Gibilisco S. Everyday Math Demystified: A Self-Teaching Guide

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4. The square of b/4, where b represents any rational number, is

(a) b

(b) b

/16

(d) b

/256

5. Figure 1-5 is called a Venn diagram. It denotes sets as circular regions

(including their interiors) on a surface. What does the shaded region in

this illustration represent?

(a) A þ B

(b) A \ B

(d) A  B

6. In Fig. 1-5, the objects labeled P, Q, and R represent individual points.

Which of the following statements is true?

(a) P 2 (A [ B)

(b) Q 2 (A \ B)

(d) P  A

7. Suppose q, r, and s represent real numbers. Which of the following

statements is not true in general, unless constraints are imposed on

one or more of the variables?

(a) q

(r þ s)

¼ q

(b) q/s  r/s ¼ (q  r)/s

(d) (qr)s ¼ q(rs)

8. When we write 34.34343434 ..., we are writing a representation of

(a) an irrational number

(b) a nonterminating, nonrepeating decimal

(d) a rational number

Fig. 1-5. Illustration for Quiz Questions 5 and 6.

CHAPTER 1 Numbers and Arithmetic 31

9. The 1/2 power of a number is the same thing as

(a) the square root of that number

(b) half of that number

(d) nothing; it is not deﬁned

10. Suppose x, y, and z are positive real numbers. What is the value of

(yz)

when y ¼ z?

(a) 0

(b) 1

(d) It is not deﬁned.

PART 1 Expressing Quantities

CHAPTER

How Variables

Relate

Tables and graphs portray relationships between changeable quantities known

as variables. Tables and graphs show how the values of variables compare

with, and in some cases aﬀect, one another. When a table or graph is well com-

posed, it can reveal phenomena that would otherwise be impossible to see.

This versus That

Consider the following statements. Each of them represents a situation that

could occur in everyday life.

The outdoor air temperature varies with the time of day.

The time the sun is above the horizon on June 21 varies with the

latitude of the observer.

The time required for a wet rag to dry depends on the air temperature.

All of these expressions involve something that depends on something else. In

the ﬁrst case, a statement is made concerning temperature versus time; in the

second case, a statement is made concerning sun-up time versus latitude; in

the third case, a statement is made concerning time versus temperature.

Here, the term versus means ‘‘depending on.’’

INDEPENDENT VARIABLES

An independent variable can change in value, but its value is not inﬂuenced by

anything else in a given scenario. Time is often treated as an independent

variable. A lot of things depend on time.

When two or more variables are interrelated, at least one of the variables is

independent, but they are not all independent. A common and simple

situation is one in which there are two variables, one of which is independent.

In the three situations described above, the independent variables are time,

latitude, and air temperature.

DEPENDENT VARIABLES

A dependent variable can change in value, but its value is aﬀected by

at least one other factor in a situation. In the scenarios described above,

the air temperature, the sun-up time, and time are dependent variables.

When two or more variables are interrelated, at least one of them is

dependent, but they cannot all be dependent. Something that’s an independent

variable in one instance can be a dependent variable in another case. For

example, the air temperature is a dependent variable in the ﬁrst situation

described above, but it is an independent variable in the third situation.

SCENARIOS ILLUSTRATED

The three scenarios described above lend themselves to illustration. In order,

they are shown crudely in Fig. 2-1.

Figure 2-1A shows an example of outdoor air temperature versus time

of day. Drawing B shows the sun-up time (the number of hours per day in

which the sun is above the horizon) versus latitude on June 21, where points

south of the equator have negative latitude and points north of the equator

have positive latitude. Drawing C shows the time it takes for a rag to dry,

plotted against the air temperature.

PART 1 Expressing Quantities

The scenarios represented by Figs. 2-1A and C are ﬁction, having been

contrived for this discussion. But Fig. 2-1B represents a physical reality; it is

true astronomical data for June 21 of every year on earth.

RELATIONS

All three of the graphs in Fig. 2-1 represent relations. In mathematics,

a relation is an expression of the way two or more variables compare or

interact. (It could just as well be called a relationship, a comparison, or an

interaction.) Figure 2-1B, for example, is a graph of the relation between

the latitude and the sun-up time on June 21.

Fig. 2-1. Three ‘‘this-versus-that’’ scenarios. At A, air temperature versus time of day; at B,

sun-up time versus latitude; at C, time for rag to dry versus air temperature.

CHAPTER 2 How Variables Relate 35

When dealing with relations, the statements are equally valid if the vari-

ables are stated the other way around. Thus, Fig. 2-1B shows a relation

between the sun-up time on June 21 and the latitude. In a relation, ‘‘this

versus that’’ means the same thing as ‘‘that versus this.’’ Relations can always

be expressed in graphical form.

FUNCTIONS

A function is a special type of mathematical relation. A relation describes how

variables compare with each other. In a sense, it is ‘‘passive.’’ A function

transforms, processes, or morphs the quantity represented by the independ-

ent variable into the quantity represented by the dependent variable.

A function is ‘‘active.’’

All three of the graphs in Fig. 2-1 represent functions. The changes in the

value of the independent variable can, in some sense, be thought of as

causative factors in the variations of the value of the dependent variable. We

might re-state the scenarios this way to emphasize that they are functions:

The outdoor air temperature is a function of the time of day.

The sun-up time on June 21 is a function of the latitude of the observer.

The time required for a wet rag to dry is a function of the air

temperature.

