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.
I'm not sure about that last bit. The procedure you described is known as "forward mode automatic differentiation", while for backpropagation you'd need "reverse mode automatic differentiation". Reverse mode can also be implemented by providing a special type that keeps track of the value of a variable and the derivative of the loss function with respect to that variable, but this second number can only be computed in an additional backward pass.
@@Michael-vs1mw Multi-dimensional dual numbers (i.e, R[epsilon_1,epsilon_2,..., epsilon_k], with epsilon_i*epsilon_j=0) certainly work to compute a whole gradient, but computing the gradient this way takes about k times longer than computing the original function, and also k times more space. With reverse-mode differentiation, you can take a gradient with respect to a million variables (say, weights in a modest-size neural network) in only about twice the running time of the forward function and with space proportional to the number of computations (you need to remember intermediate values of computations). Computing Hessians is impractical for large numbers of variables because the matrices are huge, but one can multiply the Hessian times a vector in about four times the original running time, by computing the gradient of (the gradient of the function dot-times the vector). For some algorithms (like the conjugate gradient method) that's all you would need. Disclaimer: I haven't thought about this in a few years, so I might have said something incorrect.
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.
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
Yes it even makes sense rigorously in a setting called smooth infinitesimal analysis. Here your number line R is filled with infinitesimals, where some have the property e^2 = 0. Here every function is infinitely differentiable and the derivative f‘ of f: R -> R is uniquely determined by f(x+e) = f(x) + ef‘(x), where e is a nilsquare infinitesimal (e^2 = 0). This is basically the Taylor expansion where the higher order terms are neglected as you said.
@@zswu31416 actually you cannot order C in the same way. to be an ordered ring does not just mean that you have a ring with an understood order- it implies two extra axioms about how the ordering interacts with the two operations.
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…
@@FranFerioli he is the guy who says "wow, this person revolutioned the cience, but is an asshole, he is worthless and x person is better because was a nice person"
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 ..."*
Real numbers are their own conjugates -- Duality. Complex roots come in pairs -- automorphisms (duality). Duality (thesis, anti-thesis) synthesizes reality (non duality) -- the Hegelian dialectic. "Philosophy is dead" -- Stephen Hawking.
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!)
Thatd be dope, do you know of any videos like this guy that have about it, I am not very advanced math yet but this video was easy to follow (except for summation part, I am not familiar with it lol)
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.
I think if I had ordered brown chalk on a whim, I would probably use it for dividers and boxes, but I don't think I would use it much for actual writing
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).
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.
I think geometric algebra was first discovered but the Clifford guy mentioned here. Unfortunately he died young and his discoveries were forgotten until being revived in modern fashion by the people you mention. As others mentioned in the comments, geometric algebra pretty much follows from these concepts.
I remember reading about dual quaternions as a good way to do vertex skinning on the GPU, being a compromise between full rot/trans matrices and normal (real) quaternions. It would be good because dual quaternions could encode rotation *and* translation, allowing for smooth blending between the transforms in vertex skinning.
LInkedIn has two Geometric Algebra groups. One is called "Geometric Algebra", and the other is "Pre-Univerisity Geometric Algebra". Your participation will be welcome in both. I, myself, have several videos on GA that are meant primarily for self-learners like me.
Yes definitely!:) Geometric Algebra is basically build on the generalization of complex, dual and split-complex numbers (e^2 = -1, 0, 1 respectively) and adds higher grade objects as well as the geometric product to connect everything.
Most vector algebras struggle with multiplying vectors. GA lets you multiply vectors and it gives... a complex number‽ And people previously thought complex numbers _were_ vectors!
@@angeldude101 complex numbers are vectors. A k-vector is a vector, matrices are vector, polynomials are vector. Anything could be a vector is you choose to think of it as a vector and obviously has vector properties.
@@rajinfootonchuriquen Yay for vector meaning both a one dimensional geometric object as well as an abstract mathematical object. The former is the latter, but not all instances of the latter are the former. Now I'm curious what you get of you try forming a Clifford Algebra around an abstract vector space that isn't a simple quadratic form. What would it mean to take the wedge product of two functions?
@@angeldude101 I don't now. I don't advocate for using Clifford algebra. I don't know much about it but for instance, i know that you can get a nice maxwell equation, but it seems odd that the magnetic field is a bivector field. I don't know how to picture a bivector field. If you try to take a wedge product between that type of vectors, I think it should span out of your vector space, so it's undefine.
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.
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.
This is my new favorite channel on TH-cam. I love making some very simple assumptions and seeing where it leads. Its all very simple, logical, and easy-to-follow.
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.
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)
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.
Just found this channel. Excellent video; concise, interesting, and well thought out. Really loved the old school blackboard style. Reminds me of undergraduate lectures
It would be interesting to see how integration over the dual numbers is done, and whether it generates something akin to differentiation, as what happens with grassmann numbers.
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.
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!
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)
Correct me if I'm wrong but: isn't exp(2ε) ≠ exp(ε)² incorrect? exp(2ε) =(def)= 1 + 2ε exp(ε)² =(def)= (1 + ε)² = (1 + ε)*(1 + ε) = 1*1 + 1*ε + ε*1 + ε*ε = 1 + 2ε using the binomial formula you can generalise this to exp(nε) = exp(ε)^n for all natural n
A "dual angle" is really just a distance along a straight line. In the same way, normal angles and hyperbolic angles are basically just distances along a unit circle or unit hyperbola respectively.
