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Number Theory Calculating The Nth Rational Number

moyo

Member
Aug 10, 2017
30
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.
 

MountEvariste

Well-known member
Jun 29, 2017
92
Your question is not very clear. What are you referring to when you say "two consecutive terms", "nth term" and "progression"?
 

Country Boy

Well-known member
MHB Math Helper
Jan 30, 2018
692
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 [tex]\{a_n\}[/tex] such that to every natural number n, we have a rational number [tex]a_n[/tex] and every rational number is on that list, then, for any n we could determine [tex]a_{n+1}- a_n[/tex]. 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.