Yes, which means "a half turn around the unit circle in the complex plane is -1". Try explaining that in words without saying "half" or something equivalent to it.

Opposite of 1 on the unit circle in the complex plane is -1. Pi just means "enough of a turn to get back to the straight line".

You mean a vector in the opposite direction. Clear as mud.

That's ambiguous. Which straight line? Forwards or backwards? It's a half-turn.

...

e^(τi) = 1

so still better.