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 think this is related to the exponential of dual numbers. Much like how exponentiating an imaginary number makes you walk along the unit circle, exponentiating ε makes you walk along a straight line (which is why I sometimes call them and the complex numbers "flat" numbers and spherical numbers respectively due to their relation to the corresponding geometry).
R(ε) is just the ring R[x]/(x²) Also, I believe all thr 2D R-algebras are (isomorphic to) either C, R², or R(ε). They can all be written in a uniform way: R[x]/(x²+1), R[x]/(x²-x), R[x]/(x²), respectively. Notice how the quotient polynomials have different signs for the discriminants (negative, positive, zero). I believe that's the invariant that tells you which of these your 2D R-algebra is isomorphic to.
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?
I think so. You can observe that the polynomial x² is not irreducible, therefore the quotient will not be a field, and since x² is not prime it will not be an integral domain. It is a nice way to construct these numbers.
More generally, ℝ has (up to isomorphism) three different quadratic extensions (of commutative rings): ℝ[x]/(x²+1) ≅ ℂ ℝ[x]/(x²) ≅ ℝ[ε] ℝ[x]/(x²−1) ≅ ℝ × ℝ For an arbitrary quadratic extension ℝ[x]/(f) of ℝ, the isomorphism class is determined by the sign of the discriminant Δ of the quadratic polynomial f: ℝ[x]/(f) ≅ ℂ if Δ < 0 ℝ[x]/(f) ≅ ℝ[ε] if Δ = 0 ℝ[x]/(f) ≅ ℝ × ℝ if Δ > 0
1) The brown chalk is sufficiently conspicuous to be used on standard chalk boards. 2) Algebraic toys are interesting. 3) It will be nice to see modular functions and a lecture on grids. Thank you Michael.
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
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.
It’s an open container and content. It’s open because it’s the field in a rotation. The container can be defined a couple ways like 0,1,0,0 or 0,0,-1,0 for 2x2 matrices of elements a,b,c,d. These containers are linear to contain angular and vice versa. I use these to define potentials at the roots of imaginary numbers of the forms 0,-1,1,0 and 0,1,-1,0 and their conjugates. In this way you can define the roots of quaternions.
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.
multiplying non-zero dual numbers doesn't generally produce rotations since the matrix representation of the non-zero dual numbers generally aren't invertible and rotations are invertible (another way to say it is that information is lost by collapsing the entire dual line to 0 under square, the dual line can be seen as an eigenvector with eigenvalue 0, but rotations have imaginary (and non-zero) eigenvalues)
This is how you define the "tangent space " or derivative in algebraic geometry! If X is a variety, and spec k --> X a point, lifts to spec k[eps] --> X are tangent vectors. Keywords: Zariski tangent space, lifting criterion/formally etale, etc
The way one can define this space is in fact analogous to C: you can view C as the quotient ring R[x]/ and the dual numbers as R[x]/ The fact that x²+1 is irreducible is enough to ensure that C is a field, while x² is trivially reducible, and in fact ε is not invertible. Suppose it is: ε = ε^-1 * ε² = 0
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.
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
this is quite interesting piece of maths! it reminds me of Galois extension field, but this number system has a big problem, you cannot factorize polynomials properly, sorry I was not aware comments should be about brown chalk :)
Sub groups are dual to sub fields -- the Galois correspondence. Real numbers are their own conjugates -- Duality. Complex roots come in pairs -- automorphisms (duality). Duality (thesis, anti-thesis) synthesizes reality (non duality) -- the Hegelian dialectic. "Philosophy is dead" -- Stephen Hawking.
But it doesn’t need to be R(ε), it can very well be C(ε) and nothing would have changed. Let me apply something I’ve learned in the algebra course this semester: First of all, this an artinian ring, with the only ideal I being . Moreover, the ring is communicative, therefore coincides with the Jacobson radical of the ring, where its only irreducible modules are R[ε]/ = R. However R[ε] is an indecomposable module of itself.
Fun fact (that others have pointed out as well): Unlike C, R[epsilon] can be ordered by simply assuming epsilon is greater than or less than 0 (an infinitesimal). In either case, it's similar to the ordering of finite and infinite ordinals.
This construction is isomorphic to the quotient of the polynomial ring in one variable (say, x) over the reals, modded by the ideal generated by x^2. i.e. the quotient ring R[x] / (x^2). Such elements can be written as the representative of their class (a + bx) for some real numbers a,b, and multiplication is defined in the same way as for dual numbers with instead x^2 = 0.
