Given that imaginary numbers exist perpendicular to a number line, intersecting at 0, does i = 0/0?

Question

If imaginary numbers bisect real numbers at zero, am I correct in thinking $0i = 0$?

And if so, does that mean $i = 0/0$?

EDIT:

I’ve just realised my careless error in my thought process.

I immediately just “removed” the $0$ from both sides without thinking about the fact that I was actually simplifying $\frac{0i}{0}=\frac{0}{0}$ to $i=\frac{0}{0}$ despite $\frac{0}{0}\neq1$, meaning $\frac{0}{0}i \neq i$

Answer 1

An interesting way of looking at this question is to ask the following (to expand on Lulu's answer: If we have $0\cdot 1 = 0$, why is $1\neq \frac{0}{0}$? The answer is, of course, that $\mathbb{R}$ is a field, and the zero element has no inverse. Therefore, from a pure algebraic perspective, $\frac00$ makes no sense.

Now we can ask ourselves the same question in $\mathbb{C}$. Is $\mathbb{C}$ a field? What does that imply for your question?

Answer 2

Yes, $0i=0$, but the prohibition on dividing by zero remains valid in the complex plane, and one does not conclude $i=0/0$.

Answer 3

It's also true that $0\cdot 1 = 0$. Does that mean $1=\frac00$?

It's also true that $0\cdot 2 = 0$. Does that mean $2=\frac00$?