The complex numbers are formed by adding an element i to the real numbers such that i² = − 1. We can create other number systems by adding other elements to the reals.
One example is dual numbers. Here we add a number ε ≠ 0 with the property ε² = 0. Dual numbers have been used in numerous applications, most recently in automatic differentiation.
Another example is double numbers [1]. Here we add a number j ≠ ±1 such that j² = 1. (Apologies to electrical engineers and Python programmers. For this post, j is not the imaginary unit from complex numbers.)
(If adding special numbers to the reals makes you uneasy, see the next post for an alternative approach to defining these numbers.)
We can find analogs of Euler’s formula
for dual numbers and double numbers by using the power series for the exponential function
to define exp(z) in these number systems.
For dual numbers, the analog of Euler’s theorem is
because all the terms in the power series after the first two involve powers of ε that evaluate to 0. Although this equation only holds for dual numbers, not real numbers, it is approximately true of ε is a small real number. This is the motivation for using ε as the symbol for the special number added to the reals: Dual numbers can formalize calculations over the reals that are not formally correct.
For double numbers, the analog of Euler’s theorem is
and the proof is entirely analogous to the proof of Euler’s theorem for complex numbers: Write out the power series, then separate the terms involving even exponents from the terms involving odd exponents.
Related posts
[1] Double numbers have also been called motors, hyperbolic numbers, split-complex numbers, spacetime numbers, …