Consider the field F of rational functions defined in Example 2.6.

 

(a) Which is larger, x² or 3/x?

(b) Which is larger, x/(x + 2) or x/(x + 1)? There is one more algebraic property of the real numbers to which we give special attention because of its frequent use in proofs in analysis, and because it may not be familiar to the reader.

Example 2.6

For a more unusual example of an ordered field, let F be the set of all rational functions. That is, F is the set of all quotients of polynomials. A typical element of F looks like

where the coefficients are real numbers and b_k ≠ 0. Using the usual rules for adding, subtracting, multiplying, and dividing polynomials, it is not difficult to verify that F is a field. We can define an order on F by saying that a quotient such as above is positive iff a_n and b_k have the same sign; that is, an ⋅ b_k > 0. For example,

We have not proved that Q ⊆ R, but this relationship should come as no surprise to the reader. A rigorous proof may be found in Stewart and Tall (1977).

The verification that “>” satisfies the order axioms is left to the reader (Exercise 11). It turns out that the ordered field F has a number of interesting properties, as we shall see later in the chapter.