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

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.

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

fun fact: unlike C, R[eps] is an ordered ring. it has two valid orderings, defined by either eps>0 or eps

Maybe they should call these "The Physicists' Numbers", as this is how they do maths anyway: *"Blah blah blah, and ε² is negligible so we may regard ε² = 0 ..."*

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.

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…

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.

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!)

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.

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).

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

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

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.

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.

Love the brown chalk, it's like a highlighter for the chalk board. Not good on its own but it makes other things pop.

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

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

So, what silliness can one get by mixing dual and complex numbers?

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!

You know that this guy is a hardcore mathematician when starts talking about his chalk.

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.

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

never encountered these before, what a cool and interesting concept. the derivative result is especially cool. thanks michael

I like white chalk, white as an angel.

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"

The brown chalk is perfect for "dividing lines" as you used it, easy to distinguish, yet not distracting.

That application at the end really blew my mind! In the first semester of my maths bachelor we learned how to compute Matrix exponentiations as well as matrix exponentials quickly, both in the diagonal and Jordan forms as well as by bringing the matrix into such a form. I vaguely remembered how it worked, but I never really understood why it really worked that way, I knew how to write it down with formulas, but there wasn't any deeper understanding. But now, with the translation of the problem into the dual numbers, it was like a dozen lightbulbs went off in my head at once! Really great application and connection between the two areas.

Such a great video Mike. I’ve noticed some people get weird when talking about non-conventional numbering systems like the dual numbers. Great work again on your part.

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

The diversion to briefly discuss brown chalk at 10m is surreal. And then the comments on the video are a random mixture of high-level math and opinions about brown chalk. I love this place.

This is such a great video Mike. Thank you.

Love the chalk, your writing is very easy to read and you explain things very well. Thank you

You're right about brown. I enjoy the advanced topics, rock climbing videos, and number theory ('cause I know nothing about number theory). I enjoy contest problems, slightly, but not as much.

I love the BROWN chalk. You won me over with that. You are fantastic. You're teaching is amazing.

Awesome video! Thank you!

The brown chalk bit cracked me up. Thanks for this

Dig the brown chalk for boxes and dividers. Good color for just separating meaningful pieces of the work :)

Nice! I really enjoyed this discussion.

This is a whole new world I was unfamiliar with. Fascinating. Thank you! Fred

Delightful content and presentation. Brown chalk is a plus for me because it reads softly though clearly.

This is great timing. I just started exploring split complex numbers.

Thank you. I had forgotten this.

the brown chalk is great, you should use it for boxing, crossing things out when they cancel and such. sometimes i find all the bright colors you use a little distracting, but this color is really nice imo

Thank you! I Really love your videos

Oh that's quite a powerful application! Nice!

Thank you. This is very inspiring.

Good job! Keep it going

Loved the video !

10:16 that's correct, very good for borders, not so for writing

Hey! Could you cover the dual numbers? Amazing content as always

Very interesting. I worked on a math minor, but never came across Dual numbers. Re the brown chalk, it came across as sort of a dark orange, which worked well for lines and boxes as you noted.

Very cool, thank you for the nice video!

Nice! Suggestion for follow-up video: Show that given any rule like "i^2 = a+bi", where a and b are real numbers, R(i) is isomorphic to exactly one of "i^2 = -1" (complex numbers), "i^2 = 0" (dual numbers) or "i^2 = 1" (I'm sure it has a name)

you teach so accessibly!

Very interesting, thanks for the introduction

Very interesting. I work quite a lot with stochastic calculus and see some parallells to that area as well. We could introduce another symbol, say q, which squared give the epsilon. Then the epsilon would represent the standard differentials dt and q represent the stochastic differential dW for a brownian motion W. No idea if anything new would come out of this, but I clearly see the analogy.

This is known in physics as Grassmann directions, in the special case of 1 number. They are used to build the bases of superfields.

Great video about a neat topic. Huge admiration for keeping the part at 7:13 in.

brown chalk looks good for the applications you described 👍

Yeah I basically agree with you on the brown chalk. Might also be good for shading or other small details on diagrams if you don’t want it to get too cluttered idk The dual numbers are cool btw I’ve never seen this set before

I like your channel. You're a great man!

Awesome! Any more?

you should talk about the split complex numbers too

very good introduction to the topic

I like the brown for boxes and divider lines; it reminds me of old computer games like Rogue

awesome prof... greetings from colombia i ever stay tuned with the channel... i love maths.... great board and pretty nice colors

So what happens when you raise e to a dual number?

Semantic suggestion: In complex numbers we have the "real" part and "imaginary" part. For dual numbers, how about the "real" part, and "secondary" part? We need a word that's different from "dual", I think.

thanks alot for this video

!! What an underated piece of math. Very very cool.

Great video. Thanks

Very cool video! Will you continue to post about Witt’s algebra?

