Certainly in your previous experience with mathematics you have encountered a variety of objects which have been described as numbers: counting numbers, whole numbers, integers, fractions, decimal numbers, real numbers, imaginary numbers. Though we won't start a lengthy philosophical discussion about numbers, here are some questions worth considering initially on our way toward the study of the calculus. "What is a number? How do we decide when something is a number? How do we distinguish numbers? How do we come to have knowledge about numbers?" Though these questions may seem simple, they are not easy to answer with certainty. In fact, numbers have been the source of much deep and probing discussion by philosophers from Plato and Aristotle to Immanuel Kant and Bertrand Russell and the American philosopher Willard V.O. Quine. Well, if these names don't mean much to you now, you might do a little research on them in the library or on the internet. The point is that the concept of number is worth a little careful thought, not just now, but throughout your encounters with mathematics. Number Examples: One simple but powerful way to begin to explore a concept informally is to look at examples. So let's recall some of the different kinds of numbers with the notations traditionally used to represent them. First perhaps are the counting numbers: 1,2,3,...These numbers have so many uses that they appear in some form in most cultures where any kind of number distinctions have developed. The notation we use for the counting numbers developed using HinduArabic numerals brought to Western Europe in about the 12th century. Some authors use the term natural number to describe the counting numbers. We will follow the more common convention of mathematics today and use natural number to describe the counting numbers together with the number 0. We'll denote the set of natural numbers with the letter N, so that N = { 0, 1, 2,... }. The nonzero natural numbers are the counting numbers which are also described as the positive natural numbers and are denoted N_{+}. Integers: To make sense out of contexts where numbers are used to represent changes in quantity, the opposite of counting numbers, called negative numbers, were developed. These numbers, 1, 2, 3, ... , can be considered as solutions to very simple problems such as "what number added to 5 will give a result of 2 ?" Of course, the answer, developed to answer this question, is the negative number 3, because 5 + (3) = 2. The integers, denoted Z, consist of the natural numbers together with their opposites, i.e., the numbers 0,1,1,2,2,3,3,... . So we have another way to describe the counting numbers, this time as the positive integers, denoted Z_{+}. We will use a letter usually from the list i, j, k, l, m, n, p, q when we want a symbol to represent an unknown or arbitrary integer (or a natural number) 

Irrational Numbers: We certainly have not exhausted the examples of numbers that you have encountered. Many numbers were first recognized in studying algebra and geometry. One such number is the square root of 2, denoted `sqrt{2}`. This number illustrates some typical difficulties in the concept of number. As with the rational numbers, this number does solve and algebra problem. We can think of `sqrt{2}` is the solution to the problem, "what number has its square equal to 2?" The difficulty with this number is that `sqrt{2}` is not a rational number. [The end of this section has an argument that justifies this last statement.] So is it a number at all? and if so, what kind of number is it? Fortunately, this number has a geometric interpretation as well, namely, it is the ratio of the length of a square's diagonal to the length of its side. See Figure 1. So at least `sqrt{2}` seems to have meaning through its geometric interpretation. If a square has a side of length 1, then its diagonal will have length `sqrt{2}`. Thus it does make sense to think there is such a number. We will say more about `sqrt{2}` later. 

