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In some way, $\mathbb{C}$ completes $\mathbb{R}$, why is there nothing that completes $\mathbb{C}$? Is it just more so that we don't want anything more than $\mathbb{C}$, or is there a property of $\mathbb{C}$ that makes it complete, in some sense? I know algebraically $\mathbb{C}$ is very nice, but is there nothing else we look for?

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    http://en.wikipedia.org/wiki/Quaternion, http://en.wikipedia.org/wiki/Octonion, http://en.wikipedia.org/wiki/Sedenion, http://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction – Jonas Meyer Jul 11 '14 at 03:52
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    what do you mean by "completes"? In the sense that usually means $\Bbb R$ is already "complete." If you're talking about algebraically, then $\Bbb C$ is the algebraic closure of $\Bbb R$. If you're talking about a largest normed division algebra that contains $\Bbb C$, then as @JonasMeyer says, the quaternions, $\Bbb H$, suffice. If you mean just algebraically, then the buck stops at $\Bbb C$ a per http://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra – Adam Hughes Jul 11 '14 at 03:55

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A structure $X$ has to be completed if some property or operation we'd like to have is only partially present in $X$. E.g., in ${\mathbb N}$ you can universally add, but only sometimes subtract. Therefore ${\mathbb N}$ is completed to ${\mathbb Z}$. In this larger structure subtraction is universally possible, and has the desired properties.

Similarly ${\mathbb R}$ is completed to ${\mathbb C}$, because we desire a working environment where the equation $x^2+1=0$ has a solution. Now it turns out (this is a miracle) that the smallest such enlargement, namely the set of all numbers of the form $x+iy$, with $x$, $y\in{\mathbb R}$ and $i^2=-1$, not only is a field, but contains the hoped for number of solutions to all polynomial equations whatsoever.

So if you wanted to complete ${\mathbb C}$ even further you would have to name a problem which somehow makes sense in ${\mathbb C}$ but cannot be universally solved in ${\mathbb C}$. Maybe you'd like to create a theory of "complex infinitesimals" and then would arrive at some (very large!) superstructure $\tilde{\mathbb C}$.

The simplest example of a completion of ${\mathbb C}$ consists in extending ${\mathbb C}$ to the so-called Riemann sphere by adding the single point $\infty$. This is a great idea in connection with Moebius transformations, or rational functions in general, and eliminates a lot of exception handling. But this particular completion of ${\mathbb C}$ is of no much help when we are dealing with the function $t\mapsto e^{it}$ or similar complex-valued functions occurring in mathematical physics.

Christian Blatter
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