Adventures in pure math
Introduction — what is math?
During my undergrad, Abstract Algebra was my Nemesis. My undoing.
The program I was pursuing was oriented toward theoretical physics, which makes a lot of use of some pretty esoteric and abstract math. It included a lot of pure math at the core.
Mathematics is a weird subject. It’s probably the only familiar subject where it’s possible for a good student to go into it after high school or CEGEP without really even understanding in advance what the subject is about.
Most people think of mathematics as being about computation — about finding new mathematical objects and tools that can be used to solve practical problems in science and elsewhere. Calculus helps us model and compute systems with continual change. Linear algebra helps us model geometric transformations and solve systems of linear equations and inequalities in practical subjects like operations research and econometrics. Statistics lets actuaries and financiers manage uncertainty and risk. Computer scientists use graph theory to model things like networks, dependency relations, and inheritance hierarchies.
This is how most people think of math, and how most students encounter math up through high school, CEGEP, or freshman level university studies — as an essential tool to facilitate structuring situations in the real world for computation. Students at better-than-average high school programs will get a healthy exposure to the notion of mathematical proof and the importance of rigor. But even there it mostly comes across as a matter of reading the fine print in the user manual of the toolsets.
Now, to be fair, that is an essential part of math. Applied math is a thing, and is that part that really focuses on giving tools to solve the world’s problems. There is a reason why math has been, is, and will be one of the best undergraduate minors out there to complement any remotely quantitative subject, but particularly in physics, economics, computer science, and linguistics. And again, to be fair, math is often inspired and cross-fertilized by theoreticians in various practical fields; the theoretician puts together something hand-wavy with glue and duct tape to understand a problem, and then mathematics later formalizes and sanctifies it on the holy altar of rigor.
Pure math
But, nevertheless, at the core of Mathematics are areas of study and people who operate, proudly and defiantly in towering abstract worlds of pure Platonic forms, studying the structure of mathematical objects and their inter-relations, not as a means to an end, but as an end in itself.
I was always better at the applied math parts. Calculus, Linear Algebra, Vector Calculus, Complex-valued Functions, Differential Equations — I enjoyed and did reasonably well in those courses. But there’s no real admission to the sacred fraternity of Mathematics without a solid comfort with Pure Mathematics, which means Analysis and Abstract Algebra.
Analysis was tough, but it was basically just a super formalized and rigorously precise version of something I had already studied and understood — Calculus. Algebra was something else entirely.
Algebra — it’s more than x and y (much more)
Algebra is sort of the poster-child for what I said above about most people outside of university math programs not even knowing what math really is.
Algebra for most people they hear it and think — “Oh, OK. Letting letters stand for numbers and solving equations with that. I did that in high school. What’s so hard about that? You run out of letters?”
But that’s not what Abstract Algebra is about. High school algebra uses some ideas in algebra, but implicitly. The abstract algebraic properties of integers, rational numbers, real numbers, and complex numbers are why high school algebra works.
To understand the point a little better, take a high school algebra problem.
x² + 4x + 3 = 0
Now, a decent high school student can use factoring to recognize that this is equivalent to:
(x + 3)(x + 1) = 0
The high school student will do this automatically, usually because their teacher did a drill and kill of a method and they recognize the pattern. This quadratic pattern factors into a product of two monomials. The two constants have to multiply to give 3 and add to give 4, which means 1 and 3. But why does that work?
Behind the scenes, you’re using several properties of an algebra system you’re working in.
The set of polynomials with integer coefficients has certain rules for multiplication and addition, and certain properties of allowing a unique factorization, so that if you factor, that is the right way to do it, similar to how integers have a unique prime factorization. Certain polynomial chunks break down, like composite integers, and some don’t, like prime numbers.
Furthermore, the factoring works because the variables represent real numbers, which commute and have associative and distributive properties.
Finally, the high schooler solves the problem by observing that if
(x + 3)(x+1) = 0, then either
- (x + 3) = 0, => x = -1 or
- (x + 1) = 0, => x = -3
Because non-zero integers don’t multiply to give zero.
Here’s where things get a little weird
This seems almost too obvious to mention — that’s just how numbers work, you say— but in reality this doesn’t work in some systems that behave a lot like normal integers, but just don’t have this property.
For example, “clock arithmetic” or “modular arithmetic.” Hands go around the clock, stepping up by one, until it gets back to 12, after which it returns to 1. The 12 serves the same function as the number 0 in the integers, and the numbers cycle back around. So if you call 12 ≡ 0 in this system, you can make a self-contained system with the numbers (actually, classes of numbers, to be precise) {0, 1, 2, 3 …. 10, 11}.
