A hemireal number
z can be written
z = r + mμ + nν
n are real, and the special numbers
μ*μ = ν*ν = 0, μ*ν = ν*μ = 1.
Addition, subtraction, and any operation involving real numbers are
defined "the obvious way," and the conjugate of
z is just
Multiplication of general hemireals is commutative but not
associative. Hemireals with
ν=0 are the same as dual numbers.
The motivation for inventing/rediscovering (?) the hemireals was to solve, using finite numbers, what would otherwise be singular equations.