Tag Archives: algebra

Mathematics Program, Excursion I: Polynomials

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A photograph of a stack of seven small math books on a wooden table (authors and titles in post).

I know like two of you care about my pretensions, but (as much for me) here is a math study program update: I still have a long row to hoe before I am done with my textbooks on algebra and pre-calculus. I am actually pretty unhappy with how slow I am going. But I feel like I am at least over the hump of regaining lost understanding. But this far in, I have had three thoughts: (1) I am really enjoying polynomials; and (2) while I have done okay with these textbooks, I feel like what I have gained so far is merely a grab-bag of a few dozen tricks, not understanding; (3) since polynomials was once one of the primary areas that drove mathematicians crazy for literally hundreds of years, and solving its problems was the origin of group theory and abstract algebra, I feel like I should not treat this as merely on-the-way to calculus, but should spend a little time engaging with the subject of polynomials on its own. So I have gathered together this little pile of books for a study program on polynomials. They are:

  • Irving, Ronald S., Beyond the Quadratic Formula (Mathematical Association of America, 2013)
  • Andreescu, Titu and Dorin Andrica, Complex Numbers from A to … ℤ (Birkhäuser, 2005)
  • Barbeau, E.J., Polynomials (Springer, 2003)
  • Uspensky, J. V., Theory of Equations (McGraw Hill, 1948)
  • Stewart, Ian, Galois Theory (Chapman and Hall, 5th Edition, 2022)
  • Victor J. Katz and Karen Hunger Parshall, Taming the Unknown: A History of Algebra from Antiquity to the Early Twentieth Century (Princeton University Press, 2014)
  • Livio, Mario, The Equation That Couldn’t Be Solved: How Mathematical Genius Discovered the Language of Symmetry (Simon & Schuster, 2005)

Some brief notes on these books:

  1. The first five are mathematics books proper, and the last two are histories of mathematics.
  2. All of these books either claim in their introductions, or have been assessed by reviewers, to be at the level of an ambitious high school student.
  3. I think I am considering Barbeau’s Polynomials to be my primary text, with Irving’s Beyond the Quadratic Formula as a supporting text.
  4. Uspensky’s Theory of Equations is here for antiquarian interest (theory of equations is what the study of polynomials used to be called). Older books tend to have very different problems and I want to see how people used to think about polynomials.
  5. I dipped a little bit into Uspensky’s book when I first got it and quickly found myself overwhelmed by issues of complex (imaginary) numbers, so added Andreescu and Andrica’s Complex Numbers from A to … ℤ to my study program as a preliminary.
  6. Stewart’s Galois Theory will take me from the core study of polynomials, up through the beginning of abstract algebra. Again, I have read that Stewart’s book is so measured and stepwise, that even an intermediate reader can follow his presentation.
  7. Katz and Parshall’s book is a complete history of algebra, from cuneiform tablets to modern abstract algebra. Livio’s book focuses on the quandary of the quintic equation (a polynomial of degree 5) and, the two geniuses of Niels Henrik Abel and Évariste Galois, and the advent of group theory and abstract algebra.

Of course there is the whack-a-mole problem of prerequisites. One of those is already addressed by the book on complex numbers. But complex numbers at least seem core to polynomials. My real problem is that all of these books include in almost every one of their problem sets, questions asking you to prove this or that theorem. And being the completist that I am, I am not about to skip a whole class of problem. And besides, proofs seem integral to the depth of understanding I seek. It had been my intention to put off learning the techniques of proofs until somewhere in the midst of calculus. But I guess after my basic algebra books, I will work through a book or two on proofs before proceeding to this program.

I shouldn’t be disappointed. A significant motivation for this project has been the stature a certain collection of books hold in my mind. And one of those books, I book a spend a long time wondering over on a particularly memorable day spent in the University of Washington book store, with its amazing collection of math books (oh, to be able to go back), is Chartrand, Polimeni and Zhang’s Mathematical Proofs: A Transition to Advanced Mathematics (2nd ed., Pearson, 2007). I don’t even know if it is a good book on proofs. I just know that it has loomed in my imagination for 30 years. I am going on vacation next week and it is coming as one of my vacation books.

My Undergraduate Mathematics Study Program

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A photograph of three textbooks fanned out on a wooden table. They are Algebra 2 by Ron Larson and Laurie Boswell, College Algebra by Robert Blitzer and Precalculus by David Cohen.

I think my project for myself for the next couple of years (probably decade) is to work through the undergraduate sequence in mathematics. My goal is to get to the point where I can read a book on general relativity and on quantum field theory, as well as a few other math-heavy topics. So what I would like to do is:

  1. Calculus I, II, III
  2. Proofs
  3. Linear algebra
  4. Ordinary differential equations
  5. Abstract algebra
  6. Real analysis
  7. Partial differential equations
  8. Complex analysis

I don’t know where I am going to run out of steam, but if I want to get to general relativity, I will need to do a little more than this, at least some differential geometry. But I don’t know. That’s next level.

The standard sequence starts with calculus. I have taken calculus I & II twice in my life, once in college and once about 15 years ago. I started calculus III in college before temporarily dropping out. So even starting calculus over again is backing up for me. But when I took calculus, I was acutely aware that I wasn’t doing as well as I should have been because my algebra and trigonometry were too weak. So this time I want to make sure those skills are solid before proceeding. At first I decided that I wasn’t even ready for pre-calculus: I needed to go back and work on an algebra book first. The book I got is a college algebra book. Its first group of chapters is a review of intermediate algebra. After working through most of those chapters, I felt like I was too weak in even my basic algebra to proceed with this college algebra material. I could probably proceed and continue to improve basic skills as I went along, but having been burned by inadequate preparation the first time through, I want to make sure I am solid in one skill before moving on to the next. So now I am working through an intermediate algebra book.

It is kind of frustrating that I am moving backwards so quickly. At this rate I am never going to make it back to calculus. At this rate it’s going to take forever to get through calculus. It’s going to be years to get back to where I was thirty years ago. I am desperate to get to something new to me. I remember taking calculus and it just being so mind-blowing. I imagine since I did so little calculus III, it may have a similar effect (I remember the first time I saw a curve rotated about the x axis in the z plane and being, again, mind-blown).

I have spent my entire life wanting to be good at math. When I was a teenager, I wanted to be a physicist. In my early 20s, I wanted to be an economist. But for some reason I convinced myself that I couldn’t hack the math for either of these. Which is weird, because while I was never in the top math class in school, I never struggled with math.

But in the last few years I have decided that math isn’t an innate capacity, or at least requiring some core of innate ability. Well, maybe for like the top one percent of achievers there is some brain difference, but for middling achievement, what I am aiming for, it is a skill, like plumbing or painting: something that you get better at by practice. It is my plan to focus on doing the exercises and grind my way to understanding.

The problem is that I have been living my life at quarter speed. I have always just thought that there will be time later. But I am 50 years old now and there is not time. At this point in my life, one of the limitations on this project is that I am on the downside of my cognitive ability. At some point relatively soon, I am not going to be able to do this. I am spending the last 15 or maybe 20 years where it is still possible doing something I should have done thirty years ago.