In general, there are four possible variants for what we call intervals. The parenthesis 
and 
are related to the strict inequality 
, while these ones 
and 
are related to the weaker 
. So, when we want to denote intervals, we use them as follows
$$\{x \text{ such that } a<x<b\}=(a,b)
\{x \text{ such that } a\leq x<b\}=[a,b)
\{x \text{ such that } a<x \leq b\}=(a,b]
\{x \text{ such that } a \leq x \leq b\}=[a,b]$$
You might also see 
for 
, that is, the reversed 
are used just like parenthesis.
There is also what we call "rays" (which are also intervals), which involve a "one sided" inequality:
$$\{x \text{ such that } a<x\}=(a,\infty)
\{x \text{ such that } a\leq x\}=[a,\infty)
\{x \text{ such that } x \leq b\}=(-\infty,b]
\{x \text{ such that } x < b\}=(-\infty,b)$$
and what we usually denote by the real line
$$\{x \text{ such that }x\in \Bbb R \}=(-\infty,\infty)$$
Answer from Pedro on Stack ExchangeLike 10/2- imagine a 10 square foot box, saying 10 divided by 2 is like saying “how many 2 square foot boxes fit in this 10 square foot box?” So the answer is 5.
But if you take the same box and ask “how many boxes that are infinitely small, or zero feet squared, can fit in the same box the answer would be infinity not “undefined”. So 10/0=infinity.
I understand why 2/0 can’t be 0 not only because that doesn’t make and since but also because it could cause terrible contradictions like 1=2 and such.
Ah math is so cool. I love infinity so if anyone wants to talk about it drop a comment.
Edit: thanks everyone so much for the answers. Keep leaving comments though because I’m really enjoying seeing it explained in different ways. Also it doesn’t seem like anyone else has ever been confused by this judging by the comment but if anyone is I really liked this video https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:foundation-algebra/x2f8bb11595b61c86:division-zero/v/why-dividing-by-zero-is-undefined
In general, there are four possible variants for what we call intervals. The parenthesis and are related to the strict inequality , while these ones and are related to the weaker . So, when we want to denote intervals, we use them as follows
$$\{x \text{ such that } a<x<b\}=(a,b)\{x \text{ such that } a\leq x<b\}=[a,b)\{x \text{ such that } a<x \leq b\}=(a,b]\{x \text{ such that } a \leq x \leq b\}=[a,b]$$
You might also see for , that is, the reversed are used just like parenthesis.
There is also what we call "rays" (which are also intervals), which involve a "one sided" inequality:
$$\{x \text{ such that } a<x\}=(a,\infty)\{x \text{ such that } a\leq x\}=[a,\infty)\{x \text{ such that } x \leq b\}=(-\infty,b]\{x \text{ such that } x < b\}=(-\infty,b)$$
and what we usually denote by the real line
$$\{x \text{ such that }x\in \Bbb R \}=(-\infty,\infty)$$
The notation 
refers to the set of all real numbers 
such that 
. Another common notation for this set is 
; which is more common often depends on the language in which the author was educated.
In general, there are four possible variants for what we call intervals. The parenthesis and are related to the strict inequality , while these ones and are related to the weaker . So, when we want to denote intervals, we use them as follows
$$\{x \text{ such that } a<x<b\}=(a,b)\{x \text{ such that } a\leq x<b\}=[a,b)\{x \text{ such that } a<x \leq b\}=(a,b]\{x \text{ such that } a \leq x \leq b\}=[a,b]$$
You might also see for , that is, the reversed are used just like parenthesis.
There is also what we call "rays" (which are also intervals), which involve a "one sided" inequality:
$$\{x \text{ such that } a<x\}=(a,\infty)\{x \text{ such that } a\leq x\}=[a,\infty)\{x \text{ such that } x \leq b\}=(-\infty,b]\{x \text{ such that } x < b\}=(-\infty,b)$$
and what we usually denote by the real line
$$\{x \text{ such that }x\in \Bbb R \}=(-\infty,\infty)$$
Answer from Pedro on Stack ExchangeVideos
It can be called the Unit Interval
In mathematics, the unit interval is the closed interval [0,1], that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted I (capital letter I). In addition to its role in real analysis, the unit interval is used to study homotopy theory in the field of topology.
Since you seem to be primarily interested in rational numbers, a good candidate is proper fraction.
Mostly it is based on convention, when one wants to define the quantity $\binom{n}{0} = \frac{n!}{n! 0!}$ for example. An intuitive way to look at it is $n!$ counts the number of ways to arrange $n$ distinct objects in a line, and there is only one way to arrange nothing.
In a combinatorial sense, $n!$ refers to the number of ways of permuting $n$ objects. There is exactly one way to permute 0 objects, that is doing nothing, so $0!=1$.
There are plenty of resources that already answer this question. Also see:
Link
http://wiki.answers.com/Q/Why_is_zero_factorial_equal_to_one
http://en.wikipedia.org/wiki/Factorial#Definition