Euler’s formula for dual numbers and double numbers

The Endeavour 2025-01-28

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.)

We can find analogs of Euler’s formula

$\exp(i\theta) = \cos(\theta) + i \sin(\theta)$

for dual numbers and double numbers by using the power series for the exponential function

$\exp(z) = \sum_{k=0}^\infty \frac{z^k}{k!}$

to define exp(z) in these number systems.

For dual numbers, the analog of Euler’s theorem is

$\exp(\varepsilon x) = 1 + \varepsilon x$

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

$\exp(j x) = \cosh(x) + j \sinh(x)$

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.

[1] Double numbers have also been called motors, hyperbolic numbers, split-complex numbers, spacetime numbers, …

The post Euler’s formula for dual numbers and double numbers first appeared on John D. Cook.

Euler’s formula for dual numbers and double numbers

The Endeavour 2025-01-28

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