1  Quantity, function, space

This book presents calculus in terms of three concepts central to the study of change: quantities, functions, and spaces. Those words have everyday meanings which are, happily, close to the specific mathematical concepts that we will be using over and over again. Close … but not identical. So, pay careful attention to the brief descriptions that follow.

1.1 Quantity vs number

A mathematical quantity is an amount. How we measure amounts depends on the kind of stuff we are measuring. The real-world stuff might be mass or time or length. It equally well can be velocity or volume or momentum or yearly corn yield per acre. We live in a world of such stuff, some of which is tangible (e.g., corn, mass, force) and some of which is harder to get your hands on and your minds around (acceleration, crop yield, fuel economy). An important use of calculus is helping us conceptualize the abstract kinds of stuff as mathematical compositions of simpler stuff. For example, crop yield incorporates mass with length and time. Later, you will see us using the more scientific-sounding term dimension instead of “stuff.”

Chapter 15 covers units and dimension.

Most people are inclined to think “quantity” is the same as “number”. This is understandable but misguided. By itself a number is meaningless. What meaning does the number 5 have without more context? Quantity, on the other hand, combines a number with the appropriate context to describe some amount of stuff.

The first thing you need to know about any quantity is the kind of stuff it describes. A “mile” is a kind of stuff: length. A meter is the same kind of stuff: length. A liter is a different kind of stuff: volume. A gallon and an acre-foot are the same kind of stuff: volume. But an inch (length) is not the same kind of stuff as an hour (time).

“Stuff,” as we mean it here, is what we measure. As you know, we measure with units. The appropriate units depend on the kind of stuff. Meters, miles, and microns are all appropriate units of length, even though the actual lengths of these units differ markedly. (A mile is roughly 1.6 million millimeters.)

Only after you know the units does the number have meaning as a quantity: a number is only part of specifying a quantity.

Example: You cannot add feet and acres. They describe different kinds of things: length versus area. Yet you can multiply feet and acres. The unit “acre-foot” is widely used in water resource management.

Here’s the salient difference between number and quantity when it comes to calculus: All sorts of arithmetic and other mathematical operations apply to numbers: addition, multiplication, square roots, etc. But for quantities, only multiplication and division are universally allowed. For addition and subtraction, square roots, and such, the operation makes sense only if the dimensions are suitable.

The mathematics of units and dimension are to the technical world what common sense is in our everyday world. For instance (and this may not make sense at this point), if people tell me they are taking the square root of 10 liters, I know immediately that either they are just mistaken or that they haven’t told me essential elements of the situation. It is just as if someone said, “I swam across the tennis court.” You know that person either used the wrong verb—walk or run would work—or that it wasn’t a tennis court, or that something important was unstated, perhaps, “During the flood, I swam across the tennis court.”

1.2 Functions

Other examples of relationships:

  • the input is the altitude on your hike up Pikes Peak; the output is the air temperature. Typically, as you gain altitude the temperature goes down.
  • the input is the number of hours past noon; the output is the brightness of sunlight. As the afternoon progresses, the light grows dimmer, but only to a point.

Functions, in their mathematical and computing sense, are central to calculus. The introduction to this Preliminaries Block starts, “Calculus is about change, and change is about relationships.” The idea of a mathematical function gives a definite perspective on this. The relationship represented by a function is between the function’s input and the function’s output. The input might be day-of-year1 and the output cumulative rainfall up to that day. Every day it rains, the cumulative rainfall increases.

A function is a mathematical concept for taking one or more inputs and returning an output. In calculus, we will deal mainly with functions that take one or more quantities as inputs and return another quantity as output.

Pay careful attention to our use of “input” and “output.” We avoid using the word “variable” because it is too vague. (For instance, it does not distinguish between what goes in and what comes out of a function.) There are two contexts in which we will use “variable,” neither of which has to do with inputs to functions. In talking about data, we will use “variable” in the statistical sense, meaning “a type of quantity” like height or pH. And in the final part of the text, involving systems whose configuration changes in time, we will use “variable” in the sense of “a quantity that varies over time.” Try to put the word “variable” out of mind for the present, until we get to discussing the nature of data.

But sometimes we will work with functions that take functions as input and return a quantity as output. And there will even be functions that take a function as an input and return a function as output.

In a definition like \(f(x) \equiv \sqrt{\strut x}\), think of \(x\) as the name of an input. So far as the definition is concerned, \(x\) is just a name. We could have used any other name; it is only convention that leads us to choose \(x\). The definition could equally well have been \(f(y) \equiv \sqrt{\strut y}\) or \(f(\text{zebra}) \equiv \sqrt{\strut\text{zebra}}\).

Notation like \(f(x)\) is also used for something completely different from a definition. In particular, \(f(x)\) can mean apply the function \(f()\) to a quantity named \(x\). You can always tell which is intended—function definition or applying a function—by whether the \(\equiv\) sign is involved in the expression.

