- #1
exmachina
- 44
- 0
After a rather interesting discussion in #math on freenode, I'm perplexed by the fact that there is no way to construct lines from points.
For example, I initially thought of constructing a line between 0 and 1 by repeatedly dividing into two smaller halves, eg:
0,1
0,0.5,1
0,0.25,0.5,0.75,1
..etc
But obviously this approach will miss the irrationals and cover only rationals of the form (1/2)^n thus, missing all others with such as (1/3)^n, (1/5)^n, (1/m)^n where m is prime.
So, in the hypothetical case where I do cover all the points arising from
(1/2)^n
(1/3)^n
(1/m)^n ...
I would STILL miss the irrationals.
For example, I initially thought of constructing a line between 0 and 1 by repeatedly dividing into two smaller halves, eg:
0,1
0,0.5,1
0,0.25,0.5,0.75,1
..etc
But obviously this approach will miss the irrationals and cover only rationals of the form (1/2)^n thus, missing all others with such as (1/3)^n, (1/5)^n, (1/m)^n where m is prime.
So, in the hypothetical case where I do cover all the points arising from
(1/2)^n
(1/3)^n
(1/m)^n ...
I would STILL miss the irrationals.