But what IS Algebra?

[@normanklein3155]

This is meant as constructive criticism, but I would add the Fundamental Theorem of Algebra from Gauss stating there are an equal number of roots as the highest order coefficient to the Classical Algebra half.

[@jdbrown371]

I agree that it should be stated and used but proving is an exercise in analysis.

You are right, unfortunately I forgot to mention that. While the core of the proof is analytic in nature the statement is clearly an algebraic one that would have fit in perfectly with the discussion of the n > 4 case.

Could you please add the word "complex" (roots) to your comment?

[@sumdumbmick]

why are you citing Gauss for this? it was established 170 years before his birth. also, Gauss lived in the Modern Algebra period anyway, so citing him while making the claim that you're making is completely nonsensical in absolutely every way. Galois theory requires it, and Galois died nearly 20 years before Gauss published one of his proofs of it. do you understand how time works?

[@sumdumbmick]

I think in general, mathematicians do not remotely grasp how time works. for instance, it is nearly universally claimed that the Naturals are the kinds of numbers known to Euclid, even though the actual definition of the Naturals was invented in the 1880s, and includes contradictions which induce Godel Incompleteness. importantly, Euclidean geometry is not subject to Godel Incompleteness specifically because it does not include the Naturals. that's some fantastic rigor you idiots are using there.

[@lecro8974]

It's great that the basic structure is explained and not started straight in!

Thanks to Pinter's book on algebra. It has such an inspiring introduction to the roots of the subject that I thought I should make a video about that before diving right in.

[@claycurry5782]

underrated comment. just boosting in hopes more teachers will take notice

[@Riboflavaflav]

Thanks for the introduction. I've attempted several times to learn abstract algebra. Hopefully, this time, things will click a little better for me.

Especially in the first lessons I'm proceeding extra slow and careful, so hopefully this will help you with things clicking better this time. Keep me posted about your progress.

P.S.: Try to do as much of the exercises on the problem sets as you can - that really helps to get a better grasp of the new concepts.

[@namantenguriya]

Thank u for such detailed overview. I needed this one. ❤

You are very welcome! Thanks for commenting.

[@nicholasmaniccia1005]

Thank you this was a great introduction. After learning basic Calc and some Linear Algebra (Still working on matrix operations and their applications), I bought The Princeton Companion to Mathematics. I loved the 2nd chapter with all the histories of the major modern math branches like number theory, analysis, geometry, and algebra, etc. As I learnt more math I kept brushing up against this term 'abstract algebra' (amongst others) and thought it was some seperate or parallel subject to classical algebra. But TPCtM really cleared that up and showed the ancient roots up to Galois and beyond, but you did a great job of introducing what exactly Galois discovered. It was nice to read a more historical approach and then have this bridge over to the theory and application side. BTW Cardano did have knowledge of the quadratic formula. His work just didn't focus on it and instead built upon that knowledge. From the way TPCtM most of these guys were thinking in parallel and often were rediscovering work due to not publishing as a norm like you said, and of course the limitations of pre modernity. They also often were like Edison and the lightbulb, smart and often considered genius but kinda just doing trial and error and seeing what happens either haphazardly or systematically. the perfect example of this is Descarte. Fermat, I believe is also credited with what we now refer to as the cartesian coordinate system. And Descarte evn brushed up on imaginary numbers but didn't really "discover" them. Of course Fibonacci and I believe a guy before him actually brought over Arabic numerals, or their were a few attempts before it stuck. Kinda like how your guy Diophantus was way ahead of his time and Al Khwarizmi rediscovered algebra on his own. Or like how Pythagoras didn't discover the theorem after his own name. But either he, or someone who like him enough to attribute it to him, was the first to document it in such a way that it became easily consumable to spreadable knowledge. Not to discount any of the genius of these historical but it was often a complex slow moving, torch passing, group effort and even something like "ideas through osmosis". of course there is even the considerations of women and as modernity approaches different non european ethnic minorities living or interacting with the west contributions being overshadowed. Very often women like Ada Lovelace existed and they became "spinsters" or "old maids" but were high educated and often ignored and unrecognized outside of their immediate families and probably had to have work accredited to male members if it was eve even discovered. War and Peace by Tolstoy has a character like this it is one of the main male character's sister, and it is based on archetypal or stereotypical but real people just like every character in that book. I could continue but I believe the point is made haha.

