I’ve been working most of today on the foundations of my Modern Algebra course for the fall, which you will remember is being done without a required textbook. The last time I posted about this course, I made the point that not having a textbook around forces the professor to think about certain deep issues that often get glossed over — for example, what the course is all about and how the topics connect together and reinforce the main questions of the subject. Good college professors always think about these things regardless of textbooks; the absence of a textbook, then, forces the professor to think like a good professor. And that can’t be bad.
Along those same lines today, it occurred to me that there are two particularly useful ways to consider the entire subject of Modern Algebra.
1. Modern Algebra can be thought of as the study of why algebra, as the students know it, works. For example, we all “know” that if 0x = 0 for all x. But how many college students have thought about why this is true? It’s easy enough to give a hand-wavy explanation that ends up sounding like a 60’s hippie anthem (e.g. “Nothing times x can’t give anything but nothing!”), but is there a more solid mathematical explanation? The answer is yes, and surprisingly the thing that makes it work is the distributive property. So we find that the distributive property of multiplication over addition is a deeper property of the real numbers than any supposedly mystical properties of the number 0.
This view of Modern Algebra makes the subject immediately applicable to education majors, who make up the majority of the students I’ve had who take this course. Some smart kid in their class someday is going to ask why 0x = 0. Don’t insult that kid’s intelligence by offering half-baked mysticism — it’s a real question, so give them a real answer. Or at least convey to them the idea that there are real reasons why things in math actually are true, and we don’t have to rely on blind faith or the instructor’s (or textbook’s) authority.
2. Likewise, Modern Algebra can be thought of as the study of finding the largest possible collection of structures for which algebra, as the students know it, works. I mentioned 0x = 0 above. But I didn’t say initially what kind of thing x is, nor what I meant by “0”. There are lots of things we denote with “0” in math — the zero vector, the zero matrix, the zero polynomial… What kind of thing is it now? And more importantly: Does it matter, or is this a property that is true for any situation where “0” and multiplication make sense?
On the other hand, everybody “knows” that if ax = bx and x is not zero, then a = b. Right? Well, what we find out is that it depends. It depends on what the a, b, and x represent. If they are integers, real numbers, or polynomials with real coefficients, then a = b always in this situation. But if they are 2×2 matrices, then it’s quite possible for a ≠ b — just let a and b be any nonsingular matrices and x any singular matrix. So it does matter here exactly what kind of object a, b, and x represent. So we make an important discovery: There is a limit to how far we may apply the notion that ax = bx implies a = b. It doesn’t always work. Which leads to the next question — under what conditions will this implication hold? What algebraic conditions have to be in place to use this simple cancellation idea? This is the right kind of line of questioning that a junior/senior in mathematics, especially somebody in math education, ought to be pursuing.
Taking Modern Algebra and casting it in terms familiar to students like I’ve done here seems to be crucial to crafting a course that isn’t just going to be written off by students once the exam’s over.
: 0x = (0 + 0)x = 0x + 0x. Subtraction gives 0x = 0. This proof works in any instance where multiplication and addition are defined, additive inverses exist for each object, and multiplication distributes over addition. The ring is the kind of structure we usually think of here.