Talk:Complex number/Draft

Definition

I reworked the text a bit. So this is why.

• I think ${\sqrt {-1}}$ is an objectionable notation...
• The definition hardly matches my understanding... The imaginary unit can be really understood only within the field of complex numbers (defined independently). Otherwise, what is "i"? A square root of (-1)? Then which one? (there are usually two square roots; BTW, have you ever seen an independent definition of a square root of a negative number?). So let's define it by "i^2=1". Then, does it exist? Does it deserve to be called a number? (operations are possible?) The same question arise if we define "i" as a solution of "x^2+1=0". In practice we can use any of these well known properties, but how can we understand it as a definition?

At best, we can say "i" is "just a formal symbol" with no meaning. We define some operations on formal sums "a+bi". Basically, that's OK. The point is that it explains nothing and it can be done in a more elegant way, where we really define all is needed in terms of elementary well-known objects:

Complex numbers are just ordered pairs of reals -as simple as this - with appropriate addition and multiplication. BTW, these operations are enlisted in the article with the "formal" use of "i". Then i=(0,1). And for computational convenience we discover that i^2=-1, and use it.

I think your revision is a good one. I had considered using the term "formal expression" for $a+bi$, but decided not to. But, in truth, I didn't spend a great deal of time on this. It just seemed an obvious omission, giving that there was already an article on real numbers! A possible revision/addition I had considered was adding a section on how the definition can be formalized by saying $\mathbb {C}$ is the splitting field of $x^{2}+1$ over $\mathbb {R}$. Without context, though, that seems like a bit of overkill. Of course, it's formally the same as the definition of algebraic number fields such as $\mathbb {Q} [{\sqrt {-1}}]$ or $\mathbb {Q} [{\sqrt {2}}]$. But I suppose that's a topic for another article. Greg Woodhouse 06:21, 2 April 2007 (CDT)

The bottom line is that I do not object use of "i" in the informal intro, just to give an outline of the idea, there must be, however, a definition that really explains where it logically comes from. --AlekStos 03:01, 2 April 2007 (CDT)