Wow, thank you for this extensive comment with all that valuable background information. Amazing! I've never heard of the TPCtM before - seems like a book well worth reading. Hope you will follow this algebra-series - the next video (appearing saturday) will give a glimpse into matrix representations of symmetries.

I just browsed through the online version of the PCM sites.math.rutgers.edu/~zeilberg/akherim/PCM.pdf What an amazing book (imagine what it must be like to really understand all the different areas mentioned therein) and perfect reference for historical facts. Thanks again for telling me about it.

[@nicholasmaniccia1005]

Your welcome! I already subbed and will definitely be following this series.

@@nicholasmaniccia1005 👌👌

[@abc36196]

Extremely clear exposition, awesome!

Thank you so much!

[@ogiritmat]

Your dedication to teaching has made a significant impact on my academic journey. Thank you sir✌️📚

I'm delighted to hear that - thank you very much. And the best of luck to you with your ongoing academic journey!

[@idrissaso8020]

sehr schön erklärt und der historische Aspekt ist sehr interessant. Danke

[@nafisfaisal5817]

Excellent video. Thanks a lot for the overview of all those algebraic structures!

Thank you, glad you like it.

[@ihmejakki2731]

Seems like a very cool course, hopefully I will have time to follow it through! :)

I hope so too. If not at first try, the videos will be waiting for you... :)

[@TranquilSeaOfMath]

Your handwriting on the board is impressive! I love chalkboards.

Thank you. I do too, I always admired the blackboard writing of some of my high school teachers. Too bad chalkboards are being replaced by "smartboards".

[@karanshome]

You have such beautiful handwriting

Thank you very much. How about the size of the letters - is it easy to read or too small?

[@karanshome]

I thought they were easy to read, though to be completely fair I knew most of the words already, I'm not sure what someone who was seeing these words for the first time would think.

Thanks, that's helpful. The next couple of videos are already finished, but after that I'll try to zoom in a little more.

[@karanshome]

Yeah of course! Glad to be of help!

[@mnqobimsizi4328]

WHO IS THIS MATHEMATICIAN 🔥🔥🔥🔥 KEEP IT GOING

[@johnwarren8032]

Thank you.

[@yakpopo]

freut mich sehr dass das video aufmerksamkeit bekommt!

Danke sehr - mich auch. Bin tatsächlich gerade ziemlich geflashed, wie viel Aufmerksamkeit ... (hab in den letzten zwei Tagen so viele Abonnenten gewonnen wie sonst in einem halben Jahr 😮)

[@tonyb8660]

a friendly introduction to algebra is like a warm greeting from a gallon of Golytely

🤣

[@murdock5537]

Many thanks for the "friendly" introduction, Sir! Great vid. In my opinion, there is no need for a German version. Go ahead! 🙂

Thank YOU, sir, for the kind words. 👍😀

[@JoannBaron]

Really good video !

Thank you so much, glad you like it.

[@fzzdi4435]

Just to flesh it out a bit: algebra becomes abstract when we are element-agnostic w.r.t the set under consideration. The nature of the elements of the set is left unspecified except for the constraints given through axioms. That's why abstract algebra is so widely applicable both within mathematics and to other disciplines. This ties in with Russell's remark that mathematics is the subject where we don't know what we are talking about; only here, not talking about anything in particular means talking about everything in general, in a sense.

Wow, that's deep - thanks. Love Russell's remark.

