When I write that $\frac{1}{x} - \frac{1}{x} = 0$, should I include the fact $x\neq0$?

Question

Welcome to Mathematics Stack Exchange. $1/x$ is not defined when $x=0$ — J. W. Tanner, Dec 22 '19 at 18:58
@Tucker: That is irrelevant. $1/x$ is already undefined when $x=0$, so there is no need to consider $1/x-1/x$. — TonyK, Dec 22 '19 at 19:07
"I haven't seen that remark when looking at similar expressions." ... Where have you been looking? It's possible that the context of the discussion makes clear that $x$ is non-zero (perhaps as the side-length of a non-degenerate triangle). Of course, it's best to make these exclusions explicit, but sometimes writers can be informal and trust that their readers will fill in logical gaps. On sites like Math.SE or even Wikipedia, it's important to keep in mind that we aren't writing formal, peer-reviewed journal articles; oversights happen. — Blue, Dec 22 '19 at 19:10
Unless there's some formal context that forces your hand one way or the other, you would generally write $x\neq 0$ if you want to draw attention to what happens if $x=0$ and omit it if you don't want to draw attention to that case (e.g. the bigger argument only needs minor adjustment for $x=0$). — Milo Brandt, Dec 22 '19 at 19:12
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score 3 · Answer 1 · answered Dec 22 '19 at 19:00

Yes you have to write $x\neq 0$. But why? Well division is really just multiplication. By definition, $x/y:=x\cdot \frac{1}{y}$ and multiplication is an operation defined on $\mathbb{R}$. To be precise, is is the map $\cdot: \mathbb{R}\times \mathbb{R}\to \mathbb{R}$, $(x,y)\mapsto x\cdot y$. Therefore, in order to even talk about multiplying two elements, you need both to be in $\mathbb{R}$. But $\frac{1}{x}\not\in \mathbb{R}$ if $x=0$ (in fact it is not even defined).

score 1 · Answer 2 · answered Dec 22 '19 at 19:09

An equation is what is called a statement form; it includes a placeholder, here it's $x$.

Now we call any number a solution if the statement which is produced by substituting the placeholder by that number is a statement which is true.

Take $2x=6$, e.g. By substituting $x$ by $5$ the statement $2\cdot5=6$ is achieved. But as that statement isn't true, $5$ is no solution of that statement form.

Now there may be substitutions which don't produce a statement at all, but rather something incomputable (something that is not a number). Incomputability arises in our case if you plug in $x=0$ as $1/0$ is not a number.

Hence $0$ is neither a solution nor not a solution of the given statement form. In case you substitute any other number a statement which is true is produced.

When I write that $\frac{1}{x} - \frac{1}{x} = 0$, should I include the fact $x\neq0$?

2 Answers2