It is a new year, so we might as well have a new discussion. Many of you are probably familiar with the complex numbers, where we took the good ol' real numbers then introduced a new number i with the property i^2=-1. Then Sir William Rowan Hamilton set out to create a number system that could be used to represent points in space, just as points in a plane could be represented by complex numbers. In 1843 the quaternions, and the idea of hypercomplex numbers, were born. Quaternions are a four dimensional number system in the form a+bi+cj+dk, where a, b, c, and d are real numbers and i, j, and k have the following properties:

i^2 = j^2 = k^2 = -1
ij = -ji = k
jk = -kj = i
ki = -ik = j

Basically what he did was take the complex numbers and added a new imaginary unit j. Then k can be defined as the product of i and j. As you can see, these units obey the usual laws of multiplication except for commutativity. Some of this might look vaguely familiar if you have learned about the cross product of vectors, and that is no coincidence. The unit vectors i, j, and k used in vector analysis were introduced to mimic the quaternions. In fact, a quaternion can even be thought of as a scalar (the real part) added to a vector (the imaginary part, bi+cj+dk).

There is also another number system that mimics the complex numbers called the split-complex numbers. These numbers are written in the form x+yj, where j^2=1. In this case, j should not be thought of as ±1, but as a quantity independent of the real numbers. This number system is notable for having "zero divisors", which means that two nonzero quantities can be multiplied to have a product of zero, as is the case of (1+j)(1-j).

But why stop there? What happens when you take the complex numbers and give it another unit j (as in the quaternions), but instead set j^2=1? Why, then you would have a new set of hypercomplex numbers, the split-quaternions! Being a 4-D number system, multiplication is not commutative. This number system has its own quirks that set it apart from the others, such as the lack of symmetry among the units i, j, and k (where k=ij). For instance, ki=j and ij=k, but jk=-i.

Taking this even further, we can take the quaternions and introduce yet another unit l (lowercase L). If we let l^2=-1, then we have arrived at the octonions! There are now one real unit and seven imaginary units: i, j, k, l, li, lj, and lk. As is the case in other 8-Dimensional number systems, multiplication not commutative and not associative; that is, (ab)c does not in general equal a(bc). If instead you wanted l^2=1, you would then have the split-octonians (my personal favorite!).

Yeah.