The triple line equals sign symbol ‘≡’ means “congruent to.” We say two values are congruent modulo some value if they have the same remainder when dividing by that value. For example, we say 14 ≡ 2 (mod 12), or “14 is congruent to 2, modulo 12,” because they are both 2 more than a multiple of 12.
So in this system, the “numbers” are groupings of integers based on the remainder on dividing by 12. So:
- {0, 12, 24, 36 …} all belong in the class 0
- {1, 13, 25, 37 …} belong in the class 1
- {2, 14, 26, 38 …} belong in the class 2, and so on.
These “congruence classes” {0, 1, 2, … 11} partition the set of integers into 11 equal subsets in a way analogous to how “even numbers” and “odd numbers” partition the integers into two equal subsets. Mathematically, we call this set of classes Z/12Z, the integers mod 12. The division notation represents the reality that we’re dividing the set of integers into equal subsets using 12Z, the set of multiples of 12.
In fact, evens and odds are congruence classes, modulo 2. Even numbers are congruent to 0, mod 2, and odds are congruent to 1, mod 2. Evens and odds are actually Z/2Z.
You can add, subtract, and multiply these classes as if they are integers. For example, if you take any integer that leaves a remainder of 2 on dividing by 12, and add it to any integer that leaves a remainder of 3, you get a number that leaves a remainder of 5. 2 + 3 ≡ 5 (mod 12). And similarly, 2 * 3 ≡ 6 (mod 12).
In fact, you can show it all works a lot like normal integers, as long as you take into account how things work when you go all the way around the circle and loop around again. Again, there is an analogy to our common understanding of odds and evens in this closure under addition and multiplication for congruence classes. An odd + an even always equals an odd, an odd + an odd is an even, and an even + an even is an even. Similar for multiplication. Even * even = odd * even = even, and odd * odd = odd.
So, for example:
- 6 + 7 ≡ 1 (mod 12), because 6 + 7 = 13 = 12 + 1, and 12 ≡ 0 (mod 12).
- 6 * 7 ≡ 6 (mod 12), because 6 * 7 = 42 = 3*12 + 6.
Solving equations in modular arithmetic
I think most people can follow along with this well enough after a lifetime of using clocks, but imagine you took the same algebraic equation as before, but modulo 12:
(x + 3)(x + 1) ≡ 0 (mod 12)
In this system, the integers modulo (mod) 12, you can’t pull off the same trick of concluding that one of the two terms has to be zero.
Don’t misunderstand: the solutions that come from that approach are still valid — x ≡ 9 = 12–3 and x ≡ 11 = 12 -1 are solutions.
But they’re not the only solutions, modulo 12.
Because in modulo 12 arithmetic, we have things like:
6 * 4 = 24 ≡ 0 (mod 12)
8 * 6 = 48 ≡ 0 (mod 12)
12 * 10 = 120 ≡ 0 (mod 12).
In modular arithmetic, a lot of the time, non-zeros can divide zero, or zero divisors. Which complicates things.
But … there’s another fun twist to this story. If the modulo is a prime, then everything works out nicely as you would expect. For example, if you look at Z/5Z, the equation solves normally, with solutions x = -1 ≡ 4 (mod 5) and x = -3 ≡ 2 (mod 5), and that’s it, because there are no zero divisors when the modulo is prime.
Moreover, modulo 5 (or any other prime number), things work actually even better than you would expect for an integer. For every non-zero value in Z/5Z, there is a multiplicative inverse — something that multiplies with it to give 1. So:
- 1 * 1 = 1 ≡ 1 (mod 5)
- 2 * 3 = 3 * 2 = 6 ≡ 1 (mod 5)
- 4 * 4 = 16 ≡ 1 (mod 5)
This means you can always do division (except for division by zero) without leaving the set. For normal integers, this only works when you extend the integers to the rational numbers. So this is weird and interesting stuff.
Introducing … Abstract Algebra
Abstract Algebra in general studies abstract types of objects with operations that obey certain general properties.
So, for example:
- A Group is a set of objects with one binary operation and certain properties so that it behaves similarly to integers under addition. Examples: integers, integers modulo n, symmetries of a polygon or polyhedra, permutations of n items, matrices under addition, 2 and 3 dimensional rotations
- A Ring is a set of objects with two binary operations, and certain properties so that it behaves similarly to integers under addition and multiplication.
