# Number TheoryCalculating The Nth Rational Number

#### moyo

##### Member
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
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
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.