Properties of the
Number Systems of
Undergraduate
Mathematics
CHAPTER 9
Throughout most precalculus and elementary calculus courses, as well as
in several earlier portions of this text, a basic assumption is that a mathema-
tical object called the real numbers is familiar to students. Indeed, we would
guess that most students would respond "somewhat familiar," if not "very
familiar," to a multiple-choice question asking for a rating of their familiar-
ity with the real number system R. If this guess is accurate, before reading
further you might engage in an interesting exercise by trying to write an
explanation of why R is familiar. In fact, you should at this point pause
to write down all that you actually know about R.
Assuming you have spent some time on this exercise, let us speculate
on your answers. Various algebraic properties of R probably came to mind,
such as "anything times zero equals zero,... the product of a positive with
a negative real number is negative,... and... a nonzero factor can be can-
celed from both sides of an equation." On another level, you may have
focused on the representation of R as the set of points on a line. Theorems
such as "between any two real numbers lies a third real number... and...
there is no largest real number" suggest themselves in this context. Or else,
you may have alluded to the role of the rational numbers within R and the
existence of real numbers, such as a, that are not rational.
Any number of other responses are, of course, possible and of equal value
to those just suggested. But consider the following: How many of these
responses, as well as the ones you gave, apply equally well to the question,