Why Read This Book? - Index of

0.2.3 Other Assumptions
0.2 Assumptions about the Real Numbers 7
(L1) For every a ∈ A, we have that a ≤ L, and
(L2) If N is any upper bound for A, it must be that N ≥ L.
The real numbers are indeed a complicated set. The final real number properties
we mention are not standard assumptions, and they deserve your attention at some
point in your mathematical career. In this text, we assume them.
(A21) The real numbers can be equated with the set of all base 10 decimal representations.
That is, every real number can be written in a form like
338.1898 ..., where the decimal might or might not terminate, and might
or might not fall into a pattern of repetition. Furthermore, every decimal
form you can construct represents a real number. Strangely, though, there
might be more than one decimal representation for a certain real number.
You might remember that 0.9999 ...= 1. The repeating 9 is the only case
where more than one decimal representation is possible. We will assume
this.
Our final assumption concerns the existence of roots of real numbers.
(A22) For every positive real number x and any natural number n, there exists a
real number solution y to the equation y n = x. Such a solution y is called
an nth root of x. The common notation n√ x will be addressed in Section 3.9.
Notice we make no assumptions about how many such roots of x there are, or
what their signs are. Nor do we assume anything about roots of zero or of negative
real numbers. We will derive these from assumption A22.
One final comment about assumptions in mathematics is in order. In a rigorous
development of any mathematical theory, some things must be assumed without
proof; that is, they must be axiomatic, serving as a starting place for the mathematician’s
thinking. In a study of the real numbers, some of the assumptions A1–A22
are standard. Others would be considered standard assumptions only for some
subsets of the real numbers, perhaps for the whole numbers. The mathematician
would then very painstakingly apply assumptions made to the whole numbers in
order to expand the same properties to all of the real numbers. One assumption
in particular, A21, is a most presumptuous one. But let us make no apologies for
this. After all, many of the foundational issues in mathematics were addressed
very late historically, and this is not a course in the foundations of mathematics. It
is a course to teach us how mathematics is done and to give us some enjoyment of
that process. We choose assumptions here that likely coincide with your current
idea of a reasonable place to start. In some cases, we will dig more deeply as we
go, though some of the foundational work will come in your later courses.