[@MDNQ-ud1ty]

That isn't really a great way to put it. It is true but Abstract algebra is a generalization of "Concrete algebra". It's putting well developed algebra in an abstract framework so they are more applicable things that are not numbers. After all, the integers are element agnostic. The point of it all is to develop and see the internal structure that exists in numbers(which happen to exist throughout the universe because, well, numbers are fundamental to structure). Also, it is true of all generalizations in mathematics. The very act of generalization means you are going to treat things more abstractly. E.g., Topology is "element-agnostic". Category theory is element-agnostic. The point of mathematics is to understand the patterns in structure(else it becomes some type of applied field). To do this one has to abstract away the details to reveal the underlying structure and then do that until one see's enough of the machine underlying a valid theory that it becomes general enough to apply to other things. E.g., Lawevere has a great analysis: A map of sets is a process for getting from one set to another. We investigate the composition of maps (following one process by a second process), and find that the algebra of composition of maps resembles the algebra of multiplication of numbers, but its interpretation is much richer. Basically abstract algebra is the adult version of the kids version of algebra. it was a natural process to learn to generalize as teachers would constantly be teaching algebra and over time seeing deeper patterns. So I wouldn't say it is "abstract because it's element-agnostic". It is abstract because it uses an abstract language. The very word "element" is abstract. As I said, the integers are abstract. What makes the theory work is that it creates a template that can be used in other places. If you use the language of "algebra" and talk about integers and independent variables then can't apply that to rationals or reals or operators or whatever. By abstracting the theory of algebra one has a much more general and useful tool. At worse it is only applicable to the the original theory, at best it is applicable to much more and provides a general machine, that once one knows it, can be used to solve other problems. It is an algebraic machine and so the other problems will become more familiar since they will then be solved "algebraically"(using the ideas of zeros, polynomials, etc). Really, what your are talking about is set theory itself. The power of set theory is that it abstracted the idea of "element" and "container"(things that were not abstract before and one can't even talk about sets and elements without being abstract). People without mathematical institution will typically understand only tangible or concrete things and everything will be different from everything else to them. They won't understand that a collection of 5 books has a relation to a collection of 5 apples because to them those are very different things. When you think abstractly and ignore the specifics then what you have is a collection of things or sets and elements. Then you study the abstract relationships between those things(such as cardinality, what it means for something to be apart of something else, recursive usage of the terms, what are the axioms underlying our usage, etc). Pretty much once you start doing math you are actually doing set theory. When you are doing arithmetic you are implicitly working with sets. Over time that implicit structure becomes explicit and you start thinking in terms of variables(which are just sets or types) and equations of unknowns become more generalized arithmetic equations (x + 3 = 5 is both an arithmetic problem and an algebraic problem). Everything evolves naturally and it is from the specific to general. But modern mathematics is really learning how to properly generalize things. Category theory is the codification of this and really is the epitome of mathematics. This is really true of all things. If you are a programmer, engineer, etc... it has to do with how knowledge works. As you learn more you put pieces together and like a puzzle the more you do it the more the picture is revealed. It's impossible to escape except be lobotomized or not to learn anything(which is impossible but it can be slowed it down greatly). The way I see abstract is the same way I see a meta language. E.g., in many programming languages you have a way to program at a higher level so you can do things that are very difficult or ugly to do in the main language. This is just like algebra. 1 + 3 = 5, 2 + 3 = 5, -5 + 3 = 5, etc is represented as something like ℤ + 3 = 5. in a more abstract sense. It is easier to write and sometimes easier to understand. So abstract algebra is the meta algebra language. Category theory is the meta meta language of algebra and also of topology, number theory, etc. Just as abstract algebra is is a meta language of number theory too as it is a meta language of symmetries, etc. The point being is that it's far more than just "element agnostic".

[@bluefly1]

First time watching, it's good

Thanks - hope you'll enjoy the complete series as well. Next video is coming in a few hours.

[@campbellmorrison8540]

I did Abstract Algebra 101 in my maths degree and I never got a good understanding of it, but I scraped through so I await your next video in the hope it clarifies some of my worries about this subject :)

That would be wonderful - if at least some "worries" could be clarified through this series. Keep me posted. As a beginner, I struggled as well and only after returning to this subject after a year or so, things became a lot clearer. Did you find the playlist? (Almost) every saturday there will be a new lesson.

