Duplex Numbers

Displayed in depth-first order (following minus before plus).

Consider the duplex number a ± b ± c

• the mean of the resulting distribution is a.
• the variance of the resulting distribution is b² + c².

As such, any symmetrical distribution of numbers can be represented as the underlying set of a duplex number.

Duplex numbers can be used to find the eigenvalues for a 2x2 matrix. Consider the matrix:

    | x  z |
    | v  y |
            

    a±b = (x,y) and c±d = (v,z)

    a±b = (x+y)/2 ± |(x+y)/2-x|
    c±d = (v+z)/2 ± |(v+z)/2-v|
            

       a±c

       a ± √(b² + c² - d²)

Duplex numbers may be used to represent '≠' inequalities. For example, the inequality

1 = 0

can be represented as:

0.5 ± 0.5

Duplex numbers are useful for plotting multiple parallel lines on a graph. For example, the equation: x = 5±2 results in the following graph:

Duplex numbers can be used to build an intuition about quantum calculations being performed on a quantum computer. You can think of a duplex numbers as being an additional numerical type available on the quantum computer, which can efficiently perform modulus and exponentiation operations, which would otherwise become infeasible to do as the number of simultaneously represented values increases.