@matematico314 The Wikipedia article (https://en.wikipedia.org/wiki/Dual_number) lists a lot of applications, but it misses the most common use case — perturbation theory.

In perturbation theory, you have a known effect, represented by a number 𝑎 (or several numbers, 𝑎₁, 𝑎₂, ... 𝑎ₙ, but that's similar), together with a perturbation that is written ε𝑏, where ε is thought as a very small number. So you effectively have instead of each number 𝑎 a number 𝑎 + ε𝑏, the perturbed version of 𝑎. You add two perturbed numbers component-wise, but when you multiply them, you consider ε a so small number that every term with ε² in it vanishes. The resulting multiplication rule is then

(𝑎 + ε𝑏)(𝑐 + ε𝑑) = 𝑎𝑐 + ε(𝑏𝑐 + 𝑎𝑑),

exactly the rule for dual numbers.

So the physicists have been working with dual numbers for many decades without noticing them!

(I do not know whether the YouTube video tells this.)

#DualNumbers #PerturbationTheory #Physics