Decimals: The development of notations for numbers
and the science of measurement have also contributed much to the extension
of number concepts. In about 1585 Simon Stevin proposed decimal notation
as a tool for expressing and manipulating all numbers. Later, after the work
of René Descartes (15961650), much of geometry in the seventeenth
and eighteenth centuries was reduced to the study of numbers. Thus the concept
of a real number developed through the centuries to include all
numbers that could represent measurements of length based on the choice of
some unit for measuring or could be estimated to any decimal precision with
a decimal number. School children work with these notions beginning with
manipulatives such as cuisenaire rods. The continued debates over the acceptance
of the metric system in the United States demonstrate the importance of decimals
in the choice of units for even the most common of daily experiences.
How do we know we have an accurate decimal representation
for `sqrt{2}` or `pi`?
Even such rational numbers as 3/7 require an infinite process when described
in decimal notation. A key fact is that any rational number when expressed
in decimal notation will involve a block of digits that will repeat indefinitely
in its decimal representation. But unlike 3/7 and other rational numbers,
the decimal representation of any number that is not a rational number
will not have a repeating block of digits. So how do we tell what digit is in each position of the decimal expression for `sqrt{2}` or `pi`? In fact we don't have a method for knowing these in any precise formulation comparable to the way we can describe the decimal expression for 3/7. The development of procedures to obtain better descriptions of the decimal expressions for these and other numbers has been an important and lengthy part of mathematical history. To shorten our discussion of numbers and their history, we'll stop here on the trail. In summary, the numbers that can expressed with the (possibly infinite) decimal notation, and can be interpreted as the measurement of a length using some fixed unit, are called real numbers. We will denote the set of real numbers with the symbol Â (and the positive real numbers are denoted Â_{+}). Notation: When we want a symbol to represent a real number we will usually use one of the letters a, b, c, p, q, r, s, t, u, v, w, x, y,or z. Often we will want to consider a real number that measures a change or difference between other numbers. In this case we will often denote the real number by the letters h or k, or symbols like `Delta x` or `Delta y`. [The symbol `Delta` is the Greek letter "delta." It is often used in mathematical notation to indicate a change, as in this case, where it indicates the change in the quantity measured with the variables x or y.] 
Units and Measurement: The use of units in measuring everything from distances between objects to the forces that act upon them to the duration of time is so common place in our lives that we often overlook how these units are determined and adopted as conventions to facilitate communication. New units develop and older ones fall into disuse. The need to make changes of scale between units is a consequence of the variety of units as well as the utility of having units for describing features at different levels of discourse. Inches and centimeters may be appropriate to measure computer discs or keyholes, but light years are much better for measuring the distances between stars. Grams and ounces are appropriate for measuring sugar in a cake recipe, but not for measuring the mass or weight of a dump truck or an elephant. Dollars and cents seem appropriate for measuring the cost of a meal at a restaurant, but not for measuring the budget of the United States federal government or the World Health Organization. Once a unit is established, then numbers can be interpreted in terms of that unit of measurement. Each interpretation through the unit gives a number meaning in its application beyond the abstract number itself. 5 inches, 5 centimeters, 5 light years, 5 grams, 5 ounces, 5 tons, 5 dollars, 5 cents, 5 trillion dollars,.... The connection between numbers and measuring units is one of the crucial aspects of the application of mathematics to real contexts. Try not to lose sight of this connection even when we focus on the more abstract aspects of the mathematical concepts. The interaction of mathematics with interpretations and applications is subtle and not fully understood in all its philosophical and practical details, so feel free to question these things. In your struggles to understand numbers and their uses you will arrive eventually at your own view that will allow you to make sense of this subtle connection. 
Mathematics
is sometimes described as the language of science. The language aspect
of mathematics involves notation, naming, and concept definition.
Many features of a language are organized by historical accident and
use, following precedents that seem convenient for the purpose of
communication. Other aspects are consciously determined through open
discussion of the value of certain choices to the purposes of the
language. Whether by choice or history, the language of mathematics
is dominated by conventions. You can see this conventional quality
clearly in the symbols and names we give to numbers. It is something
you should note occasionally as we proceed that we develop (mathematical)
concepts through experience, motivation, examples, and definitions
that name objects, properties, and relations. Moreover we introduce
notation for these concepts to organize and communicate our knowledge,
making further investigations and applications sensible. The notation
of mathematics has a long and illuminating history, connected to the
development of the mathematical concepts and their applications.

For
example, we can look at the concept of a multiplicative inverse. We
say that 1/3 is the multiplicative inverse of 3, using the
words to describe a relation between the numbers 1/3 and 3, while
we can discuss the multiplicative inverse of the number `pi`,
referring to a number described by its relation to the number
`pi`. The words "multiplicative inverse" name a relation or a number,
so the designation is ambiguous without establishing a context.
Looking at the number use of the words, we can articulate a definition
of words to describe a number. So we might say that a number a
will be called a multiplicative inverse for a number b if the
product of the two numbers, ab, is 1. We have a notation for
this as well. We write `1/b=b^{1}` to denote a number that is a multiplicative inverse for b.
Now to describe a number does not necessarily mean that such a number exists! We can talk about a multiplicative inverse for almost all numbers and make sense of how that number might be described in some other fashion that would allow us to acknowledge its existence. However, using conventional arithmetic for real or complex numbers, there cannot be any multiplicative inverse for 0. The reason is that the result of multiplying any number by 0 is 0, and 0 is not 1. So no number can satisfy the defining condition for it to be a multiplicative inverse for 0. You might consider other situations like `1/0` where an object or concept is defined by describing conditions under which it would be appropriate to use the term but where no actual object exists. This may happen even when a notation apparently indicates an object. In a way, mathematicians are like creative writers who can describe and name fictitious animals called unicorns when no such animal exists in our world. 
The
ambiguity of notation: What does "" mean? When you see an expression
involving the minus sign, it can mean different things depending on
the context. When discussing numbers the notation 5 indicates we
are discussing the number "negative 5" or the number which when added
to 5 would give 0 as a result. Some might even distinguish this use
of the minus sign by writing it slightly higher than the numeral 5,
as ^{}5. A second meaning for the minus sign is to
indicate the opposite of the number that follows the sign. This
can be a little confusing, since writing 5 indicates both a negative
number and the number that is the opposite (or additive inverse) of
5, but the use in this sense is easier to identify when we write an
expression like (^{}5) describing the number 5 which
is the opposite of ^{}5. Finally, the minus sign is
used to indicate the operation of subtraction in an expression such
as 53 which is another way to denote the resulting number 2. Of course
there are situations where all three interpretations can come into
play, as in the expression 10( ^{}5) which can
be found by computing 10( ^{}5) = 10 + 5 = 15..