Later in this Preliminaries Block, we will introduce the “pattern-book functions.” These always take a pure number as input and return a pure number as output. In the Modeling Block, we will turn to functions that take quantities—which generally have units—as input and return another quantity as output. The output quantity also generally has units.

One familiar sign of applying a function is when the contents of the parentheses are not a symbolic name but a numeral. For example, when we write \(\sin(7.3)\) we give the numerical value \(7.3\) to the sine function. The sine function then does its calculation and returns the value 0.8504366. In other words, \(\sin(7.3)\) is utterly equivalent to 0.8504366.

In contrast, using a name on it is own inside the parentheses indicates that the specific value for the input is being determined elsewhere. For example, when defining a function we often will be combining two or more functions, like this: \[g(x) \equiv \exp(x) \sin(x)\] or \[h(y,z) \equiv \ln(z) \left(\strut\sin(z) - \cos(y)\right)\ .\] The \(y\) and \(z\) on the left side of the definition are the names of the inputs to \(h()\).2 The right side describes how to construct the output, which is being done by applying \(\ln()\), \(\sin()\) and \(\cos()\) to the inputs. Using the names on the right side tells us which function is being applied to which input. We won’t know what the specific values those inputs will have until the function \(h()\) is being applied to inputs, as with \[h(y=1.7, z=3.2)\ .\]

Once we have specific inputs, we (or the computer) can plug them into the right side of the definitionto determine the function output: \[\ln(3.2)\left(\sin(3.2) - \strut \cos(1.7)\right) = 1.163(-0.0584 + 0.1288) =-0.2178\ .\]

We will introduce the idea of “spaces” in Section ?sec-spaces-intro. A function maps each point in the function’s input space into a single point in the function’s output space. The input and output spaces are also known respectively as the “domain” and “range” of the function.

The various mathematical functions that we will be studying in this book are in the service of practical problems. But there are so many such problems, often involving specialized knowledge of a domain or science, engineering, economics, and so on, that an abstract mathematical presentation can seem detached from reality.

The video linked here, How to shoot, breaks down a simple-sounding situation into its components. The function itself is literally a black box. The inputs are provided by a human gunner training a telescope on a target and setting control dials. The ultimate output is the deflection of the guns in a remote turret. The main function is composed of several others, such as a function that outputs target range given the target size based on knowledge of the size of the target and how large it appears in the telescopic sight.

Dividing the gunnery task into a set of inputs and a computed output allows for a division of labor. The gunner can provide the skills properly trained humans are good at, such as tracking a target visually. The computer provides the capabilities—mathematical calculation—to which electronics are well suited. Combining the inputs with the calculation provides an effective solution to a practical problem.

1.3 Spaces

We said earlier that the functions used in calculus take quantities as input and produce a quantity as output. We’ve also said that a quantity is something like “2 light-years” or “150 watts”. Now we want to connect a new concept to input and output: the concept of spaces.

A space3 is a collection of continuous possibilities. A child learning about numbers starts with the “counting numbers”: \(1, 2, 3, \ldots\). In primary school, the set of numbers is extended to include zero and the negative numbers: \(-1,-2,-3, \ldots\), giving a set called the “integers.” Counting numbers and integers are discrete sets. Between two consecutive members of the counting numbers or the integers, there is not another number of the set.

The next step in a child’s mathematical education is the “rational numbers,” that is, numbers that are written as a ratio: \(\frac{1}{2}, \frac{1}{3}, \frac{2}{3}, \ldots, \frac{22}{7}\), and so on. Rational numbers fit in the spaces between the integers. That is, between any two integers, even consecutive ones, there is a rational number. For instance, the rational number \(\frac{1}{2}\) falls between 0 and 1.

Between any two rational numbers, there is another rational number, indeed there is an infinite number of rational numbers. For instance, between \(\frac{1}{2}\) and \(\frac{2}{3}\) is \(\frac{6}{11}\) (and many others, such as \(\frac{7}{11}\) or \(\frac{13}{21}\)). It is useful to think of rational numbers as fitting in the spaces between integers.

If you didn’t stumble on the word “spaces” in the previous sentence, you are well on your way to understanding what is meant by “continuous.” For instance, between any two rational numbers there is another rational number. Think of rational numbers as stepping stones that provide a path from any number to any other number.

It is a deep question whether the rational numbers are a walkway instead of isolated stepping stones? A walkway is a structure on which you can move any amount, no matter how small, without risk of going off the structure. In contrast, a too-small move along a path of stepping stones will put you in the water.

A continuous set is like a walkway; however little you move from an element of the set you will still be on the set. The continuous set of numbers is often called the number line, although a more formal name is the real numbers. (“Real” is an unfortunate choice of words, but we are stuck with it.)

