วีดีโอ

Unfortunately, I don't have an electrical circuit problem here.

Glad it was useful!

Will do! There is the first video on Visual abstract algebra already posted, and more will be coming.

but thank you for the explanation it did make some things clear for me

You are welcome! Glad it was helpful.

Thanks, Josh! Got your email and sent a reply.

You'd need one extra variable for each additional "direction".

Thank you for the kind words, Diogo.

You can apply the Gauss-Jordan elimination algorithm to find the RREF of any matrix. This video th-cam.com/video/4k7s9uM7YCs/w-d-xo.html explains how to do it.

Once you get to RREF, you find pivot columns, which correspond to dependent variables, and non-pivot columns, which correspond to free variables. It's RREF that tells you what the free variables are. Look up early videos in my linear algebra playlist, its all explained there in detail.

@@mathflipped thank you!

You are welcome. Glad it was helpful.

You may want to revisit your understanding of continuous/discontinuous functions. There is nothing wrong in saying that a function is discontinuous at a point where it's not defined. A good example of this is the so-called removable discontinuity.

​@@mathflipped Removable discontinuities are a great example which has made me think again. Looking it up, it turns out that whether or not continuity/discontinuity should apply to points not in a function's domain is actually a matter of convention - for example there are multiple wikipedia articles using each convention. As you would expect, the difference is quite technical, and one first needs to recall that limits of a function _f_ : _D_ → *R* are only defined at so-called _limit points_ (also called cluster points) of its domain, which are points _a_ ϵ *R* (which need not necessarily be in the domain _D_ ) such that there are points of _D_ arbitrarily close to (and not equal to) _a_ . To make things agree with topology one needs ones definition of continuity to imply that functions on a discrete set (a set of isolated points) are always continuous, e.g. any function _f_ : *N* → *R* - one way of doing this is to say " _f_ : _D_ → *R* is continuous at _a_ if _a_ is an isolated point of _D_ or [ the usual limit-equals-value definition ]". Therefore whenever I say "the limit as _x_ → _a_ ..." from now on I mean "either _a_ is an isolated point of _D_ , or the limit as _x_ → _a_ ...". With that recalled, the two conventions are as follows: - For the convention you use -- in which removable discontinuities are actually discontinuities -- one takes the notion of continuity/discontinuity to apply to the _closure_ of a function's domain (its domain plus the limit points thereof). One says that _f_ : _D_ → *R* is continuous at _a_ ϵ cl( _D_ ) if the limit of _f_ as _x_ → _a_ exists and equals _f_ ( _a_ ) - in particular _a_ ϵ _D_ , so if _f_ is not defined at _a_ -- i.e. _a_ ϵ cl( _D_ )∖ _D_ -- then by this convention _f_ is discontinuous at _a_ (and the discontinuity may be either removable or not removable). - For the other convention, the rationale is that the _limit_ of a function as _x_ → _a_ is a "punctured-local property" -- i.e. it concerns punctured neighborhoods of _a_ (neighborhoods of _a_ with _a_ deleted) so the behaviour at _a_ is irrelevant -- while _continuity_ of a function at a point _a_ is a local property -- i.e. it concerns neighborhoods of _a_ including the behaviour at _a_ itself -- and therefore using this convention, if a function _f_ is not defined at _a_ then we can't talk about (unpunctured) neighborhoods of _a_ and thus it doesn't make sense to talk about continuity or discontinuity of _f_ at _a_ . This is the convention my analysis lecturer used, and I suppose that if one uses this convention then to be consistent one should use the alternative term "removable singularity" instead of removable discontinuity (since by this convention such singularities aren't discontinuities). To summarise, the two different conventions define continuity as - " _f_ : _D_ → *R* is continuous at _a_ ϵ cl( _D_ ) if _a_ ϵ _D_ and the limit as _x_ → _a_ ...", and - " _f_ : _D_ → *R* is continuous at _a_ ϵ _D_ if the limit as _x_ → _a_ ..." respectively, and since discontinuity is just the negation of continuity, discontinuity is applicable for the same values of _a_ , i.e. the domain or the closure of the domain respectively. So using the first convention (x^2 - 1)/(x-1) = (x+1)(x-1)/(x-1) would be discontinuous at x=1, the discontinuity being removable, and using the second convention one would say it has a singularity at x=1, the singularity being removable (and one would not say it is discontinuous because 1 is not in the domain). For a step function which jumps at x=0, using the first convention one would say it is discontinuous at x=0 whether or not it is defined there, whereas using the second convention it depends whether it is defined at x=0; e.g. one would say that sign: *R* → *R* with sign(0) := 0 has a jump discontinuity at x=0 whereas sign: *R* ∖ {0} → *R* has a singularity at x=0, which I suppose one would call a "jump singularity". Finally, for the function 1/x, using the first convention one would say it is discontinuous at x=0 (the discontinuity being neither removable nor a jump discontinuity), whereas using the second convention one would say it has a singularity at x=0. Like all conventions, which one decides to use is a matter of taste, but having written all this out it is clear to me why both are in use: the first convention is simpler terminologically as one doesn't need to make an (often pointless) distinction between discontinuities and singularities, whereas the second convention is simpler definitionally since it doesn't have the quirk of the other convention of defining (dis)continuity for values _a_ ϵ cl( _D_ ) and then immediately restricting to _a_ ϵ _D_ - this makes the second convention cleaner for proofs, but the quirk of the first convention is what allows one to describe removable discontinuities and jump discontinuities as "discontinuities" (which is desirable because intuitively discontinuities are "jumps" in the value of a function).

