Abstract algebra/Hypercomplex numbers

Hypercomplex numbers are numbers that use the square root of -1 to create more than 1 extra dimension.

The most basic Hypercomplex number is the one used most often in vector mathematics, the Quaternion, which consists of 4 dimensions. Higher dimensions are diagrammed by adding more roots to negative 1 in a predefined relationship.

Quaternions

A Quaternion consists of four dimensions, one real and the other 3 imaginary. The imaginary dimensions are represented as i, j and k. Each imaginary dimension is a square root of -1 and thus it is not on the normal number line. In practice, the i, j and k are all orthogonal to each other and to the real numbers. As such, they only intersect at the origin (0,0i, 0j, 0k).

The basic form of a quaternion is:

• $q=a+bi+cj+dk$

where a, b, c and d are real number coefficients.

For a quaternion the relationship between i, j and k is defined in this simple rule:

• $i^2=j^2=k^2=i\times j\times k=-1$

From this follows:

• $i\times j=k$, $j\times i=-k$
• $j\times k=i$, $k\times j=-i$
• $k\times i=j$, $i\times k=-j$

As you may have noticed, multiplication is not commutative in hyperdimensional mathematics.

They can also be represented as a 1 by 4 matrix in the form

real	i	j	k
1	1	1	1

...

...

The quaternion is a 4 dimensional number, but it can be used to diagram three dimensional vectors and can be used to turn them without the use of calculus.

see also: Wikipedia's Article on Quaternion

Octonion

8-dimensional. See: Wikipedia's Article on Octonion

Sedenions

16-dimensional. See: Wikipedia's Article on Sedenion