If we follow the arrows in Fig.6 as shown (and skip the common fractions like 1:1 is the same as 2:2 or 3:3 and so on ; 2:1 is the same as 4:2 or 6:3 and so on and so forth)

we can again use the correspondence principle used earlier. Number each arrow-stop using the natural numbers :-

Each fraction has a ‘unique’ natural number associated with it and the entire grid can thus be covered exhaustively. Both sets therefore have the same ‘infiniteness’, that is they are both ‘countably infinite’ and have **א _{0} **elements.

- If we think of the rational numbers, say just between ‘0’ and ‘1’ (we can also write this as [0,1] ) , and arrange them by size, it seems this interval is completely filled by these fractions. But this is not so, between two rational numbers, however close, there will always be another rational number. To appreciate this take (here the 6 recurs endlessly and we can write instead 0.41‾6) 12

and we take the fraction (here the entire sequence 428571 cycles)

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