Exercise [16.07]

robin · 15 May 2010, 08:27

We wish to define a way of ordering the fractions. We can do this by consider the fractions as the ordered pairs $(a,b)\in\mathbb{Z}\to\mathbb{Z}$ , such that $b\neq 0$ , and with the equivalence $(a,b)\equiv(\lambda a,\lambda b)$ for any $\lambda\in\mathbb{Z}\backslash\{0\}$ .

In Exercise 16.8 we found a 1-1 mapping $f:\mathbb{N}\times\mathbb{N}\to\mathbb{N}$ . Thus there exists a well-defined inverse function $g = f^{-1}:\mathbb{N}\to\times\mathbb{N}$ .

We first construct the sequence $\{a_n\}, n\in\mathbb{N}$ via:
$a_0=g(0), a_1=-g(0), a_2=g(1), a_3=-g(1), a_4=g(2), a_5=-g(2), \dots$

Note that we needed to include the negative values because the fractions take both positive and negative values. Now we edit the sequence to remove duplicates and values which do not represent fractions. Namely we go through the sequence $\{a_n\}$ term by term starting from $\{a_0\}$ . If a_k=(p,0)

or $a_k\equiv a_r$ for some r<k

then we delete the term. Otherwise we leave it in, identifying a_k=(p,q)

with the fraction p/q

.

After relabelling indices, this provides a new sequence $\{b_n\}, n\in\mathbb{N}$ which defines an ordering for the fractions.