67 cm = 79.1875
�
�� cm� ≈ 79.2 cm .
calculator reports
In fact, if this were to be the final computation in our report, we might write simply
6.25 cm × 12.67 cm ≈ 79.2 cm2 .
30 In fact, what we really have here in 123+1 is some x + y, where x = 123 ± 1 and y = 1 ± 1, so we really have
x + y = 123 + 1 ± 2 = 124 ± 2, so the final digit can be argued to be not quite significant either. This is often
glossed over in the literature, and we accept the answer 124 to be, more or less, having a tolerance of ±1 though,
mathematically, it is more like ±2. There are several explanations, but in practice it is often left to the judgment
of the one reporting the result, or the reader reviewing the computations.
There is even some disagreement regarding how to translate a number like 125 into a two significant digit
truncation. Some argue to “round up” if the last digit is in {5, 6, 7, 8, 9} and “round down” if the last digit is in
{0, 1, 2, 3, 4}, while some argue that 125 is half-way between 120 and 130 and there is no compelling reason to
round up instead of down. Some will even use the odd or even nature of the preceding digit, in this case 2, to
decide, reasoning that in a large pool of data this should the roundings up and down should be roughly equal in
number.
In this textbook we will always round up if the last digit is 5 or greater, and down otherwise, and so 125 ≈ 130, if
we wish to change the three significant digit number to a two-digit representation instead. This kind of truncation
is often done in the last step of a series of computations, where the calculator outputs more digits than we can
really call significant.
2.3. ARITHMETIC OF UNITS; SIGNIFICANT DIGITS
97
Since the first factor only had three significant digits, the final answer can not have more than
three. However, we do not truncate the second factor first, because we would be needlessly
discarding information about that second quantity if we replace it with 12.7, which assumes a
tolerance of ±.1, where the original, 12.67, assumes a tolerance of ±.01, and is closer to the
actual value than is 12.7. Indeed, if we round first, we get 6.25 × 12.7 = 79.375 ≈ 79.4, and our
last digit differs from the previous compuation by 2.
In general, it is better to assume the quantities are exact, and allow the calculator to use
all of its digits of accuracy, rounding only its final output to reflect the uncertainty of the final
result, that uncertainty being inherent in the original, usually measured, inputs.
This can be a difficult lesson for students of physical sciences, because it is often easier to
write out the intermediate computations in some rounded manner, but this introduces needless
error. It is better to solve for the variable in question, input the numbers. This is true whether
the operations involved are multiplication and division, addition and subtraction, trigonometric
functions, exponential and logarithmic functions, or any combination of these or other functions.
When intermediate results must be written down for future computations, it is better to write
them in as many significant digits as possible, even if they are “reported” in a truncated manner.