@@schweinmachtbree1013 Glad you dove into details of this phenomenon. I hope this was instructive.

You are welcome, Jorge. Glad it was useful.

Thank you, Whitney.

In what sense?

Kinda hard to explain.. Its color saturation seems more extreme than all the other C colors compared with their B and D counterparts--or put another way, none of the other colors get so pure as yellow-c

I guess pure magenta and pure cyan are missing, but yellow-c looks like a pure_yellow

@@dmcdouga07 I see what you mean. Personally, I don't care for the standard Manim colors and use my own custom scheme.

Thank you for the constructive feedback! A lot more will be coming in this series. And good luck with your entry. I think it should definitely make it to the short list, and possibly end up as one of the winners.

@@mathflipped Thank you, and good luck with your series!

The voice is that of my collaborator for this series, Matthew Macauley. He is credited in the video description.

Thank you for the feedback, Stafford. A lot more is coming in this series, and these and other questions will be answered in the future videos. For now, we showed a historic motivation for the notion of a group and played a bit with some more advanced notions such as generators and relations.

Isnt it like saying what is the point of logarithm when you literally defined something for equations that you cant solve, of course in math you start with an idea that seems like have no application but you can expand the idea itself for more application in the future

@Stafford It is not exaggerating to say that groups are one of the main tools in mathematics. When there are groups, you have a way to harness structure to understand your object, and conversely, when studying an object with useful structure, at some point you will find a group which encodes the information. Almost all of mathematics is studying very nice groups, albeit not always with the very general perspective one associates with abstract algebra. Of course, the applications are more advanced than what is covered in lecture one (or here, the introductory video). Three conceptually simple classes of examples for useful groups are 1) groups which directly correspond to other objects. There are a few areas where you can define groups whose subgroups have a nice bijection to the substructures you want to understand (examples: Galois group of field extension, fundamental group of space) 2) groups are rather nice to study, so it is sometimes helpful to assign an object you want to understand a group which captures some information about it - this is the idea of using groups as invariants: If the groups are different, then the object are different too, which is a powerful tool to distinguish them. The prime „example“ for this is the idea of (co)homology. 3) when you have *really* nice groups, you can use them as a framework for further work. What I have in mind here are vector spaces, which are fundamental for huge parts of mathematics in their own right, but still primarily special groups. And of course, mathematics has applications to describe nature! What do you mean, manmade ideas cannot describe the physical world!? ;) Physicists are quite good at using mathematics (read here: groups) to make predictions and enable technologies.

Lots of amazing responses in this thread! To add to Stafford's remark that groups are constructed by humans. It is an open philosophical question whether mathematics is created or discovered. But when it comes to groups, I believe they are discovered because there are too many patterns and structures in nature that fit the notion of a group. I plan to address this exact question in one of the future videos in this series.

The fundamentals of particle physics actually stems from certain kinds of group theories. So it's actually types of symmetries from which fundamental interactions that make all matter and energy as we know it exist. For example, Electromagnetic theory comes from the U(1) group which is basically the circle rotation group, and it explains why stuff has charge. More abstract groups like SU(2) and SU(3) explain the weak and strong forces, and the generators of these groups correspond with force particles like photons, W bosons, Z bosons, gluons, etc.

Thank you, Tom! We should collaborate sometime in the future.

They should be the other dihedral groups, right? I forgot about the colours part initially and just saw a cyclic transformation of arbitrary length along with an additional self-inverse transformation. So f^2 = 1 and r^n = 1 for some n (for the triangle, n = 3). I'm not certain what the general form of the commutator looks like though.

Yes, they are dihedral groups. The general relation is rf = fr^{-1}.

@@mathflipped That makes sense. Since r^n = 1, r^(n-1)r = 1, so r^(n-1) = r^-1.

Right, and the relation with the inverse can be generalized to the case when the r^n = 1 relation is dropped. I will probably talk about this in one of the future videos.

@@mathflipped I'm looking forward to the rest of your series on visual algebra.

You are welcome! I'm glad this video was useful for you.

Because it corresponds to a non-pivot column in RREF.

You are welcome!

Thanks, Tom.

Thanks, Lee. Any resemblance of colors and symbols is purely coincidental. 😇

Not quite. Hyperreal numbers come from nonstandard analysis. They are an extension of real numbers which includes infinitely small and infinitely large numbers. I was fascinated by nonstandard analysis some 10 years ago, and read several books on the subject.

@@mathflipped if you identify constant functions with real numbers, you get that there are elements of this field (for example x) that are greater than any positive real number, such as infinite hyperreals, and other elements of this field (for example 1/x) that are positive but less than any positive real number, such as infinitesimal hyperreals. It seems to me that identification can be made. I am also very interested in non-standard analysis: I have been teaching analysis with this formulation to my pupils for several years now, rather than with the limit-based formulation.

@@VideoFusco From this point of view, yes, there are elements that are smaller or larger than any positive real number. But would the transfer principle hold? I've never seen rational functions used as a model of hyperreals in place of the standard ultrapower construction.

Ha ha

Thanks, Whitney!