1st-order hypercomplex numbers are made up of two real numbers coupled in the following ways. They can be generally catagorized as complex, dual and double numbers.

We may represent the ways that two real are coupled by writing them in the form

a + b i

where a and b are reals and i is as yet undefined, but a member of this set of numbers.

We will make three demands of the algebra (where z1 and z2 are elements of this new number system):

1) a (b + ci) = ab + aci

(b + ci) a = ab + aci

2) (a z1) (b z2) = (ab) (z1 z2)
3) z1 (z2 + z3) = z1 z2 + z1 z3

(z1 + z2) z3 = z1 z3 + z2 z3

(i.e. multiplication is distributive over addition)

This leads to the general formula for multiplication:

(a + bi) (c + di) = ac + adi + bci + bd i2

and here we have not assumed what i*i is yet. But we can at least make the assumption that it should be in general another number of the form p + qi.

Rewriting the above formula using i*i = p + qi we get:

(a + bi) (c + di) = (ac + bdp) + (ad + bc + bdq) i

Choosing what value i*i is defined what kind of number we have defined. This might seem like this means that there should be an infinite number of different kinds of 2nd-order hypercomplex numbers, but it turns out that they can all be reduced to three species.

Value for i*iNumber systemDivision system?
i*i = -1Complex numbersYES
i*i = 0Dual numbersno
i*i = 1Double numbersno

We have added the category "division system" at the end to signify one thing that is special about the complex numbers: it is a system which allows for a definition of division. In general, division is not defined for the dual and double numbers. Most work with physical theories are done over algebras which are division systems. Complex numbers are therefore unique in being the only division system which can be made up from two real numbers being coupled in the form a + bi.