There is an interesting application for this in contact geometry (i e geometrical questions about whether lines and circles are in oriented contact), which i used in my master thesis. here is a "short" introduction. first, lets define the "crossratio" of 4 numbers cr(a,b,c,d) = (a-b)(c-d)/[(b-c)(d-a)]. this can be done for real numbers, complex numbers or dual numbers (although obviously you would have to deal with 0s in the denominator, but lets just ignore that here). one interesting and fairly well-known fact is that 4 points (in the plane) lie on a circle if and only if their crossratio (when interpreted as complex numbers in the usual way) is *real*. this fact is independent of the order of a,b,c,d (although the precise value of cr(..) obviously isnt) this can be used in a neat way to prove incidence theorems about circels. a good example is Miquels six circle theorem. the statement is: if 5 circles share 4 points of *triple* intersection, then the remaining 4 *double* intersection points also lie on a circle (look it up on wikipedia for an illustration). proving this with the above goes as follows: you have 5 groups of 4 points which each share a circle, you can mentally connect them to get 5 faces of a (combinatorical) cube (1 face per circle). now we know that the crossratios corresponding to each face are real. further one can see that by multiplying the crossratios in an elegant way way, most terms will cancel (i.e. if (a-b) appears in the numerator of one face, you put it in the denominator on the neighbouring face and vice versa, which is possible because the order of the points in the crossratio doesnt influence whether the result is real) and what you are left with is the crossratio of the sixth face, which then also has to be real (because its the product of real numbers), implying that the 4 corresponding points also lie on a circle. one interesting fact is, that this and similar results generalize to the spherical plane (just use stereographic projection, which preserves circles) and the hyperbolic plane (just use the poincare model, in which hyperbolic circles are also euclidean circles) ok, so where do dual numbers come in? they are important, when you want to *dualize* this (or similar) statements. i e you exchange points lying on a circle with (oriented) lines in (oriented) contact with circles. to visualize this for the dual of miquels theorem, imagine 8 lines, which intersect such that you get a 3x3 grid of quadrilaterals. now "color" the grid like a checkerboard. lets say the corner quadrilaterals are white. now if each of the white quads has a circle inscribed, then there is also one "big" circle inscribed in the quad formed by the 4 sides of our grid. obviously you could also leave out one of the "white" circles and instead be given the big circle, in which case the theorem would imply the existence of the missing circle. to see how the proof works, lets first look at the "easy" case, which is the spherical one. in that case each line is the intersection of the sphere with a plane and you can thus identify the line with a unit length normal of that plane (choosing which normal to use according to orientation of the line). now its fairly easy to prove that 4 (spherical) lines are in oriented contact with a circle if and only if the corresponding 4 normals are on a (spherical) circle, if and only if their images under stereographic projection are on a euclidean circle, if and only if their crossratio is real. now the proof works as above. now the same doesnt work in the euclidean plane, however here is what we can do. first we write an (oriented) line as L(n,d) = {x | =d}, where n is assumed to be unit length, where the sign of (n,d) signifies orientation. now we identify L(n,d) with the point (n,d). these points are on the cylinder in R³ with x²+y²=1. now it can be shown that 4 lines are in oriented contact with a circle if and only if the corresponding points on the cylinder are in a (euclidean) plane. now you use stereographic projection from the cylinder through the point (0,0,1) into the plane with x=-1 and identify the point (a,b,-1) with the dual number a+b*eps. and now it can be shown (which is actually one of the things i did in my thesis), that the cross-ratio of 4 dual numbers is *real* if and only if the corresponding points on the cylinder lie on a plane (which is true if and only if the corresponding lines are in oriented contact with a circle, as stated above). so again you can use cross ratios to prove the dual of Miquels theorem and similar results. which actually means that (most) results of this type (involving groups of 4 points lying on circles) dualize to lines. you can use a similar technique to identify lines in the hyperbolic plane with so called double numbers (which works similar to the other two concepts, just instead of i or eps you have e with e²=1) and you have the same result: 4 lines are in oriented contact with a circle if and only if the cross-ratio of the corresponding double numbers is real. (so Miquels theorem dualizes in the hyperbolic plane as well) also, note that while we "technically" use complex/dual/double numbers, there is no need to calculate any actual numbers. if you want a rigorous proof you probably need to write down the crossratios and how they cancel, but when trying to figure out the combinatorics, it is easier to do it visually, for example by connecting the four points (or lines) lying on /touching a circle into a quadrilateral, where you use alternating colors for the sides and one color represents a difference in the denominator, the other in the numerator. then you just have to make sure, that two points are connected by the same number of lines of each color, such that the terms cancel. oh, and as a little extra, this is a result i proved in my thesis with this technique: you again have a checkerboard-grid of quads, this time size 5x5, again color the corners white and the rest accordingly. now if all the "white" quads are inscribed, except one of the corner quads, then the existence of that circle follows. the proof idea is as follows: due to the dual of miquel, it is sufficient to show the existence of the "big" circle in the 3x3 grid in the corner of the missing circle. also, from miquels we know that the big circles exist in the 3x3 grids of the the other 3 corners. now you can combine the crossratios of the 3 given big circles and 8 of the small circles (you dont need the one in the middle or the ones in the corner anymore, they were only needed to show the existence of the big circles) to get the cross-ratio of the missing big circle, which then has to be real (since its the product of the others, which we know to be real by what we already know), implying the existence of that big circle and thus the existence of the missing white circle in the corner. however in this case it is not as easy to see how you have to combine the crossratios, such that all the terms cancel and you are left with the one you want. ok, that wasnt too short after all, but there probably wasnt a way to make this shorter while conveying all the information.
@@ginanjarutama well. technically. can give you a github link. although i dont know if that works on youtube, because last time i tried linking something it got deleted for some reason. (link: github.com/sur-taka/publicstuff/blob/main/ma.pdf ) but its in german^^ so unless you are fluent in german, you wont understand most of it. maybe you can still have a look at the formulas. i prove in theorem 3.1 that "4 lines tangent to one circle corresponding crossratio is real", by first interpreting the lines as points on a quadric (as described above) and then showing that the determinant of their projective coordinates is 0 if and only if the corresponding crossratio is real. maybe you will be able to understand the formulas with this explanation. 3.3 and 3.4 are the incidence theorems for the 3x3 and 5x5 grid. its mostly formulas, so maybe you can understand that as well. if you want to learn more about dual and double numbers in this context, you can also take a look at the book "complex numbers in geometry" by yaglom. maybe you can find it online somewhere
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.
