# If $i^2=-1$, then what about $(-i)^2=-1$?

• How to tell $i$ from $-i$?
#### Solutions Collecting From Web of "If $i^2=-1$, then what about $(-i)^2=-1$?"
There is some interesting mathematics bound up in your question. There is an automorphism of $\mathbb C$ as a field extension of $\mathbb R$ which is defined by sending $i$ to $-i$. Provided you are consistent (including very great care with signs) everything algebraic works nicely. This reflects the fact that in building $\mathbb C$ from $\mathbb R$ we make an arbitrary choice of a square root of $-1$ to call $i$.
For amusement there is an automorphism of $\mathbb Q(\sqrt 2)$ as an extension of $\mathbb Q$ which sends $\sqrt 2$ to $-\sqrt 2$ – for much the same reason. However, interesting things happen to the metric (distance between points) – so though the algebraic properties are retained, it is not true that “everything” remains the same.