Is there any way to verify that $1/i = -i$ without using relation $i^2 = -1$?

Question

Is there any way to show it without using calculations involving expressions $i^2 = -1$ or $\pm i = {\sqrt {-1}}$?

Answer 1

We can.

We know that $i = (0,1)$ and $1/i$ is the multiplicative inverse of $i$. We are trying to find, say $1/i= (a,b)$, such that $(0,1)\times(a,b) = (1,0)$, i.e. $$(-b, a) = (1,0)$$ and got $b = -1, a = 0$. The other side of invers can be check similarly. Thus $1/i = (a, b) = (0,-1) = -(0,1) = -i$.

Answer 2

You need some definition of $i$ to prove anything about it. Without the definition, it is an undefined character. The usual definition is that $i^2=-1$. If you do not want to use that, you need to provide an alternative. If your definition is consistent with usual practice, $i^2=-1$ becomes a theorem that is proved from your definition. Your question would then make sense, asking if we can get from your definition to $\frac 1{i}=-1$ without passing through $i^2=-1$. What is your definition of $i$?