Algebras we love

I had a semester of abstract algebra as an undergrad, and I&#x27;ve always been surprised by how many dividends that semester has paid back over the years. The tools of higher algebra are very powerful in the right situation. This article is a great development of algebraic concepts and introduction to thinking algebraically, which is a really powerful tool to have in your toolbox.<p>(Closely related is the idea of invariants: properties that are preserved by particular operations or functions. Often invariants are related to some algebraic structure of the system, but can be easier to identify and support a lot of the same insights. Reasoning about invariants of systems is another great way to make progress on hard problems.)<p>I find that very few engineers (especially in hardware) have had exposure to this stuff. Being the only one in the room who&#x27;s had an abstract algebra course means I&#x27;ve occasionally been able to provide a completely different line of attack on hard problems. This has been good for my career!<p>As an example, I once helped a friend debug a complex system that was not behaving correctly. There was an input state, a nasty sequence composed of simple operations applied to that state, and an output state, which was not behaving as expected. A bit of algebraic thinking showed that each of the simple operations preserved an invariant, so <i>no</i> sequence of valid operations, no matter how complex, could produce that output. This meant that debugging attention could be directed at the implementations of those simple operations, which led to finding bugs in short order. This saved a lot of work because the actual sequence came from elsewhere and would have been difficult to audit!

I&#x27;ve had 4 semesters of abstract algebra because I was a math major. I found even the first two semesters difficult courses. Math had always come easy to me when I was young so I didn&#x27;t really understand that at the university level mathematics would require some serious study; naturally, I would have gotten a lot more out of the early algebra classes if I had put more work into them.<p>I had much better intuition for analysis than for algebra. The results in algebra are just more surprising than those in analysis for me. My attempts at proofs in algebra were kind of like random-walks where I would eventually stumble upon the answer and then I would have to reconstruct the logical steps to get there without all of the unnecessary circumlocution.<p>Years later, as a much better student, I took a graduate course in group theory and really enjoyed it because I actually spent some time studying the subject.<p>I really love the way that abstract algebra deals with such simple, almost meager entities: sets with just a few basic operators. The theorems about these completely abstract, virtual and not actual things, reveal properties that are foundational for all math and somehow underly our reality.

When thinking about re-learning math I always think of doing it through some Computer Algebra System. It seems to me like a way better way to learn math, allowing me to tinker with things and treat subjects as black boxes (what&#x27;s the ouput of X process if I change the input to Y?) as opposed to just try to understand the static examples presented on paper.<p>The problem is the embarrassment of riches [1].<p>Compounding the problem is the awful marketing some of this products have for casual and hobby learners. At this point, I&#x27;m willing to spend a few hundred dollars for a personal edition of a product like Wolfram&#x27;s Mathematica, Matlab or Maple, but I&#x27;m not sure what would be the best investment for my time and money.<p>Could you recommended any courses or books using a CAS to teach math concepts and applications?<p>1: <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;List_of_computer_algebra_systems" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;List_of_computer_algebra_syste...</a>

I really dislike the standard definition of a group - that it&#x27;s a set S together with a binary operation * such that a bunch of properties hold. The definition doesn&#x27;t build intuition, and doesn&#x27;t motivate the introduction of the concept of &quot;group&quot;.<p>For me, a group is the set of isomorphisms of a graph. If you expand the definition of &quot;graph&quot; a little to include continuous spaces, that is sufficient to define all groups. And yet, this nice, intuitive definition of a group <i>might</i> show up at the <i>end</i> of a course in group theory - if you&#x27;re lucky.<p>It really is a shame how much intuition is stripped from mathematics teaching in the name of formalism.<p>[0] <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Frucht%27s_theorem" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Frucht%27s_theorem</a>

As someone who have struggled to self-study abstract algebra, I do find subject lacking in &#x27;motivation&#x27;, Eg: why do we define groups, fields, in the way that they are defined? Why do we need them at all?<p>Also most treatments do not cover categories, which seems to have its own separate literature.<p>Perhaps having some geometric intuition will greatly help, viz coordinate systems in 3d Eucliden space, answering interesting questions like when is it legal to take dot product of two vectors? and if two unit vectors are parallel are they the same vector?

&gt; Additive and multiplicative notations are used, when your set A can have 2 different algebras defined for it, e.g. real numbers are group for addition: (R,+,0) and multiplication: (R,∗,1). 2 different notations help us keep trace which group we are talking about at the moment.<p>(R,∗,1) isn&#x27;t a group, (R_+,∗,1) or (R\{0},∗, 1) are, but that doesn&#x27;t really work as an example of two different algebras defined on the same set. Having your operations obey the distributive property is incompatible with that kind of structure other than the zero ring (0=1).

This is a fantastic introduction to abstract algebra. Highly recommended.

I&#x27;m surprised how good of an investment learning algebra is. I never learned any branch of math that restructured my thinking process as radically as algebra. It even surpasses discrete math and probability. I think every undergrad should take a standard algebra course with groups, rings and fields.