Chris wrote:
Hi Mike, I hope you don't feel piled up on.
The irrational numbers just sit between the rational numbers. You can give them any useful name, and you can estimate their location on a number line in relationship to another number. After all, we know that "pi" sits between "3" and "4" on an integer line, and is between "3.125" and "3.25" on a rational number line.
Because you can't express irrational numbers algebraically, you have to arrange them in relationship to real numbers and other irrational numbers, but they're still countable/estimate-able.
I agree with chenille that metaphor of the dart is about chance - not physicality. Vi is trying to express how odds/probability are effected by the multiple types of infinities contained on a single number line. Here's a simplified example:
In this integer set, I'm counting by tens: -10,0,10
There are three items, and my chance of hitting my choice is 1/3. (33%)
My odds of hitting any of the three is 3/3. (100%)In this integer set, I'll count by ones:
-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10There are 21 items and my odds have changed.
Now my odds of hitting any one I choose is 1/21. (5%)
My odds of hitting any of the original 3 is 3/21. (14%)
My odds of hitting any of the new numbers is 18/21. (86%)So that's just from changing a short number line with easily counted numbers, and Vi is talking about how you fill the gaps with infinities.