Alternative definition of complex number, showing it is equivalent to the tradidional one. The author of a book makes an alternative definition of the complex numbers, later he shows that this definition is equivalent to the ordinary definition where we define $i^2=-1$.

Here is his definition:



Here he shows that they are equivalent:



Now comes my questions:


Has he really shown that they are equivalent. It seems to me that maybe he has shown that starting with the ordered pair definition he gets the tradisional definition of $i^2=-1$. But isn't that only an implication and not an equivalence. Should he have also shown it the other way around, starting with the original definiton, and showed that it implies the ordered pair definition?
Why is it enough to show what he did. If I were asked to show it I would have thought that I must also show that both definitions are equivalent when you add or multiply. Why hasn't he showed that? Or has he showed that implicitly?
They are the same thing, just written in an other way! Complex numbers in the plane can be represented as an ordered pair, and there they are homeomorphic to $R^2$. If in $R^2$ we define the addition and multiplication of two vectors, as in the definition given, than $R^2$ and complex numbers will be algebraically the same, so they would be equivalent.

P.S. the other way of the given proof is trivial: $i=0+1i=(0,1)$.
They are the same thing, just written in an other way! Complex numbers in the plane can be represented as an ordered pair, and there they are homeomorphic to $R^2$. If in $R^2$ we define the addition and multiplication of two vectors, as in the definition given, than $R^2$ and complex numbers will be algebraically the same, so they would be equivalent.

P.S. the other way of the given proof is trivial: $i=0+1i=(0,1)$.