A relation can be a function only when every element in the set of its

independent variables has at most one correspondent in the set of dependent

variables. If a given value of the dependent variable in a relation has more

than one independent-variable value corresponding to it, then that relation

might nevertheless be a function. But if any given value of the independent

variable corresponds to more than one dependent-variable value, that

relation is not a function.

REVERSING THE VARIABLES

In graphs of functions, independent variables are usually represented by

horizontal axes, and dependent variables are usually represented by vertical

axes. Imagine a movable, vertical line in a graph, and suppose that you

can move it back and forth. A curve represents a function if and only if it

never intersects the movable vertical line at more than one point.

Imagine that the independent and dependent variables of the functions

shown in Fig. 2-1 are reversed. This results in some weird assertions:

The time of day is a function of the outdoor air temperature.

PART 1 Expressing Quantities

The latitude of an observer is a function of the sun-up time on June 21.

The air temperature is a function of the time it takes for a wet rag

to dry.

The ﬁrst two of these statements are clearly ridiculous. Time does not depend

on temperature. You can’t make time go backwards by cooling things oﬀ or

make it rush into the future by heating things up. Your geographic location is

not dependent on how long the sun is up. If that were true, you would be at a

diﬀerent latitude a week from now than you are today, even if you don’t go

anywhere (unless you live on the equator!).

If you turn the graphs of Figs. 2-1A and B sideways to reﬂect the

transposition of the variables and then perform the vertical-line test, you’ll

see that they no longer depict functions. So the ﬁrst two of the above

assertions are not only absurd; they are false.

Figure 2-1C represents a function, at least in theory, when ‘‘stood on its

ear.’’ The statement is still strange, but it can at least be true under certain

conditions. The drying time of a standard-size wet rag made of a standard

material could be used to infer air temperature experimentally (although

humidity and wind speed would be factors too). When you want to determine

whether or not a certain graph represents a mathematical function, use the

vertical-line test, not the common-sense test!

PROBLEM 2-1

Figure 2-2 represents ﬂuctuations in the prices of two hypothetical stocks,

called Stock X and Stock Y, over a portion of a business day. Do either

or both of these curves represent stock price as functions of time?

SOLUTION 2-1

Both of the curves represent stock price as functions of time. You can

determine this using the vertical-line test. Neither of the curves intersects a

movable, vertical line more than once. Thus, both curves represent functions

of the independent variable (time).

PROBLEM 2-2

Suppose, in the situation shown by Fig. 2-2, the stock price is consid-

ered the independent variable, and time is considered the dependent

variable. Do either or both of the curves represent time as functions of

stock price?

SOLUTION 2-2

To determine this, plot the graphs by ‘‘standing the curves on their ears,’’

as shown in Fig. 2-3. (The curves are rotated 90 degrees counterclockwise,

CHAPTER 2 How Variables Relate 37

and then mirrored horizontally.) Using the vertical-line test, it is apparent

that time can be considered a function of the price of Stock X, but not a

function of the price of Stock Y.

Simple Graphs

When the variables in a function can attain only a few values (called discrete

values), graphs can be rendered in simpliﬁed forms. Here are some of the

most common types.

Fig. 2-3. Another illustration for Problem 2-2.

Fig. 2-2. Illustration for Problems 2-1 and 2-2.

PART 1 Expressing Quantities

VERTICAL BAR GRAPHS

In a vertical bar graph, the independent variable is shown on the horizontal

axis and the dependent variable is shown on the vertical axis. Function

values are portrayed as the heights of bars having equal widths. Figure 2-4

is a vertical bar graph of the price of the hypothetical Stock Y at intervals

of 1 hour.

HORIZONTAL BAR GRAPHS

In a horizontal bar graph, the independent variable is shown on the vertical

axis and the dependent variable is shown on the horizontal axis. Function

values are portrayed as the widths of bars having equal heights. Figure 2-5

is a horizontal bar graph of the price of the hypothetical Stock Y at intervals

of 1 hour. This type of graph is used much less often than the vertical bar

graph. This is because it is customary to place the independent variable on

the horizontal axis and the dependent variable on the vertical axis in any

graph, when possible.

HISTOGRAMS

A histogram is a bar graph applied to a special situation called a distribution.

An example is a portrayal of the grades a class receives on a test, such as is

shown in Fig. 2-6. Here, each vertical bar represents a letter grade (A, B, C,

D, or F). The height of the bar represents the percentage of students in the

class receiving that grade.

Fig. 2-4. Vertical bar graph of hypothetical stock price versus time.

CHAPTER 2 How Variables Relate 39

In Fig. 2-6, the values of the dependent variable are written at the top

of each bar. In this case, the percentages add up to 100%, based on the

assumption that all of the people in the class are present, take the test, and

turn in their papers. The values of the dependent variable are annotated

this way in some bar graphs. It’s a good idea to write in these numbers if

there aren’t too many bars in the graph, but it can make the graph look

messy or confusing if there are a lot of bars.

In some bar graphs showing percentages, the values do not add up to

100%. We’ll see an example of this shortly.

NOMOGRAPHS

A nomograph is a one-dimensional graph that consists of two graduated

scales lined up directly with each other. Such graphs are useful when the

Fig. 2-5. Horizontal bar graph of hypothetical stock price versus time.

Fig. 2-6. A histogram is a bar graph that shows a statistical distribution.

PART 1 Expressing Quantities