Blog: Articles filed under math

Tonight a friend on Facebook asked me: why do you like math? I knew that any suitable answer to that question would be a long one, and as I was cooking at the time and logged into Facebook chat on my phone, and so I deferred. After dinner I began typing a response on Facebook, but then I realized that this is worth its own blog post. I think it’s evident from this blog that I do love math, but I seldom pause to discuss why I love it.

This is what I said three years ago:

For those who don’t understand how someone can be so excited about math, the best way I can describe it is like being closer to God. I don’t necessarily believe in God, but I imagine that what I feel when I’m exploring mathematical concepts is the same feeling pious people get when they do whatever it is pious people do to feel closer to God. And math truly is the language of the universe. If God does exist, in one form or another, then understanding math helps one understand the universe and, in a way, get closer to God and creation.

Well, in the intervening time I have crossed that dark gulf between agnosticism and atheism, but the metaphor still holds. Mathematics is, ultimately, the most powerful tool we have for understanding and interacting with existence itself. There is mathematics behind anything you care to name: music, art, poetry, prose; there’s math in the swing of a baseball bat or in the spiral of a football. So the idea that we can express the fundamental nature of existence through mathematics is incredibly compelling—and also, I think, incredibly beautiful.

Mathematics is a form of communication. When I tell people that my two teachable subjects are math and English, they almost invariably furrow their brows and say something like, “Those aren’t a common combination!” And that might be true, but the implication—that math and English are somehow polar opposites—is not. Both are languages; both are about expressing ideas using an agreed-upon vocabulary, syntax, grammar. One just happens to be the modern world’s lingua franca, while the other has been placed on this pedestal: “Oh, I can’t do math! I just don’t have that kind of brain!”

I don’t recall any particular event that triggered my love of math. I remember favouring it over many of the other subjects when I played school as a child; and of course, it probably helps that I am rather good at it. It’s true too that some people have a talent for math while others struggle—I’m never going to be a star athlete—but I reject the idea that there is a “mathematical brain” as a social construct rather than a neurological edict. After all, I like to read and write too: there is more to me than my left hemisphere, thank you very much. My ability in mathematics helps, but it’s not the sole reason I love math. I love math for the same reasons I love philosophy or physics—for their deeper thought and what they can say about this world, about all possible worlds—and that I majored in math rather than physics or philosophy is perhaps more of a fluke than anything else.

Of course, even though I laud math for its role in our relationship with the physical world, I make no secret of the fact that I love “pure” mathematics. I prefer the dialect of rings and groups over that of differential equations or probability densities. I love the really abstract stuff, the ideas that verge upon being philosophy of mathematics instead of mathematics itself; I love discussing the theories behind the theories. That sort of love isn’t something you can really justify in words. It’s like asking writers where they get their ideas: we can provide a multitude of answers, but the real answer is that we don’t know. Everywhere, and nowhere.

Maybe it’s genetic. Maybe it’s environmental. Maybe the government put a chip in my head. Why do I like math? I don’t have a damn clue.

With the first term nearing its end, here’s a little review of my second year so far.

That is the best way to describe how I spend most of my time now. With three math classes, I spend nine hours a week listening to math lectures. I have three assignments due each week, so I work on those in my time between school and work. Every second week until the middle of November, I went to a practice sessions for math competitions for an hour on Fridays. Even when I’m not doing my own math, I like to help other people with their math. I am living and breathing math.

AND IT’S FRELLING AWESOME.

For those who don’t understand how someone can be so excited about math, the best way I can describe it is like being closer to God. I don’t necessarily believe in God, but I imagine that what I feel when I’m exploring mathematical concepts is the same feeling pious people get when they do whatever it is pious people do to feel closer to God. And math truly is the language of the universe. If God does exist, in one form or another, then understanding math helps one understand the universe and, in a way, get closer to God and creation.

When I first started university, I was worried that I wouldn’t be able to keep up. I had this deep, dark fear that I’d fail to understand crucial concepts and I‘d never be able to graduate in the math program. Fortunately, so far that isn’t the case. I am learning, and it is a challenge—some of these concepts are really complicated! But I build on what I learned before, and that allows me to understand concepts that a year ago I would have been unable to grasp.

The more I learn, however, the more I‘m able to comprehend just how much more there is I don’t understand yet. I‘m starting to get an idea of where my interests lie, however. I’m really enjoying ring theory—we’ll see if my interest continues next term, when we learn group theory. Abstract algebra appeals to me because it focuses on the reason I love mathematics. Abstract algebra involves constructing and proving the fundamental aspects of math. It’s the fundamentals of the fundamentals. I‘m discovering that I love doing proofs.

I’m getting the sense that most of my peers don’t have the same white-hot passionate love for math that I do. But that’s fine. I’ll show them. I’ll show them all! Muwahahaha! Muwahaha—er … right. Moving on.

When I tell people who know me that I want to teach high school, most of them react with scepticism. Apparently I walk around with the word “Professor” stamped on my forehead. Working for the first time with a new hire at the gallery, I made an allusion to Sisyphus, and my boss said, “Ben’s our resident Einstein.” And the new girl replied, “Yeah, I’m getting that vibe.”

Apparently I give off a vibe now….