[@campbellmorrison8540]

I didn't but I shell recheck thank you. Yes Im 70 now and completed my degree around 50 extramurally. I enjoy watching the lessons now available, things seem so much clearer now I am at least familiar, or even just aware, with the subject. I had no idea what abstract algebra was when I did the paper so just getting my head around the concepts was enough let alone actually having to do an exam. I worked most of my life in electronics and IT and only did the maths degree later in life out of interest and thinking if I ever did a Physics degree, which is what I really wanted to do, it would be the maths I would fail in. However I enjoyed it and it definitely widened my horizons but it also demonstrated I was no mathematician however I did get pleasure being able to at least read some of the text books I had never been able to decipher previously .

@@campbellmorrison8540 Here's the playlist: kzbin.info/www/bejne/ap-Uc5uNr8mJqbc Thanks for sharing your story - it was very interesting for me to read. The best of luck to you in your ongoing journey into maths and physics.

[@MDNQ-ud1ty]

It's worth going back through. I also did rather poorly in college. It is generally not taught well and most of the concepts are not given proper context and purpose. If you go through and work it out slowly you will see how nice it really is. It's worth learning well because a lot of mathematics uses group theory. It's really not difficult. You already pretty much know everything. The issue is that you don't know the language. It is very much like learning a natural language. It is difficult at first but gets easier the more you use it. Also, typically, many concepts that are useful are not understood until later. E.g., "Why is an ideal important, why should I care about it" or "What is the point of a nil potent element". But when you start dealing with algebraic problems you will start, AFTER you learn the concepts, to see them everywhere. It is best to actually just get a good book and work through it. Allenby is a good book but it starts with ring theory. But it explains things in context and if you are familiar enough with modulo arithmetic then it might be ok. It's best to have a few books and worth through them all at once and continually go back. E.g., read chapters 1-3, then start over and read more carefully and go through 1-5 skimming 4-5 then start over again. This may seem like it will take much longer but it won't because each iteration you will understand it better and retain the definitions better.

[@hakonbh]

Excellent!

Thank you!

[@georgesadler7830]

Thank you for the video.

Thank you for watching and commenting.

[@delhatton]

Good start

[@manunitedred8912]

Hello Professor where do we find the full English lectures. It says the videos are hidden ? Thank you for giving this wonderful knowledge to the world for free

Thank you very much for your kind words. Too much honor, though - I'm no professor but simply a school teacher. Every week, 1-2 new videos (plus solution videos) will get released, so that beginners have enough time to take in the new concepts and to work through the problem set of the week. So you have to be a little patient and follow the course every week. 😀

The new lesson is always released saturdays at 3 p.m. (German time zone).

[@vitalic_drms]

we may never know 🤔

😀 ... at least what algebra is going to become in the next decades.

[@roninlviaquez]

Is it accurate to say that algebra is the language of math?

That's a bit too narrow. A lot of branches of math use algebraic terminology, but you also have to "speak" the language of analysis, topology (which uses a lot of algebra) etc. to be a "complete" mathematician.

[@SvenErik_Lindstrom3]

2+x = 9?

Hilarious 🤣 That IS (one aspect of) algebra.

[@gargetang7620]

u are wrong sorry! El Jabr means not the reunion . it means difficult . u gathering up all fragments is a time-costing tedious job so it means difficulties.

Thanks for clarifying. I do not speak Arabic at all, so all I could do was to rely on my sources, e.g. Pinter's book or hsm.stackexchange.com/questions/12988/why-did-al-khwarizmi-use-al-jebr-the-reuniting-of-broken-parts-to-signify-alg

[@Unordinary-lg4yt]

As much as I would love to understand this upper level of math, the reality is it takes several years of going from elementary algebra/trig/geo, calc i-iii, linear, analysis, and topo.

That is true. But abstract algebra is a good starting point because you don't need much prerequisites (except for the matrix-stuff) and the new concepts in this course are thoroughly explained.

[@CharlieChan-l6u]

The he says he learnt from India

Who do you mean?

[@dieterdreier7109]

I cant tell you why you follow this trashy math.?

Why write trashy comments instead of just watching something else?