9:40 that part deserves a bit more time to see that, first : since epsilon is a 0 divider, it has no inverse (numbers like 45+3e/17e don't have a meaning, cuz when you square them you get 1/0 error), and… that's interesting : if ad =bc, (a+be)/(c+de) = a ?? Sure this has an use…
This was awesome! I was wondering what you get when you combine dual and complex numbers. Something of the form: a + b · i + c · epsilon? {a,b,c} \in R, i^2 = -1, epsilon^2 = 0?
@@b43xoit well you can use R[epsilon] to do a little bit of non-standard real analysis (see e.g. the wikipedia article on non-standard analysis, although if you want to do proper non-standard analysis then you need the hyperreal numbers), so I guess you could use {a + b i + c epsilon + d i epsilon} = {(a+bi) + (c+di)epsilon} = C[epsilon] to do a bit of non-standard complex analysis :D
In the Julia Programming language, there is a package named ForwardDiff, that allows differentiation of arbitrary function routines via Dual numbers. You could create a complex number where the real and imaginary parts are itself Dual numbers, and it is useful for differentiation of routines that have intermediate complex steps. I, for example, use it to differentiate along a cubic root polynomial solver
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 ?
Brown chalk would be good for any "supplementary" text, providing some contrast using different brightnesses (compared to contrast from different colors of similar brightness)
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"
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.
Thanks a lot. Great video. You have forgotten to remove a repetition at 7:09. You can do that in the TH-cam panel, and you don't even need to change the url of the video or lose comments+likes.
Wow. That's really cool! You can even use this to find formulas for the product rule, quotient rule and even chain rule very simply. (Not a proof, but very neat)
16:22 Purposely forgetting to write the squared in the formula, so that when you act like you realized it was missing, it draws more user attention to the formula.
Brown chalk brings a very elegant flavor to the layout, but I agree it doesn’t pop enough for use on the blackboard. Perhaps different on a greener board
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.
That is genius to do it that way
I'm not sure about that last bit. The procedure you described is known as "forward mode automatic differentiation", while for backpropagation you'd need "reverse mode automatic differentiation". Reverse mode can also be implemented by providing a special type that keeps track of the value of a variable and the derivative of the loss function with respect to that variable, but this second number can only be computed in an additional backward pass.
seriously? that’s crazy, so cool!
@@Michael-vs1mw Multi-dimensional dual numbers (i.e, R[epsilon_1,epsilon_2,..., epsilon_k], with epsilon_i*epsilon_j=0) certainly work to compute a whole gradient, but computing the gradient this way takes about k times longer than computing the original function, and also k times more space. With reverse-mode differentiation, you can take a gradient with respect to a million variables (say, weights in a modest-size neural network) in only about twice the running time of the forward function and with space proportional to the number of computations (you need to remember intermediate values of computations).
Computing Hessians is impractical for large numbers of variables because the matrices are huge, but one can multiply the Hessian times a vector in about four times the original running time, by computing the gradient of (the gradient of the function dot-times the vector). For some algorithms (like the conjugate gradient method) that's all you would need.
Disclaimer: I haven't thought about this in a few years, so I might have said something incorrect.
You described autodiff better in a comment than all the online tutorials imo.
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.
hehe autodiff
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
Yes it even makes sense rigorously in a setting called smooth infinitesimal analysis. Here your number line R is filled with infinitesimals, where some have the property e^2 = 0. Here every function is infinitely differentiable and the derivative f‘ of f: R -> R is uniquely determined by f(x+e) = f(x) + ef‘(x), where e is a nilsquare infinitesimal (e^2 = 0). This is basically the Taylor expansion where the higher order terms are neglected as you said.
You did study engineering at University of Pisa in the early '90s, didn't you? :)
Yes, just like what we had done in perturbation theory.
You cannot simply say an infinitesimal squared is negligible, a mathematician would cringe.
I thought the same. As a physicist, the concept is pretty clear to me on an intuitive level. Never have I seen it formalized like that though.
fun fact: unlike C, R[eps] is an ordered ring. it has two valid orderings, defined by either eps>0 or eps
where do i learn more about this mysterious epsilon???
You can "order" C as well, in the same way, but neither ordering is a well-ordering.
so is it a lexicographic ordering?
@@zswu31416 actually you cannot order C in the same way. to be an ordered ring does not just mean that you have a ring with an understood order- it implies two extra axioms about how the ordering interacts with the two operations.
@@shortcutz6491 Oh yeah now I realize what "ordered ring" actually means. How stupid of me.
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 don't like him, just by reading his overall personality in wikipedia. Maybe Euler was better.
19:17?
@@ΕχιΜιμζ what's wrong with Clifford personality in general and his Wikipedia page in particular?
@@FranFerioli he is the guy who says "wow, this person revolutioned the cience, but is an asshole, he is worthless and x person is better because was a nice person"
Also, Euler is better than any mathematician, I dont doubt that
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 ..."*
Those pesky physicists get away with murder!
@@lina31415 "assume the victim is a non-rotating sphere..."
@@SeanCMonahan In a vacuum
Real numbers are their own conjugates -- Duality.
Complex roots come in pairs -- automorphisms (duality).
Duality (thesis, anti-thesis) synthesizes reality (non duality) -- the Hegelian dialectic.
"Philosophy is dead" -- Stephen Hawking.
Or maybe they should be called "The Physicists' Numbers" because they are the mathematical foundation of supersymmetry...
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!)