The underlying metaphor here is space. Between any two points in space, there is another point in space. We will be working with several different spaces, for instance:

Your spatial intuition of lines, planes, etc. will suffice for our needs. Mathematicians as a class value precise definitions; we won’t need those. Widely accepted mathematical definitions of continuous sets date from the 1800s, 150 years after calculus was introduced. For instance, it has been known for more than 2000 years that there are numbers—the irrational numbers—that cannot be exactly expressed as a ratio of integers. We know now that there is an irrational number between any two rational numbers; the rational numbers are indeed analogous to stepping stones. But the distinction between rational and irrational numbers will not be one we need in this book. Instead, we need merely the notion of continuous space.

  • the number line: all the real numbers
  • the positive numbers: the real numbers greater than zero
  • the non-negative numbers: this is the tiniest little extension of the positive numbers adding zero to the set.
  • a closed interval, such as the numbers between 5 and 10, which we will write like this: \(5 \leq x \leq 10\), where \(x\) is a name we are giving to the set.
  • the Cartesian plane: all pairs of real numbers such as \((5.62, -0.13)\). Other metaphors for this: the points on a piece of paper or a computer screen.
  • three-dimensional coordinate spaces like our everyday three-dimensional world, generally written as a set of three real numbers such as \((-2.14, 6.87, 4.03)\).
  • higher-dimensional spaces, but we won’t go there until the last parts of the book.

The specialty of calculus is describing relationships between continuous sets. Functions such as \(\sin()\) or \(\text{line}()\), which are typical of the functions we study in calculus, take numbers as input.

Every function has a set of legitimate inputs. For the functions studied in calculus, this set is continuous: a space. The name given to a function’s space of legitimate inputs is the function domain. Functions such as \(\sin()\) and many others have the entire set of real numbers as the function domain. The square-root function has the non-negative numbers for its domain. The logarithmic function, \(\ln()\), has a domain of the positive numbers.

Just as a “domain” is the set of legitimate inputs to a function, the function’s range is the set of values that the function can produce as output. For instance, the range of \(\sin()\) is the numbers between \(-1\) and \(1\) which we will usually write in this format: \(-1 \leq x \leq 1\). Another example: the range of \(\ln()\) is the entire space of real numbers.

1.4 All together now

The three mathematical concepts we’ve been discussing—quantity, function, space—are used together.

A quantity can be a specific value, like 42.681\(^\circ\)F. But you can also think of a quantity more broadly, for instance, “temperature.” Naturally, there are many possible values for temperature. The set of all possible values is a space. And, using the metaphor of space, the specific value 42.681\(^\circ\)F is a single point in that space.

Functions relate input quantities to a corresponding output quantity. A way to think of this—which will be important in Chapter ?chap-graphs-and-graphics —is that a function is a correspondence between each point in the input space (domain) and a corresponding point in the output space (range).

1.5 Exercises

Exercise 1.01

For each of the following, say whether the quantity is a pure number or has units. If it has units, say what kind of physical quantity is being represented.

  1. 17.3
  2. \(\pi\)
  3. 9 feet
  4. 8 \(\mu\)m
  5. 12.2 gm
  6. 37\(^\circ\) C
  7. 43 sec

Exercise 1.02

Here are some function definitions in the format we will use in this book. For each, say what is the name of the function being defined, what are the names of the inputs to the function, and what are the names, if any, of the parameters used in the definition.

  1. \(h(x) \equiv 3 x + 2\)
  2. \(\text{line}(x) \equiv m x + b\)
  3. \(\text{wave}(t) \equiv A \sin(2 \pi t/ P)\)
  4. \(\text{plane}(x, y) \equiv a + bx + cy\)
  5. \(f_1(z) = z^n\)

Exercise 1.03

For each of the following function definitions, what is the input name?

Part A Input name in \(g(t) \equiv 2 t^2 + 8\)?

t u v w x y z

Part B Input name in \(\line(z) \equiv a z + b\)?

t u v w x y z

Part C Input name in \(h(t) \equiv 2 t^2 + 8 w\)?

t u v w x y z

Part D Input name in \(f(u) \equiv a u + b + u^2\)?

t u v w x y z

Part E Input name in \(g(w) \equiv x + 4\)?

t u v w x y z

Exercise 1.04

  1. Demonstrate that between any two integers \(a\) and \(b\) there is a rational number. (Hint: Do some arithmetic with \(a\) and \(b\) to construct such a rational number.)

  2. Demonstrate that between any two rational numbers \(a/b\) and \(c/d\) there is another rational number. (Hint: Construct such a rational number by arithmetic.)


  1. “Day-of-year” is a quantity with units “days.” It starts at 0 on midnight of New Year’s Eve and ends at 365 at the end of day on Dec. 31.↩︎

  2. Sometimes, we will use both a name and a specific value, for instance \(\sin(x=7.3)\) or \(\left.\sin(x)\Large\strut\right|_{x=7.3}\)↩︎

  3. Which can also be called a “set.”↩︎