danago said:
Is this effectively what a tensor is? I have never looked at tensors before, but i remember somebody saying a tensor is a generalization of matrices/vectors etc.
Yes, that is the best answer to what you're describing, although it's not quite right to think of real numbers as a starting point.
A scalar (i.e. a one-dimensional number in the normal sense) is called a "rank 0 tensor"; a vector in n dimensions, which can be represented by a 1xn matrix, is a "rank 1" tensor; then you have quantities of higher rank, such as the inertia tensor representing the distribution of mass in a body, that can be represented by nxn matrices and are "rank 2" or higher tensors.
The important thing about all these tensors is that they are not simply arrays of numbers. What really defines them is the way they behave under rotations of the underlying coordinate systems. Not every 1xn array of numbers can be interpreted as a vector, and similarly not every nxn matrix represents a rank 2 tensor.
As for extensions beyond real numbers, have you studied complex numbers? One way to understand them is as vectors in the "complex plane", reflecting the fact that any complex number is an inherently 2-component object. They can also be represented as ordered pairs of real numbers with special rules for how to add, subtract, multiply, etc. them.
Complex numbers can then be extended to quaternions, which can be represented as ordered pairs of complex numbers, and as such have four component parts. Next is octonions, which have eight components, but that's as far as you can go. Each higher level of number loses some of the properties that make numbers useful (quaternions don't obey the commutative law of multiplication, and octonions don't obey the associative law of multiplication over addition), and going any higher become impossible (as far as I know, at least).