Thatd be dope, do you know of any videos like this guy that have about it, I am not very advanced math yet but this video was easy to follow (except for summation part, I am not familiar with it lol)
@@74bassmanjust search geometric algebra and look at the videos and playlist
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.
Also, automatic differentiation,
en.wikipedia.org/wiki/Automatic_differentiation#Automatic_differentiation_using_dual_numbers
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
If I were Michael, I probably wouldn't use the brown chalk for writing, but I would definitely use it to draw divider lines and boxes.
I think if I had ordered brown chalk on a whim, I would probably use it for dividers and boxes, but I don't think I would use it much for actual writing
There is no “brown”; there is only “orange with context”.
(Look it up.)
I concur.
@@lawrencedoliveiro9104 Tecnology Connections moment
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 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.
So, what silliness can one get by mixing dual and complex numbers?
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.
I think geometric algebra was first discovered but the Clifford guy mentioned here. Unfortunately he died young and his discoveries were forgotten until being revived in modern fashion by the people you mention. As others mentioned in the comments, geometric algebra pretty much follows from these concepts.
That's interesting, because W'pedia mentions the application to rigid bodies but does not point out that it is obsolete.
I remember reading about dual quaternions as a good way to do vertex skinning on the GPU, being a compromise between full rot/trans matrices and normal (real) quaternions. It would be good because dual quaternions could encode rotation *and* translation, allowing for smooth blending between the transforms in vertex skinning.
Yes ! For an introduction to geometric algebra, look at th-cam.com/video/60z_hpEAtD8/w-d-xo.html&ab_channel=sudgylacmoe
LInkedIn has two Geometric Algebra groups. One is called "Geometric Algebra", and the other is "Pre-Univerisity Geometric Algebra". Your participation will be welcome in both. I, myself, have several videos on GA that are meant primarily for self-learners like me.
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!
Yes definitely!:) Geometric Algebra is basically build on the generalization of complex, dual and split-complex numbers (e^2 = -1, 0, 1 respectively) and adds higher grade objects as well as the geometric product to connect everything.
Most vector algebras struggle with multiplying vectors. GA lets you multiply vectors and it gives... a complex number‽ And people previously thought complex numbers _were_ vectors!
@@angeldude101 complex numbers are vectors. A k-vector is a vector, matrices are vector, polynomials are vector. Anything could be a vector is you choose to think of it as a vector and obviously has vector properties.
@@rajinfootonchuriquen Yay for vector meaning both a one dimensional geometric object as well as an abstract mathematical object. The former is the latter, but not all instances of the latter are the former.
Now I'm curious what you get of you try forming a Clifford Algebra around an abstract vector space that isn't a simple quadratic form. What would it mean to take the wedge product of two functions?
@@angeldude101 I don't now. I don't advocate for using Clifford algebra. I don't know much about it but for instance, i know that you can get a nice maxwell equation, but it seems odd that the magnetic field is a bivector field. I don't know how to picture a bivector field.
If you try to take a wedge product between that type of vectors, I think it should span out of your vector space, so it's undefine.
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.
Great introduction. I think you should also extend it a bit more to include multiple Grassmanian variables $\epsilon_1$, $\epsilon_2$, etc.
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.
This is by far my fav YT math channel, period. Thank you Mr.Penn!
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.
This is my new favorite channel on TH-cam. I love making some very simple assumptions and seeing where it leads. Its all very simple, logical, and easy-to-follow.
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.
never encountered these before, what a cool and interesting concept. the derivative result is especially cool. thanks michael
You know that this guy is a hardcore mathematician when starts talking about his chalk.
This year's brown Hagaromo is the new yellow Hagaromo. It's an extravagance, but he just had to have it.
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)
There's not a single standard name: en.wikipedia.org/wiki/Split-complex_number
@@themattcohen Now I don't feel too bad about not remembering it 😅
Nice video suggestion!
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.
Real Part and Tangential Part? Real Part and Differential Part?
Finite and Infintessimal?
nilpotential
Dual is the best word
I think ε is a nilsquare infinitesimal. So you could
call the second part the
infinitesimal part.
Just found this channel. Excellent video; concise, interesting, and well thought out. Really loved the old school blackboard style. Reminds me of undergraduate lectures
This is known in physics as Grassmann directions, in the special case of 1 number. They are used to build the bases of superfields.
It would be interesting to see how integration over the dual numbers is done, and whether it generates something akin to differentiation, as what happens with grassmann numbers.
@@hbm293 th-cam.com/video/u58b-OgqvqQ/w-d-xo.html this video covers integrating functions of grassmann numbers for physics problems
The brown chalk is perfect for "dividing lines" as you used it, easy to distinguish, yet not distracting.
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.
Love the brown chalk, it's like a highlighter for the chalk board. Not good on its own but it makes other things pop.
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!
H
This is a whole new world I was unfamiliar with. Fascinating. Thank you!
Fred
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)
Correct me if I'm wrong but:
isn't exp(2ε) ≠ exp(ε)² incorrect?
exp(2ε) =(def)= 1 + 2ε
exp(ε)² =(def)= (1 + ε)² = (1 + ε)*(1 + ε) = 1*1 + 1*ε + ε*1 + ε*ε = 1 + 2ε
using the binomial formula you can generalise this to exp(nε) = exp(ε)^n for all natural n
@@IQuick143cz Ah yes you are right, I'll edit the comment
Nice:)
A "dual angle" is really just a distance along a straight line. In the same way, normal angles and hyperbolic angles are basically just distances along a unit circle or unit hyperbola respectively.
