MHB Calculating The Nth Rational Number

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The discussion centers on mapping natural numbers to rational numbers and the possibility of defining a "distance" between consecutive terms in this mapping. It explores whether a reasonable formula can be established to calculate the nth term of such a progression, emphasizing that the existence of such formulas depends on the specific mapping used. The participants note that since rational numbers are countable, progressions can exist, but their calculability hinges on the chosen method. A reference to an algorithm for calculating the nth term in the Calkin-Wilf sequence is also provided. Overall, the feasibility of defining a systematic approach to these calculations is the main focus.
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Hallo

If we specify a particular method for mapping the natural numbers to the rationals, could we also specify a "distance" between two consecutive terms in some general way. Also are we able to calculate the nth term in such a progression perhaps incorporating this distance function somehow within its expression.
 
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Your question is not very clear. What are you referring to when you say "two consecutive terms", "nth term" and "progression"?
 
moyo said:
Hallo

If we specify a particular method for mapping the natural numbers to the rationals, could we also specify a "distance" between two consecutive terms in some general way. Also are we able to calculate the nth term in such a progression perhaps incorporating this distance function somehow within its expression.

If "we specify a particular method for mapping the natural numbers to the rationals" is the key. If we have some function \{a_n\} such that to every natural number n, we have a rational number a_n and every rational number is on that list, then, for any n we could determine a_{n+1}- a_n. Whether there would be any reasonable formula for that function of n depends on the mapping. And asking whether "we able to calculate the nth term in such a progression" is asking whether there exist a reasonable function describing that progression.

Since the rational numbers are countable, such progressions exist but whether or not there exist reasonable formulas for calculating them depends on the progression.
 
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