# 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[edit]

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 (0,0*i*, 0*j*, 0*k*).

The basic form of a quaternion is:

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:

From this follows:

- ,
- ,
- ,

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.

## Octonion[edit]

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

## Sedenions[edit]

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