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 think this is related to the exponential of dual numbers. Much like how exponentiating an imaginary number makes you walk along the unit circle, exponentiating ε makes you walk along a straight line (which is why I sometimes call them and the complex numbers "flat" numbers and spherical numbers respectively due to their relation to the corresponding geometry).
10:16 that's correct, very good for borders, not so for writing
R(ε) is just the ring R[x]/(x²)
Also, I believe all thr 2D R-algebras are (isomorphic to) either C, R², or R(ε).
They can all be written in a uniform way: R[x]/(x²+1), R[x]/(x²-x), R[x]/(x²), respectively. Notice how the quotient polynomials have different signs for the discriminants (negative, positive, zero). I believe that's the invariant that tells you which of these your 2D R-algebra is isomorphic to.
Hello. What?
Instead of R[x]/(x²-x) you need R[x]/(x²-1), the split-complex numbers.
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.
Hi - every time I tune into this channel, I learn something. Then I realise that I have no idea what I've learned. Outstanding!
The brown chalk bit cracked me up. Thanks for this
11:42 - 12:00, pretty cool attention to detail that you won't want to gloss over.
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?
I think so. You can observe that the polynomial x² is not irreducible, therefore the quotient will not be a field, and since x² is not prime it will not be an integral domain. It is a nice way to construct these numbers.
Yes, the ring of dual numbers is exactly ℝ[x]/(x²), similar to how ℂ ≃ ℝ[x]/(x²+1).
yep th-cam.com/video/M412eVW7bf4/w-d-xo.html is an example. Not exact here but he has it on his wildegg channel. cannot find it quickly :)
More generally, ℝ has (up to isomorphism) three different quadratic extensions (of commutative rings):
ℝ[x]/(x²+1) ≅ ℂ
ℝ[x]/(x²) ≅ ℝ[ε]
ℝ[x]/(x²−1) ≅ ℝ × ℝ
For an arbitrary quadratic extension ℝ[x]/(f) of ℝ, the isomorphism class is determined by the sign of the discriminant Δ of the quadratic polynomial f:
ℝ[x]/(f) ≅ ℂ if Δ < 0
ℝ[x]/(f) ≅ ℝ[ε] if Δ = 0
ℝ[x]/(f) ≅ ℝ × ℝ if Δ > 0
1) The brown chalk is sufficiently conspicuous to be used on standard chalk boards. 2) Algebraic toys are interesting. 3) It will be nice to see modular functions and a lecture on grids. Thank you Michael.
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
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.
I like how epsilon^0 = 1 because x^0 = 1 is a deeper magic (only challenged by 0^0 being undefined).
So much breaks if you don't define x^0 to be 1. It's pretty interesting.
I think I'm pretty happy with the brown chalk.
I love the BROWN chalk. You won me over with that. You are fantastic. You're teaching is amazing.
I like white chalk, white as an angel.
I like brown chalk. Brown like an angel who afraid of height.
Brown chalk is perfectly visible but not easy to distinguish from pink or orange
Hey! Could you cover the dual numbers? Amazing content as always
It’s an open container and content. It’s open because it’s the field in a rotation. The container can be defined a couple ways like 0,1,0,0 or 0,0,-1,0 for 2x2 matrices of elements a,b,c,d. These containers are linear to contain angular and vice versa.
I use these to define potentials at the roots of imaginary numbers of the forms 0,-1,1,0 and 0,1,-1,0 and their conjugates. In this way you can define the roots of quaternions.
you should talk about the split complex numbers too
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.
multiplying non-zero dual numbers doesn't generally produce rotations since the matrix representation of the non-zero dual numbers generally aren't invertible and rotations are invertible (another way to say it is that information is lost by collapsing the entire dual line to 0 under square, the dual line can be seen as an eigenvector with eigenvalue 0, but rotations have imaginary (and non-zero) eigenvalues)
This is how you define the "tangent space " or derivative in algebraic geometry! If X is a variety, and spec k --> X a point, lifts to spec k[eps] --> X are tangent vectors. Keywords: Zariski tangent space, lifting criterion/formally etale, etc
This is such a great video Mike. Thank you.
I agree that the brown chalk should be mostly used for boxes and dividers.
The way one can define this space is in fact analogous to C: you can view C as the quotient ring R[x]/ and the dual numbers as R[x]/
The fact that x²+1 is irreducible is enough to ensure that C is a field, while x² is trivially reducible, and in fact ε is not invertible.
Suppose it is: ε = ε^-1 * ε² = 0
Great video about a neat topic. Huge admiration for keeping the part at 7:13 in.
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.
I thought epsilon was practically zero to begin with. What application was that for?
@@onradioactivewaves Morisson equations.
Although it is also used in many other fields of mechanical engineering such as finding equilibria.
The brown chalk is nice and visible. I wouldnt be afraid to use it to underline or seperate things from one another.
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
this is quite interesting piece of maths! it reminds me of Galois extension field, but this number system has a big problem, you cannot factorize polynomials properly, sorry I was not aware comments should be about brown chalk :)
We should have more of these cool new whiteboards with the dark theme ...
Sub groups are dual to sub fields -- the Galois correspondence.
Real numbers are their own conjugates -- Duality.
Complex roots come in pairs -- automorphisms (duality).
Duality (thesis, anti-thesis) synthesizes reality (non duality) -- the Hegelian dialectic.
"Philosophy is dead" -- Stephen Hawking.
But it doesn’t need to be R(ε), it can very well be C(ε) and nothing would have changed.
