[Math] Why isn’t $\langle i,1\rangle=0$ for a complex inner product

Question

This is probably trivial, but it's really bugging me.
According to the inner product on $\mathbb{C}$ (i.e. $\langle z,w\rangle=z\overline{w}$), $\langle i, 1\rangle\neq 0$. I'd like to say that $i$ and $1$ are orthogonal (specifically in the context of defining what it means for a vector in $\mathbb{C}$ to be tangent to the unit circle), but the inner product disagrees.

Answer 1

You have to write them in the vector form as

$$ 1 \equiv (1,0), \quad i \equiv (0,1).$$