Principles of Mathematics in Operations Research

130 9 Number Systems

Define f:N^Q+,

/(l) = 1, /(2) = |, /(3) = 2, /(4) = |,

Cantor's Counting Schema Another Counting Scheme

Fig. 9.1. Counting schema for rational's

Table 9.1. List of rational numbers

n m 1 2 3 4 5

12 3 4 5

1/1 1/2 1/3 1/4 1/5 •••

2/1 2/2 2/3 2/4 2/5 •• •

3/1 3/2 3/3 3/4 3/5 •• •

4/1 4/2 4/3 4/4 4/5 •• •

5/1 5/2 5/3 5/4 5/5 •••

Example 9.6.5 Q is countable. Since <Q
can be listed as a sequence {2:1,0:2,2:3, • •
listed as {—x±, —#2, —£3,.. •}•
Q = 0 Xi —Xi X 2 —X2 X3 -X3...

N= 1 2 3 4 5 6 7 ...

/ : N i-> Q can be defined in this way.

is countable, the elements of Q+

. Then, Q~ - {q : q -< 0} can be

XR, if n is even

/(n) — I — Xn^, if n is odd

(^2) 0, ifn=l.