Can you hit an exact number on a bounded number line My first question here. I am not sure if it belongs here or not.
If I were to place a number line, e.g. [0,4], and attempt to shoot at it, while aiming for a specific number. Would I be able to actually hit that number?
The question may sound absurd, but the point I am trying to make or understand is as following:
Say, I am aiming for a rational number, e.g. 1. I aim and throw multiple times. How would I with absolute certainty be able to determine, that I have actually managed to hit 1, and not simply some number in e.g. [$1-\epsilon, 1+\epsilon$]?
Now, let's say I attempt another number. This time an irrational number, e.g. $\pi$. Is it even possible to hit it? Aren't irrational numbers by definition numbers with never ending digits. Meaning, if I were to hit a number, in this case $\pi$, wouldn't it in fact be an approximation to $\pi$, as I can't actually "reach" that last digit. Meaning, I will always be able to find a smaller bound than [$\pi - \epsilon,\pi + \epsilon$], but never actually knowing if I have reached $\pi$?
 A: This is an interesting question that has no easy answer.
The probability that you hit any particular number - rational or not, is $0$. The probability that you hit some rational number is also $0$, so the probability that you hit an irrational number is $1$. This has nothing to do with the difficulty of expressing an irrational number as a decimal fraction.
Proving these things rigorously depends on the careful definition of  probability. That definition begins with the assertion that the probability of a finite interval is its length, divided by the length of the interval you are choosing from (in your case, $4$). 
It's even harder to think about probability when you are choosing a point from the whole number line rather than from a bounded interval.
Your question asks about "aiming", not directly about probability. Then the answer really depends on how well you can aim. If perfectly, then you can hit any number you aim for with probability $1$. If you can get withing $\epsilon$ repeatedly you can calculate the number of tries needed to get as close as you like. You won't be able to hit your target for sure, whether it's rational or not.
A: As I understand the question the asnwear is no. The probability to select a number from an interval is always 0. Maybe you want to be more specific in the part "aim for an specific number"
