The strange cousin of the complex numbers -- the dual numbers.

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[Aegisworn]

My favorite application is for dual numbers is automatic differentiation. If you define some basic arithmetic operations for dual numbers on a computer, then run a function defined in terms of these modified definitions on the argument of interest + epsilon, you automatically get out both the value of the function at the argument as well as its derivative. This has pretty big implications for machine learning since you can immediately do your backpropagation since you computed the derivative of the loss function in parallel with its value.

[conoroneill]

This is actually used by programmers for automatic differentiation ('autodiff') - it allows for some fast computation techniques of the derivative of complicated functions.
Autodiff isn't always presented in this way, but it's an interpretation which I find very intuitive.

[alessandrorenna]

As an engineer, It reminds me of epsilon being an infinitesimal, so that epsilon^2 is just and infinitesimal of higher order, hence negligible in the scale of simple epsilon. Given the application with derivatives, I think it makes sense

[solsolsolsolsolsolsolsol]

fun fact: unlike C, R[eps] is an ordered ring. it has two valid orderings, defined by either eps>0 or eps<0. in both cases, |eps| is a positive number that is less than every positive real number

[goodplacetostop]

10:00 I agree. Brown is okay for boxes, not so much for writing
19:12 « In 1876, Clifford suffered a breakdown, probably brought on by overwork. He taught and administered by day, and wrote by night. A half-year holiday in Algeria and Spain allowed him to resume his duties for 18 months, after which he collapsed again. He went to the island of Madeira to recover, but died there of tuberculosis after a few months » Well, damn…

[mathematicalmachinery]

I like the brown chalk because the brighter colors pop out in contrast, making it easier to focus on the writing as opposed to the organization.

[mr.soundguy]

This is also a nice perspective on how to compare them:
1. C is ring-isomorphic to R[X]/(X^2+1). Meaning that the complex numbers as a field together have the same additive and multiple structure as all polynomials with real coefficients where we identify two such polynomials if their difference is an R[X]-multiple of X^2+1
2. R(epsilon) is ring-isomorphic to R[X]/(X^2).

[tiborgrun]

I think another application for this is in numerical analysis, where for some machine precision ε you have for example multiplication of two machine numbers (a + ε) (b + ε) + ε = ab + (a+b+1)ε + ε^2 but you treat ε^2 as 0, since ε is already very small.

[mMaximus]

It would be amazing if you could cover the geometric numbers, also created by Clifford (and Grassman), which generelize the complex, duals and hyperbolic numbers, even to any dimensions, with relative ease (plus you can do calculus with it!)

[alre]

I had never heard of dual numbers before, thank you for broading my horizon.

[DrBillPezzaglia]

Great video, but somewhere you should mention the term that epsilon is a "nilpotent". Vector spaces can be often written in terms of idempotent and nilpotent basis elements (idempotents are things that square to themselves, like "1"). Application: In physics, the 4-momentum vector of a photon would be a nilpotent (interpreted as photon has no rest mass). -From a physicist that spent a (lost) lifetime studying Clifford's algebra.

[mathunt]

I've come across these in the past in the context of rotations and translations in rigid body dynamics but they're a bit old fashioned now. People tend to use *geometric algebra* now as this provides a really nice framework for rotations and translations. Geometric algebra was pioneered by David Hestenes and picked up by Anthony Lasenby and Chris Doran and they have a nice introduction to the main ideas in geometric algebra and geometric calculus.

[AylaTheQueenIdk]

I think your opinion on the brown chalk is accurate, in that it works as a divider and for boxes, but probably not for writing

[mattlawyer]

This is the first of your videos that I've ever seen and I've loved it. I've never heard of dual numbers, but they seem fascinating, and I plan to make a study of them.

[harrisonkaiser]

If you do a talk on geometric algebra and connect it to epsilon in the dual numbers that would be one I’d definitely watch!

[Zonnymaka]

This is by far my fav YT math channel, period. Thank you Mr.Penn!

[Cerealbox]

This is my new favorite channel on YouTube. I love making some very simple assumptions and seeing where it leads. Its all very simple, logical, and easy-to-follow.

[acer]

Just found this channel. Excellent video; concise, interesting, and well thought out. Really loved the old school blackboard style. Reminds me of undergraduate lectures

[praharmitra]

Great introduction. I think you should also extend it a bit more to include multiple Grassmanian variables $\epsilon_1$, $\epsilon_2$, etc.

[harleyspeedthrust]

You could use this in error analysis - say when multiplying two measurements with some error in each of them
We used to use these in physics sometimes to write down the correct differential equations describing a system - never knew they had a formalism called "dual numbers"