The general form of a hypercomplex number is simply:

(1)

where n is a fixed integer, and a₀, a₁, a₂, ... a_n are arbitrary real numbers and i₀, i₁, i₂, ... i₃ are symbols such that

if and only if

Equation of the form (1) is said to be a nth-order complex number [see note at bottom of page].

Each multiplication of two bases i_a and i_b is necessarily a member of the set of hypercomplex numbers being defined. In other words, given two real integers (from 1 to n) a and b, and real numbers p₀ through p_n, we can define a multiplication table such that

Therefore, for an nth order hypercomplex number, n*n*(n+1) number of such constants must be defined to determine the form of the algebra. (Examples: real numbers (0th-order) require none, complex numbers (1st-order) require two, and quaternions (3rd-order) require 36 total.)

Properties that all hypercomplex numbers must obey:

1.	The product of a real component a, viewed as a hypercomplex number, with any other general hypercomplex number is established via: (a + 0 i1 + ... + 0 in) (b0 + b1 i1 + ... bn in) = a b0 + a b1 i1 + ... + a bn in and (b0 + b1 i1 + ... bn in) (a + 0 i1 + ... + 0 in) = a b0 + a b1 i1 + ... + a bn in
2.	If a and b are real components and u and v are hypercomplex numbers, then (a u) (b v) = (ab) (uv)
3.	Left and right distributive laws hold: u (v + w) = uv + uw (v + w) u = vu + wu

There are the many specific comparisions between the different properties that each number system may or may not have. See the Algebraic Properties page for more details.

[Note: the K&S book calls equations of the form (1) (n+1) dimensional, but I will refer to it as nth-order, to separate it from the possibility that several hypercomplex numbers may be needed to define a space. I.e., there may be a space which can be described by m nth-order hypercomplex numbers, which would be referred to as m-dimensional. An nth-order hypercomplex number can be represented by an (n+1) dimensional space.]