For someone who has no interest in teaching mathematics, I sure have lots of opinions on how the subject should be taught. I think I know more about education than Secretary of Educatuon Betsy DeVos, but that’s not saying much.
At least President George W. Bush asked “is our children learning?” That’s a question Trump and DeVos seem to be completely unconcerned about.
There have been some rumblings that DeVos is on her way out, hopefully she’s replaced by someone at least halfway competent. Someone like Arne Duncan (President Obama’s Secretary of Education) would be too much to ask from Trump.
So, um, maybe Kaplan CEO Peter Houillon? As the chief executive of a test prep company, it's possible that he does know something about education. Even Michelle Pfeiffer would be an improvement over DeVos.
Anyway, a year ago, I expressed the opinion that fractals on the complex plane, like the Mandelbrot set, ought to be taught in school. This quickly gets into mathematical concepts that are beyond my own understanding.
Maybe rings of algebraic integers would be better? But that also gets into mathematical concepts beyond my own understanding, or perhaps outside my interest, like monomorphisms, isomorphisms, holomorphisms, homeomorphisms, etc.
Actual math teachers will let me know in the comments whether any of this is a good idea.
Very quick crash course on imaginary and complex numbers
As you already know, the square of a number is that number multiplied by itself. So 1× 1 = 1, 2 × 2 = 4, 3 × 3 = 9, and so and so forth. This basic fact enables us to solve equations like x2 − 1 = 0.
That’s easy, x = 1, Lisa Simpson (Yeardley Smith) might say. But Martin Prince (Russi Taylor) would then remind us that x = −1 is also a valid solution, since (−1)2 = 1 also. So how are we to solve an equation like x2 + 1 = 0?
If you put that question to most calculators, you’ll get an error, and you’ll have to press the C or CE key to do anything else. This suggests no solution to the equation exists.
But what if we imagine that a solution does exist? Let’s call this number we’ve just imagined i, for imaginary. All we know about this number at this point is that i2 = −1. Clearly i3 = −i. And (−i)2 = (−1)2 × i2 = 1 × −1 = −1.
Then, if we picture the “real” numbers on the horizontal real number line, and the “imaginary” numbers on the vertical imaginary number line, and numbers with both real and imaginary parts can be found at the intersections of lines emanating from those two axes.
A number like 1 + i is a solution to an equation like x2 − 2x + 2 = 0. The other solution is 1 − i.
There are also irrational numbers to think about. For instance, the equation x2 − 2 = 0 has two solutions. One is roughly x = 1.414213562373, the other is roughly x = −1.414213562373. We call one √2, the other −√2.
And likewise x2 + 2 = 0 also has two solutions: i√2 and −i√2, or, as I prefer to call them, √−2 and −√−2.
Imaginary numbers are quite real, as anyone who studies physics in depth can tell you. But the terminology has stuck and there’s no way to change it now. However, note that electrical engineers tend to use j rather than i.
Very quick crash course on integer rings
The integers …, −3, −2, −1, 0, 1, 2, 3, … form a “ring” of numbers, sometimes notated Z, a complete system “closed under” addition and multiplication, meaning that if you add or multiply any of those integers, the resulting number is also in the ring.
If we accept i as an algebraic integer in addition to the aforementioned integers, we have another complete system of numbers. Given a and b being integers of Z, all numbers of the form a + bi are in this other ring Z[i], sometimes called the domain of Gaussian integers, which is also closed under addition and multiplication.
We can do something similar with √−2: all numbers of the form a + b√−2 are in the ring Z[√−2]. This ring is also closed under addition and multiplication.
With √−3 we get to a little bit of a wrinkle: it turns out that −½ + (√−3)/2 is a solution to x2 + x + 1 = 0. This is a special number often notated ω (Greek lowercase omega).
Adjoining ω to Z gives us a ring which has unique factorization, something we should not take for granted (e.g., Z[√−5], and indeed almost all other imaginary quadratic integer rings).
The domain Z[ω] is often called the domain of Eisenstein integers. Any time d = 1 mod 4, the domain Z[√d] contains so-called “half integers” (the term is wrong but convenient).
Diagrams of Gaussian and Eisenstein integers
In Z[i] and Z[ω], besides 0, we have some numbers that are units, some that are prime and some that are composite (nontrivial products of primes). In these two domains we don’t have to worry about numbers that are irreducible but not prime (like 7 in Z[√−10]).
It’s not difficult to find diagrams of Gaussian primes and Eisenstein primes. The MathWorld page on Gaussian primes has such a diagram for Z[i], and likewise the MathWorld page on Eisenstein primes for Z[ω].
Ethan Bolker's book Elementary Number Theory: An Algebraic Approach has a much more colorful version of the Z[i] diagram on the cover. The artist is Prof. America Ferrara? I’ll have to look up her name and correct accordingly.
Diagrams of other rings
But what about other rings? The results might be boring, but without actually producing and viewing the diagrams, I can’t really say. Or I might be missing out on something very interesting.
So about five years ago, I painstakingly made diagrams for Z[√−2] and OQ(√−7) in Photoshop Elements by laboriously clicking around with the Pencil tool. Sometimes I didn’t quite click in the right spot, but I kept going without correcting.
It occurred to me back then that it would be more efficient to program a computer to make these diagrams. But it wasn’t until about four months ago that I actually started writing the program in Java.
Although I’m not ready to call it 1.0 (the current version is 0.71, available from my GitHub page), I would like to show off some of my results so far.
Indeed some of the diagrams are more interesting than others. Here’s one that really surprised me the first time I saw it:
It vaguely reminds me of a cat’s eye. The reason it surprised me is because a lot of these have a central cluster of primes and the primes mostly thin out the farther one gets away from 0.
A little reflection should reveal that a2 + 489b2 is often a nontrivial multiple of 5. That doesn’t fully explain the diagram, but in trying to look at similar diagrams, I came across this one which seems very thick with primes by contrast:
Quick explanation of the color coding:
- 0 is black.
- Units are white.
- Prime factors of d are green. This is a little bit misleading because the current version of the program does not actually do anything with ideals (principal or otherwise), but it helped me orient myself before I added the readouts at the bottom of the window.
- Primes presumed inert are cyan.
- Primes confirmed split are blue.
- The “half integer” grid lines are a dark gray.
- The “regular” integer grid lines are black.
- The background is a dark blue.
Perhaps in version 1.0 I’ll add the ability to choose different colors. I will definitely also add the ability to switch to theta notation for the readouts when d = 1 mod 3. And, perhaps most importantly of all, the ability to directly write the images to PNG files (I used Alt-PrintScreen for the images above).
The ability to drag the diagram around (like on Google Maps) might have to wait to version 2.0 or 3.0. Also, I need to complete the javadoc for all the public constants and methods before I can call it version 1.0.
Depending on your computer’s security settings, you might be able to download and run the JAR executable from my GitHub page. You’re also welcome to download the source code to your own system and compile it there if you like.
Also feel free to suggest new features besides the ones I’ve already mentioned, regardless of how easy or difficult you might think they are to implement. I’ve found that sometimes the features that seem like they would be easy to implement are actually quite difficult, and vice-versa.
Surely I’m not the first to think of making these diagrams for rings other than Z[i] and Z[ω]? If you know of earlier diagrams, please let me know in the comments.
Figure might as well plug my fiction volume Detroit Math Detectives, First Casebook, available on Smashwords. Though to be honest, this post you’ve just read kind of contains more math than that book.