I wonder if these have some sort of polar form... TLDR: - a is like the modulus and b/a is like the angle - addition in polar form is done by adding the "moduli" and taking the weighted average of the "angle" - mutliplication in polar from is done by multiplying the moduli and adding the angles, exactly like with complex numbers. Here is my derivation: (a+bε)² = a²+2abε (a+bε)(a-bε) = a² exp(x)= 1+x+x^²/2+... exp(bε) = 1+bε (a+bε)^n = a^n + n*a^(n-1)bε exp(a+bε) = exp(a)(1+bε) (derivative argument) ln(a+bε) = ln(a) + b/aε a+bε = a(1+b/a ε) = a*exp(bε/a) This suggests representing a+bε as (a, b/a) is good alternative to a polar form. a is like the modulus and b/a is like the angle We get: (a, A) + (c, C) = a*exp(Aε) + c*exp(Cε) = a+Aaε + c+Ccε = (a+c)exp((Aa+Cc)/(a+c)ε) = (a+c, (Aa+Cc)/(a+c)) (a, A)*(c,C) = (a+Aaε)(c+Ccε) = ac+(Aac+aCc)ε = (ac, A+C)

The representation also yields another (silly) application: you can "simulate" the integers under addition using dual numbers of the form 1+nε under multiplication. Granted, you don't need the matrix representation to see that (1+nε)(1+mε) = 1+(n+m)ε but it's how I noticed it

I like the brown for delineating spaces on the chalkboard, but not for writing text/calculations.

Hi, I am a retired UK systems Engineer and I never came across duel numbers before, so thanks Mr Penn. I suppose this is a topic in number theory. Just had a vague thought, has this topic got a link to convergence and limits of a power series ?

02-04-2023.Hi Michael, good to see again, and yes brown is O.K. for ONLY boxing, l use to give this Duel Numbers exercise to my student in Malta back in 1990.

Muy interesante!! Gracias por el aporte.

LOVE THE IDEA! THX i'm subsribing now

I like the Brown for the boxes

I would love a video on formalizing fast-growing hierarchies

super fascinating!

Thanks, this really clears things out. During my engineering studies I was always told we only use first order derivation epsilon because its square is practically zero. While using dual numbers you can show in a more elegant way how first order derivatives are found.

Interesting concept. My maths chops are pretty wimpy so I'm just left wondering if there is a way to produce rotations in the dual number plane, similar to the complex plane. Might make for some interesting visualizations. wrt the brown chalk: I'm reminded by something the channel Technology Connections pointed out, brown is just dark orange without context.

Dude, if I was taught this in Discrete Control course during 5th year of Electronic Engineering, everything would have been so much easier... GREAT video. In fact, I have an idea: in digital controller design, one could approach the dynamical system with a state-space methodology, and design the controlling equation with the so-called "deadbeat response". It is choosing the poles of the system so that the state-space matrix that governs the controlled system, to go to zero after "k" ticks / periods (if the system is capable of reading the sensors every 0.01 seg, then when (t = 0.01 seg, k = 1), (t = 0.02, k = 2), etc... ) The state-space matrix is then turned into what's called a NILPOTENT matrix, case in which the matrix "G" powered by some "n" is equal to zero, where "n" is the matrix size. Then: "A ^ 2 = 0". In this case, the matrix representation of the dual numbers takes into account that matrices are of size 2 (x 2). Therefore "epsilon ^ 2 = 0". So my idea is: shouldn't we divide a DUAL number into its REAL part, and its NILPOTENT part? I think that sounds accurate!

The "brown" chalk comes out as a pinkish red and is quite visible. However it is probably always best to have white on black as the contrast makes symbols most clear.

Question: are the dual numbers constructable using quotients of polynomials in the same way as the complex numbers are? Can I mod out by x^2 in order to form the dual numbers?

Huh, just as I started to learn non-standard analysis you come with a video on a related topic :D. Maybe some video on hyperreals soon?

Yes, more please! Finally advanced math presentations I can understand. You are making RU-vid GREAT.

No me pierdo ningun video de este pibe. Cada vez que lo veo me dan ganas de volver a la Facultad de Ciencias Exactas para terminar mi carrera de matematico. Pero luego recuerdo que habia que estudiar una pila y se me pasa.

Dude, totally down with the brown chalk for boxes

that question about brown chalk was really deep!

My favourite part IS-THE-SUM....😆 I love your videos!

I love the brown chalk, I had some just like it when I was a teenager, so I'll always remember sec(x) written in brown.

Brown chalk is a great addition for dividers and stuff

This can extend easily on every C1 functions on some closed interval [a,b]. One may use Weierstrass approximation for C1 functions (through the FTC). Then you can make sense of sequences of each polynomials evaluated at matrices and those matrices must converge uniformly hence pointwise, so that everything there still can make sense in the more general C1-only setting.

Thanks! This is a nice way to define differentiation purely algebraically, i.e. it works for any field (even finite ones). This is so much nicer than having to go through the nonsense of assuming AC and so-called "real" numbers.

I knew of that as the "self destructive operator" by Rannieri

Interesting! There are some connections with the non standard analysis and Liapunov function...

11:42 - 12:00, pretty cool attention to detail that you won't want to gloss over.