Let me apply something I’ve learned in the algebra course this semester:
First of all, this an artinian ring, with the only ideal I being . Moreover, the ring is communicative, therefore coincides with the Jacobson radical of the ring, where its only irreducible modules are R[ε]/ = R. However R[ε] is an indecomposable module of itself.
!!
What an underated piece of math. Very very cool.
I saved this video in my geometric algebra playlist.
Oh that's quite a powerful application! Nice!
Fun fact (that others have pointed out as well): Unlike C, R[epsilon] can be ordered by simply assuming epsilon is greater than or less than 0 (an infinitesimal). In either case, it's similar to the ordering of finite and infinite ordinals.
brown chalk looks good for the applications you described 👍
I like the brown for delineating spaces on the chalkboard, but not for writing text/calculations.
Made a pedagogical observation: The way he writes "implies" can, when badly written, be interpret as "don't imply". Other vice interesting stuff. 😊
This construction is isomorphic to the quotient of the polynomial ring in one variable (say, x) over the reals, modded by the ideal generated by x^2. i.e. the quotient ring R[x] / (x^2).
Such elements can be written as the representative of their class (a + bx) for some real numbers a,b, and multiplication is defined in the same way as for dual numbers with instead x^2 = 0.
I don’t quite understand this math. ‘i I use and understand, duel numbers cool. You’re recent and look like you know what your talking about so 👍.
I think for writing you should take 1 white chalk and then include epsilon brown chalk.
This recalls me of asymptotic analysis. The expression a + bε, with 0 < ε
There is an interesting application for this in contact geometry (i e geometrical questions about whether lines and circles are in oriented contact), which i used in my master thesis. here is a "short" introduction.
first, lets define the "crossratio" of 4 numbers cr(a,b,c,d) = (a-b)(c-d)/[(b-c)(d-a)]. this can be done for real numbers, complex numbers or dual numbers (although obviously you would have to deal with 0s in the denominator, but lets just ignore that here).
one interesting and fairly well-known fact is that 4 points (in the plane) lie on a circle if and only if their crossratio (when interpreted as complex numbers in the usual way) is *real*. this fact is independent of the order of a,b,c,d (although the precise value of cr(..) obviously isnt)
this can be used in a neat way to prove incidence theorems about circels. a good example is Miquels six circle theorem. the statement is: if 5 circles share 4 points of *triple* intersection, then the remaining 4 *double* intersection points also lie on a circle (look it up on wikipedia for an illustration). proving this with the above goes as follows: you have 5 groups of 4 points which each share a circle, you can mentally connect them to get 5 faces of a (combinatorical) cube (1 face per circle). now we know that the crossratios corresponding to each face are real. further one can see that by multiplying the crossratios in an elegant way way, most terms will cancel (i.e. if (a-b) appears in the numerator of one face, you put it in the denominator on the neighbouring face and vice versa, which is possible because the order of the points in the crossratio doesnt influence whether the result is real) and what you are left with is the crossratio of the sixth face, which then also has to be real (because its the product of real numbers), implying that the 4 corresponding points also lie on a circle.
one interesting fact is, that this and similar results generalize to the spherical plane (just use stereographic projection, which preserves circles) and the hyperbolic plane (just use the poincare model, in which hyperbolic circles are also euclidean circles)
ok, so where do dual numbers come in? they are important, when you want to *dualize* this (or similar) statements. i e you exchange points lying on a circle with (oriented) lines in (oriented) contact with circles. to visualize this for the dual of miquels theorem, imagine 8 lines, which intersect such that you get a 3x3 grid of quadrilaterals. now "color" the grid like a checkerboard. lets say the corner quadrilaterals are white. now if each of the white quads has a circle inscribed, then there is also one "big" circle inscribed in the quad formed by the 4 sides of our grid. obviously you could also leave out one of the "white" circles and instead be given the big circle, in which case the theorem would imply the existence of the missing circle.
to see how the proof works, lets first look at the "easy" case, which is the spherical one. in that case each line is the intersection of the sphere with a plane and you can thus identify the line with a unit length normal of that plane (choosing which normal to use according to orientation of the line). now its fairly easy to prove that 4 (spherical) lines are in oriented contact with a circle if and only if the corresponding 4 normals are on a (spherical) circle, if and only if their images under stereographic projection are on a euclidean circle, if and only if their crossratio is real. now the proof works as above.
now the same doesnt work in the euclidean plane, however here is what we can do. first we write an (oriented) line as L(n,d) = {x | =d}, where n is assumed to be unit length, where the sign of (n,d) signifies orientation. now we identify L(n,d) with the point (n,d). these points are on the cylinder in R³ with x²+y²=1. now it can be shown that 4 lines are in oriented contact with a circle if and only if the corresponding points on the cylinder are in a (euclidean) plane. now you use stereographic projection from the cylinder through the point (0,0,1) into the plane with x=-1 and identify the point (a,b,-1) with the dual number a+b*eps. and now it can be shown (which is actually one of the things i did in my thesis), that the cross-ratio of 4 dual numbers is *real* if and only if the corresponding points on the cylinder lie on a plane (which is true if and only if the corresponding lines are in oriented contact with a circle, as stated above). so again you can use cross ratios to prove the dual of Miquels theorem and similar results. which actually means that (most) results of this type (involving groups of 4 points lying on circles) dualize to lines.
