I guess I am missing something obvious here. I am reading about vector bundles. (What Karoubi calls 'Quasi Vector-Bundles')
An example is the sphere, where for every point $X \in S^n$ we choose $E_X$ (the fiber) to be the vector space orthogonal to $X$, and let $E$ (the total space) be the disjoint union of the $E_X$'s, which is naturally a subspace of $S^n \times R^{n+1}$
Now later it states we can have an (iso)morphism to the space E'= S^1 \times \mathbb{R} given by $g(x,z) = (x,iz/x)$
Now I must be missing something here. If $x \in S^1$ and $z \in \mathbb{C} = \mathbb{R}^2$ (identifying $\mathbb{R}^2$ with the complex numbers), then doesn't $g(i,a+i)=(i,a+i)$, and $a+i \notin \mathbb{R}$.
What am I missing? $a+i$ is in the vector space orthogonal to $i$ (which is just the tanget line at the point $(0,1)$ of the circle)
(Feel free to re-tag, wasn't sure what it should be under)