Is there a domain "larger" than (i.e., a supserset of) the complex number domain?

Solution 1:

First of all, kudos for introducing these ideas to your 10 year-old! That's an excellent way to get them interested in mathematics at an early age.

As to your question: The short answer is no. Any algebraic operation that you can do of the sort you're describing will yield a complex number. This is due to the fact that they are algebraically closed. What this means is the following.

One way that you can show that you can find a larger domain than the real numbers is by looking at the polynomial $$ f(x) = x^2 + 1 $$ You can easily see that there are no solutions to the equation $f(x) = 0$ in the real numbers (just as the equation $x + 1 = 0$ has no solutions in the positive integers). So we must escape to a larger domain, the complex numbers, in order to find solutions to this equation.

A domain (to use your term) being algebraically closed means that every polynomial with coefficients in that domain has solutions in that domain. The complex numbers are algebraically closed, so no matter what polynomial-type expression that you write down, it will have as solution a complex number.

Now, this isn't to say that there aren't larger domains than $\mathbb{C}$! One example is the Quaternions. Where the complex numbers can be visualized as a plane (i.e. $a + bi \leftrightarrow (a, b)$), the quaternions can be visualized as a four-dimensional space. These are given by things that look like $$ a + bi + cj + dk $$ where $i, j, k$ all satisfy $i^2 = j^2 = k^2 = -1$, and moreover $ij = -ji = k$. The interesting fact about the quaternions is that they are non-commutative. That is, the order in which we multiply matters!

There are also Octonions, which are even weirder, and are an 8-dimensional analogue.

Anyhow, the answer is in the end that it sort of depends. In most senses, the complex numbers are as far as you can go in a relatively natural way. But we can still look at bigger domains if we want, but we have to find other ways to build them.

Solution 2:

There are reasons to escape even from the complex number system, but perhaps not as nice and fruitful reasons as in the development from $\mathbb N$ to $\mathbb C$.

I recon any equation or system of equations in $\mathbb C$ without solutions, will have some solution for some extension of $\mathbb C$.

$z=z+1$ will have a solution in $\mathbb C^*=\mathbb C \cup \{\infty\}$, with some new arithmetic laws.

While

$ \left\{ \begin{array}{l} x^2=-1 \\ y^2=-1 \\ z^2=-1 \\ xy=z \end{array} \right. $

will have solutions in the Quaternions.

But both these cases of extensions demands significant changes of the rules. In the first case, e.g. the cancelling law has to be modified and in the second case commutativity is lost.

I'll guess almost no system of relations of the type where all parenthesis are stated, which has no solutions in $\mathbb C$ (that is, leads to contradiction in $\mathbb C$), can generate an "escape" in an associative algebraic extension of $\mathbb C$.

More to read on Wikipedia: Hypercomplex numbers

Solution 3:

I can think of hyperreal numbers and surreal numbers.

The idea is to add infinitely large and infinitely small numbers to the reals. If you have seen the debate whether $0.999...$ and $1$ are the same, you can understand it is a natural concept to introduce. :-)

Hyperreal numbers were used in the elaboration of calculus but were abandoned in favor of the concept of limits. Surreal numbers are a "larger" set that has applications in game theory. A superset of complex numbers would be the surcomplex numbers.

Solution 4:

There are lots of "bigger" domains. I want to point something out here:

Once you get to the level of the complex numbers, it becomes unclear what a number is. You can't put the complex numbers into any order that preserves the complex algebraic properties. And you can't really use them to count things either. In a funny way, the complex numbers are representations of rotations and scalings. That is definitely a funny kind of number.

So, with that in mind, you might want to start talking about linear algebra. Vectors and matrices are generalized numbers that represent linear transformations, like rotations, scaling, symmetric "flipping".