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.

Answer

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).

Given that answer, let’s discuss your points out of order:

(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.

Attribution
Source : Link , Question Author : danielsyn , Answer Author : Arturo Magidin

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