Invariant under transformation $i\mapsto -i$ implies real? When one has an expression in terms of $i$, one can send $i$ to $-i$ and, if the expression remains unchanged, one can conclude that the expression is, in fact, real. Analogous statements hold for expressions involving radicals. Why is this?
One admittedly trivial example is the expression
$$\frac{1}{x-i}+\frac{1}{x+i} .$$

If $x+iy = x-iy$ then $y=0$ (extra characters).