Equality as identity of numbers: Let's look first
at 2+3=5. The symbol "=" placed between symbols representing numbers indicates
that the symbols are representing the same number. The idea is that
a single object can be designated in more than one way. You have your
name and your social security number. Usually (but not always) if we look
at the names on a class list there will be only person represented by
that name, and the same is true for the social security numbers on the
list. So we might say that Alice Callahan = 396 23 4583 means that these
symbols designate the same person. On the left side of the equation the
symbol "2+3" represents a single number which results by adding the numbers
2 and 3. On the right hand side we have a number designated by the numeral
"5." The assertion of the equation is that these symbols represent the
same number. This interpretation works just as well to explain `2+3=35/7`.
Both statements are considered true because the symbols on each side of
the equations represent the same number. On the other hand the statement
2+3=7 is false because the symbols on both sides of the equation
do not represent the same number, though it may not always be clear how
we know this.
Equality as defining notation: When we write "let
a be the number with the property that `a^3 = 5` " or "`a=root(3)5`" the equality
sign might be replaced with º which is
sometimes used to reinforce the fact that this use is not asserting an
identity but is establishing a representation. In this case the symbol
a is being assigned to represent any one or more objects that satisfy
the criterion that when cubed the result is 5. Such an object is also
represented by `root(3)5`.
The equation is merely establishing another notational representation.
Equality of expressions involving variables: The
equation x+5=7 may be true or false depending on the interpretation
of the variable x as a specific number. If x is 2
then the equation is true, but if x = 3 (or any other number different
from 2) then the equation is false. But what does the equation mean when
we don't have an interpretation specified for x? An equation with
a variable that does not have a specified value is called an open equation.
An open equation is neither true nor false until the interpretation
of all the variables have been specified. Such an equation serves as a
symbolic form designating a relation between numbers when the variables
are specified. Generally these open equations arise as constraints
characterizing a number or numbers and the problem is to determine
any and all numbers that will make the equation true, i.e., solve
the equation.

`[2,5]={x:2<=x<=5}`  `[2,5)={x:2<=x<5}` 
`(2,5]={x:2<x<=5}`  `(2,5)={x:2<x<5}` 
`[2,oo]={x:2<=x}`  `(2,oo]={x:2<x}` 
`(oo,5]={x: x<=5}`  `(oo,5)={x: x<5}` 
10.Write a short story about a car trip. Discuss briefly those aspects of the trip that might be measured and the units and approximate size of the numbers that would arise from those measurements.
11.Write a short story about cooking dinner. Discuss briefly those aspects of the meal's preparation that might be measured and the units and approximate size of the numbers that would arise from those measurements.
12.Write a short story about going shopping. Discuss briefly those aspects of the shopping that might be measured and the units and approximate size of the numbers that would arise from those measurements.
13.Write a short description of an ecological system. Discuss briefly those aspects of the system that might be measured and the units and approximate size of the numbers that would arise from those measurements.
14.Write a short description of the human body. Discuss briefly those aspects of the body that might be measured and the units and approximate size of the numbers that would arise from those measurements.
15.Write a short story about an athletic event or sports competition. Discuss briefly those aspects of the story that might be measured and the units and approximate size of the numbers that would arise from those measurements.
16.How Ben proved he was the pope.
Ben arrived yesterday smiling quite broadly. "I can show you that I am the Pope," he declared with a laugh. "You see I and the pope are two persons. But I can show that 1=2, so I and the pope are 1 person. Thus I am the Pope." But you may ask, "How did Ben show that 1=2?" Here's the argument Ben gave me.
b. A symbol representing such a quantity. For example, in the expression , a, b, and c are variables.
4. A distinguishing characteristic or feature.