# Complex numbers as vector spaces

I know that complex numbers can form a vector space over the field of real numbers as they obey the core axioms that constitute a vector space, however in vector spaces, the only operations that are defined are vector addition and scalar multiplication, which means vector multiplication isn’t defined, if that’s the case, then how come the multiplication of two complex numbers ( two vectors) can yield another complex number (i.e vector) ?

Am I missing something obvious ?

Note that I am well aware of the dot product and the cross product, so I’m talking about the normal vector multiplication that is not defined in a vector space

• The fact that $\mathbb C$ is a vector space over $\mathbb R$ doesn't mean it can't have additional structure. – lulu Mar 24 '18 at 12:35
• What you're noticing is that $\mathbb{C}$ is not merely a vector space over $\mathbb{R}$, it is an algebra over $\mathbb{R}$. See en.wikipedia.org/wiki/Algebra_over_a_field – Joshua Ruiter Mar 24 '18 at 12:39
• A bit weaker than what Joshua said I believe: but $\mathbb{C}:\mathbb{R}$ is also a field extension. Actually field extensions become vector spaces over their smaller fields upon losing some of their structure (look into the "forgetful functor") – Andrew Tawfeek Mar 24 '18 at 12:43
• @Joshua Ruiter Thank youuuu, that makes a lot of sense – Cbb7 Mar 24 '18 at 12:43
• @Cbb7 Please remember that you can choose an answer among the given if the OP is solved, more details here meta.stackexchange.com/questions/5234/… – user Mar 27 '18 at 14:08

Complex numbers are a vector space over $\mathbb{R}$ and thus can be handled as vectors and used to solve geometric problems but they have a different "more extended" structure with multiplication, conjugate, and so on.

• A “more extended” structure ? – Cbb7 Mar 24 '18 at 12:39
• See my comment above. – Joshua Ruiter Mar 24 '18 at 12:39
• Well then if I can extend the structure of the complex numbers space to include vector multiplication, how come I can’t do it in the Euclidean space for example ? – Cbb7 Mar 24 '18 at 12:41
• @Cbb7 The structure of complex numbers includes the structure of vector space, that means that we can do with complex number what we do with vector spaces but not the converse (that we can do with vectors what you do with complex numbers). – user Mar 24 '18 at 12:44

You're correct that not all vector spaces have vector multiplication. I don't see why you'd have a problem with a vector space having vector multiplication though. Nothing says that cannot happen.

• Ok, well the question then would be, why do some vector spaces have an additional/more extended structure that define vector multiplication whereas other vector spaces don’t – Cbb7 Mar 24 '18 at 12:50
• Your question is comparable to: "Birds are, by definition, flying creatures, and flying creatures are not necessarily blue. Despite this, there are blue birds. Why?" The answer is, of course, there's nothing saying we can't have blue birds, likewise there's nothing saying we can't have a vector space with a vector multiplication. If your bird is blue, that doesn't mean it's no longer a flying creature, and if your vector space has vector multiplication, that doesn't mean it's no longer a vector space. – Kaynex Mar 24 '18 at 14:14
• Well, I do think that the wording of my question was very poor, my issue was ( it’s solved now) concerned with the why do we need to define multiplication in some vector spaces whereas we don’t do that in others, as it turns out, defining vector multiplication, say in complex number, can help a lot in solving a lot of geometrical problems and so defining vector multiplication actually become necessary. – Cbb7 Mar 24 '18 at 15:06
• Secondly, I had an issue with the deceptive arbitrariness of the way vector multiplication is defined in a certain vector space but I think it’s defined in a particular to serve a certain purpose wihtout of course violating the axioms of the vector space. Anyhow this is what I gathered from all the answers, and I very much appreciate your help. Pls feel free to correct anything wrong that I said ^^ – Cbb7 Mar 24 '18 at 15:10