# Division by zero

I came up some definitions I have sort of difficulty to distinguish. In parentheses are my questions.

1. $\dfrac {x}{0}$ is Impossible ( If it’s impossible it can’t have neither infinite solutions or even one. Nevertheless, both $1.$ and $2.$ are divided by zero, but only $2.$ has infinite solutions so as $1.$ has none solution, how and why ?)

2. $\dfrac {0}{0}$ is Undefined and has infinite solutions. (How come one be Undefined and yet has infinite solutions ?)

3. $\dfrac {0}{x}$ and $x \ne 0$, it’s okay for me, no problem, but if someone else wants to add something about it, feel free to do it.

The first question you need to ask is: What does “$a/b$” mean?

The answer is: “$a/b$ is the unique solution to the equation $bz = a$.” (I’m using $z$ as the unknown, since you are using $x$ for other things).

(3) is perfectly fine: $0/x$, with $x\neq 0$, is the solution to $xz = 0$; the unique solution is $z=0$, so $0/x = z$. The reason it’s unique is because $x\neq 0$, so the only way for the product to be $0$ is if $z$ is $0$.
In (1), by “impossible” we mean that the equation that defines it has no solutions: for something to be equal to $x/0$, with $x\neq 0$, we would need $0z = x$. But $0z=0$ for any $z$, so there are no solutions to the equation. Since there are no solutions to the equation, there is no such thing as “$x/0$“. So $x/0$ does not represent any number.
In (2), the situation is a bit trickier; in terms of the defining equation, the problem here is that the equation $0z=0$ has any value of $z$ as a solution (that’s what the “infinite solutions” means). Since the expression $a/b$ means “the unique solution to $bx= a$, then when $a=b=0$, you don’t have a unique answer, so there is no “unique solution”.
Generally speaking, we simply do not define “division by $0$“. The issue is that, once you get to calculus, you are going to find situations where you have two variable quantities, $a$ and $b$, and you are considering $a/b$; and as $a$ and $b$ changes, you want to know what happens to $a/b$. In those situations, if $a$ is approaching $x$ and $b$ is approaching $y\neq 0$, then $a/b$ will approach $x/y$, no problem. If $a$ approaches $x\neq 0$, and $b$ approaches $0$, then $a/b$ does not approach anything (the “limits does not exist”). But if both $a$ and $b$ approach $0$, then you don’t know what happens to $a/b$; it can exist, not exist, or approach pretty much any number. We say this kind of limit is “indeterminate”. So there is a reason for separating out cases (1) and (2): very soon you will see an important qualitative difference between the first kind of “does not exist” and the second kind.