you can use a similar technique to identify lines in the hyperbolic plane with so called double numbers (which works similar to the other two concepts, just instead of i or eps you have e with e²=1) and you have the same result: 4 lines are in oriented contact with a circle if and only if the cross-ratio of the corresponding double numbers is real. (so Miquels theorem dualizes in the hyperbolic plane as well)
also, note that while we "technically" use complex/dual/double numbers, there is no need to calculate any actual numbers. if you want a rigorous proof you probably need to write down the crossratios and how they cancel, but when trying to figure out the combinatorics, it is easier to do it visually, for example by connecting the four points (or lines) lying on /touching a circle into a quadrilateral, where you use alternating colors for the sides and one color represents a difference in the denominator, the other in the numerator. then you just have to make sure, that two points are connected by the same number of lines of each color, such that the terms cancel.
oh, and as a little extra, this is a result i proved in my thesis with this technique:
you again have a checkerboard-grid of quads, this time size 5x5, again color the corners white and the rest accordingly. now if all the "white" quads are inscribed, except one of the corner quads, then the existence of that circle follows. the proof idea is as follows: due to the dual of miquel, it is sufficient to show the existence of the "big" circle in the 3x3 grid in the corner of the missing circle. also, from miquels we know that the big circles exist in the 3x3 grids of the the other 3 corners. now you can combine the crossratios of the 3 given big circles and 8 of the small circles (you dont need the one in the middle or the ones in the corner anymore, they were only needed to show the existence of the big circles) to get the cross-ratio of the missing big circle, which then has to be real (since its the product of the others, which we know to be real by what we already know), implying the existence of that big circle and thus the existence of the missing white circle in the corner. however in this case it is not as easy to see how you have to combine the crossratios, such that all the terms cancel and you are left with the one you want.
ok, that wasnt too short after all, but there probably wasnt a way to make this shorter while conveying all the information.
Do you have link for your thesis?
@@ginanjarutama well. technically. can give you a github link. although i dont know if that works on youtube, because last time i tried linking something it got deleted for some reason. (link: github.com/sur-taka/publicstuff/blob/main/ma.pdf )
but its in german^^ so unless you are fluent in german, you wont understand most of it.
maybe you can still have a look at the formulas.
i prove in theorem 3.1 that "4 lines tangent to one circle corresponding crossratio is real", by first interpreting the lines as points on a quadric (as described above) and then showing that the determinant of their projective coordinates is 0 if and only if the corresponding crossratio is real. maybe you will be able to understand the formulas with this explanation.
3.3 and 3.4 are the incidence theorems for the 3x3 and 5x5 grid. its mostly formulas, so maybe you can understand that as well.
if you want to learn more about dual and double numbers in this context, you can also take a look at the book "complex numbers in geometry" by yaglom. maybe you can find it online somewhere
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.
9:40 that part deserves a bit more time to see that, first : since epsilon is a 0 divider, it has no inverse (numbers like 45+3e/17e don't have a meaning, cuz when you square them you get 1/0 error), and… that's interesting : if ad =bc, (a+be)/(c+de) = a ?? Sure this has an use…
This was awesome! I was wondering what you get when you combine dual and complex numbers. Something of the form: a + b · i + c · epsilon? {a,b,c} \in R, i^2 = -1, epsilon^2 = 0?
you would get a + b i + c epsilon + d i epsilon, because without the d term you don't have closure under multiplication
@@schweinmachtbree1013 Does it have any practical application?
@@b43xoit well you can use R[epsilon] to do a little bit of non-standard real analysis (see e.g. the wikipedia article on non-standard analysis, although if you want to do proper non-standard analysis then you need the hyperreal numbers), so I guess you could use {a + b i + c epsilon + d i epsilon} = {(a+bi) + (c+di)epsilon} = C[epsilon] to do a bit of non-standard complex analysis :D
In the Julia Programming language, there is a package named ForwardDiff, that allows differentiation of arbitrary function routines via Dual numbers. You could create a complex number where the real and imaginary parts are itself Dual numbers, and it is useful for differentiation of routines that have intermediate complex steps. I, for example, use it to differentiate along a cubic root polynomial solver
brown chalk is sweet, we got some representation for the chalk brothas
Dude, totally down with the brown chalk for boxes
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 ?
Looks like a formalization of the concept "throw away all terms higher than first order".
Brown chalk is a great addition for dividers and stuff
Brown chalk would be good for any "supplementary" text, providing some contrast using different brightnesses (compared to contrast from different colors of similar brightness)
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"
yeah the approximations just looked ridiculous at that time.
The brown looks nice, I actually thought it was red.
Dig the brown chalk for boxes and dividers. Good color for just separating meaningful pieces of the work :)
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.
Grassman numbers are very useful in quantum field theories describing fermionic particles.
Delightful content and presentation. Brown chalk is a plus for me because it reads softly though clearly.
Thanks a lot. Great video. You have forgotten to remove a repetition at 7:09. You can do that in the TH-cam panel, and you don't even need to change the url of the video or lose comments+likes.
I was very sceptical at the start but you won me over.
Wow. That's really cool! You can even use this to find formulas for the product rule, quotient rule and even chain rule very simply. (Not a proof, but very neat)
16:22 Purposely forgetting to write the squared in the formula, so that when you act like you realized it was missing, it draws more user attention to the formula.
Love the chalk, your writing is very easy to read and you explain things very well. Thank you
I like the Brown for the boxes
Brown chalk brings a very elegant flavor to the layout, but I agree it doesn’t pop enough for use on the blackboard. Perhaps different on a greener board
My favourite part IS-THE-SUM....😆 I love your videos!
This is great timing. I just started exploring split complex numbers.
Use the brown chalk for everything. I love it...