Abstract Algebra/Hypercomplex numbers

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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.


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:

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.

see also: Wikipedia's Article on Quaternion


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


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