Examples: integers, rational numbers, integers modulo n, polynomials, power series, square matrices under addition and multiplication, set of functions under pointwise addition and multiplication - A Field is a set of objects with two binary operations and certain properties so that it behaves similarly to rational or real numbers under addition and multiplication, and also allowing division
Examples: integers, rational numbers, real and complex numbers, integers modulo a prime number, the set of nth complex roots of 1, rational functions - A Vector Space is a generalization of a set of vectors with operations of vector addition and multiplication by a scalar (This is the one Abstract Algebra object that really crossed over to general STEM undergrad awareness via Linear Algebra and its applications to linear transformations, differential equations, Fourier series, and other applications)
Examples: Regular n-dimensional vectors with real or complex components, set of solutions of a system of linear equations, functions of real or complex numbers in general, polynomials of given degree with real coefficients, matrices of given size with real values, solutions of linear ordinary or partial differential equations
The beauty of it is when you get a good minimum set of core properties, and you can prove results based on those properties, the results apply to anything with the same properties. For someone familiar with Computer Science, another symbolic discipline where abstraction is important, it’s sort of analogous to abstracting away a commonly useful class or interface that is inherited by a subclass or implementing class so that you don’t repeat yourself defining common functionalities.
At a very basic level, this is not too bad to understand, and up to a certain level, I kept up with it by clinging to more common concrete examples. But it all gets very abstract very quickly, and at a certain point I found my mind just didn’t seem to be built like that. I mean, I’m partly built like that — my wife likes to joke that I’m half robot. But the real, natural pure math guys … they are yet a different breed beyond this.
So I turned toward other paths where I could combine my math and science skills with my abilities to write and communicate ideas. Which by a wandering path got me eventually into Instructional Design and Technical Writing. But I always find myself, here and there, when the time allows, poking back at pure math stuff as a pastime, just to stretch the limits of my understanding a little farther.
Love (of math) in the time of quarantine …
Fast forward a few years … and with the recent Covid-19 business, the closure of my kids’ schools, and my own added time at home. I have found myself looking for resources to help tutor my kids so they can keep their minds sharp during the downtime. I have been spending about an hour a day every couple of days for the past few weeks doing little enrichment classes with my oldest on topics in math. I want to do this for the remainder of the normal school year period. It’s just been different basic high school sorts of topics — combinatorics and binomial theorem, quadratic equation, complex numbers. Later on, we’ll probably look at some trigonometry, exponentials, maybe some probability, vectors and matrices, some baby calculus ideas, and some basic ideas of proofs (induction, contradiction, etc) and logic.
I was also hoping to sprinkle in some bits and pieces from Number Theory and Abstract Algebra, just as an introduction. Not for mastery or anything, but just so that he has seen it before going to university, a little challenge he probably won’t see in his high school classes. (Whenever those start up again).
Also just to alleviate my own boredom a little. I actually enjoy teaching math quite a bit. Back in the day, I was actually enrolled in a teacher training program for a couple of years before switching to the Educational Technology Masters program that led to my working career in Instructional Design and Technical Writing.
So to help out with this effort, I took a look on YouTube to try to see what is available in terms of short videos we could possible watch together and discuss.
In the process I stumbled upon a fantastic channel called Socratica:
This channel has educational material on a lot of subjects related to math, science, and other disciplines.
Included in this is a full playlist of 33 videos on Abstract Algebra. The videos are short, ranging from a minimum of 2 minutes up to a maximum of about 11 minutes, and are well-produced, with good visuals and examples, and brought to life by an actress narrator. The videos give a good introduction to key ideas in basic structures of Abstract Algebra like Groups, Rings, Fields, Vector Spaces, and Modules.
These videos don’t go into a huge amount of depth, but where I think they do a good job is in communicating in a simple way the background of why these abstractions are useful, and some of the historical development of why mathematicians would decide to think about it that way. This is something that often isn’t well done in university math classrooms. Often, in these classrooms, the focus is on just throwing out definitions and then mechanically proving theorems based on the definitions. The approach on this channel really brings the subject to life, and motivates deeper exploration.
For a deeper, but more traditional look at pure math on YouTube, I would recommend the YouTube channel of Michael Penn, a professor in Virginia:
He has an extensive set of playlists of videos on Number Theory, as well as playlists on Abstract Algebra.
He also does a mean backflip from time to time:
Just … because, I guess. Like I said, math people.
This other guy (Math Doctor Bob, A.K.A. Robert Donley) also has seriously impressive playlists of pure math content.
I also recommend the following playlist from the channel Mathemaniac, which gives a nice intuitive approach to some basic ideas in group theory.
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