I have wanted to teach for as long as I can remember. As I got older, however, the age group I wanted to teach got older as well. So I can’t deny that now that I’m in university, I‘m starting to understand why I would want to teach at a university. My main reason for not wanting to become a professor is that I don’t want to write math papers and do research into theories. I just wanted to do math. Now I‘m realizing that I actually like doing proofs, and it’s scary! :whoa:

Will I stick with my original desire to teach high school? Or will I fulfil everyone else’s predictions? Tune in for the exciting conclusion over the next three years!

Either way, I’m going to be a math hermit for a very, very long time.

I must confess, in general, I dislike numbers. I love math, but numbers just hurt my head. Not all numbers were created equal, however (yes, that is a really bad pun). Certain numbers are more fascinating than others. Take prime numbers, for example. Mathematicians continue to search for larger and larger prime numbers, and we just found another one.

A prime number is any integer that can be divided by only itself and one. Two is the only even prime number. Others include three, thirteen, and twenty-nine. The largest known prime number would fill over 3,000 pages. It’s two to the exponent 43,112,609 minus one. Yeah, that’s big.

What’s the big deal about prime numbers? Surely they have no application in the real world! Those silly mathematicians are too lazy to do work, so they just sit around making up numbers all day! You might have been right, once. Then someone came along and built computers, and prime numbers now have purpose!

All integers (whole numbers) can be broken down into a unique combination of primes. For example, 10 is the product of two prime numbers, 2 and 5. Factorization is the operation of finding a number’s prime factors; you probably did this in school. It’s relatively easy for small numbers. With large numbers, it becomes harder and takes longer.

I imagine that most of you have bank accounts. If not bank accounts, then Facebook accounts, email accounts, etc. What stops hackers from getting into those accounts? Prime numbers! Prime numbers are an integral part of cryptography and securing computer systems. One way to encrypt data is to take two huge prime numbers and multiply them together, producing a larger number. To decrypt the data, you need to know the prime numbers.

As computers get faster, we need to find larger and larger primes with which to encrypt data. If quantum computing ever becomes viable, it would have great implications for current cryptographic methods, since a quantum computer would be able to factor numbers in a fraction of the time it takes current supercomputers. Bye-bye bank account! Fortunately for cryptographers, quantum computing is in its infancy.

As you can see, prime numbers have real-world applications. Our ability to find larger primes and calculate prime factors has ramifications for the security of your data.

Computing prime numbers is an excellent test of computing resources, too. Incidentally, most of the largest known prime numbers have been found using a distributed computing project called GIMPS, or the Great Internet Mersenne Prime Search (Mersenne primes are a special type of prime, and they are often the easiest to find). You can run GIMPS software on your own computer and help contribute to the search for primes! There’s even prize money involved.

Of course, most of you aren’t running supercomputers at home, so your computer isn’t finding primes all by itself. It runs tests over months on a number, reports back to the mothership, and continues running tests. As the GIMPS website states:

A single test will take approximately 3 years on a Core 2 Duo computer. Your chance of success is roughly 1 in 2,000,000.

So don’t hold your breath.

After a weekend of work, Mondays are refreshing. I don’t work on Mondays, and I get to go back to school and learn.

I’ve been back for over a week now, and I’m enjoying it. This is my easy term; I only have five courses: Foundations and Issues in Education, Educational Technology in the Classroom, Differential Equations, Linear Algebra I, and Ring Theory with Applications. Yes, two education courses and three maths. I love math.

Of the education courses, the technology one is online. I knew going into it that it would be easy, but as it stands right now, it’s rather inane: for the first four weeks all we have to do is read, then we get a test. Then we have to work in groups to create a blog about teaching technologies, theories that apply to these technologies, etc. This wouldn’t be so bad, except that the reading material is full of typos, passive voice, prepositions at the end of the sentence—I‘m very glad that I don’t have to buy a textbook, but this is almost torture. Spellcheck, please!

My other education course looks like it’ll be more interesting. At least it’s mostly discussion-based. That makes it easier to sit through those hour-and-a-half classes.

Did I mention I love math? The university atmosphere works well for me. I like when someone who knows more than I do challenges me (i.e., with an assignment), then I go and teach myself what’s required to figure out the assignment. If I have trouble, I ask for help. I know that many people don’t learn this way, so the lectures are helpful too. However, I‘m glad that university gives me the opportunity to learn in the way I want to learn.

Of my three math courses, linear algebra is the easiest. It’s basically computation: vectors, matrices, and of course, linear systems. Differential equations are slightly harder. I ordered my textbook from the United States in order to get a cheaper price, and it still hasn’t arrived. That will make doing this first assignment problematic.

Ring theory interests me the most. We haven’t started discussing rings yet; last week we covered divisibility and prime numbers, and this week we are working on modular arithmetic and congruence classes. Rings are in the next chapter. This area of math fascinates me because it involves constructing the basic operations of math from scratch, allowing us to define new mathematical systems (presumably to tackle certain problems). That’s rather exciting. And there’s nothing quite like the feeling I get after figuring out a nifty proof.

I’m trying to blog more regularly—not once a day, but at least a couple of times a week. I‘d planned to blog over the weekend, but I was very tired. I found that the major reason I wouldn’t blog (other than being tired) is that I had no compelling idea for a post. Now I’m keeping a list of potential topics in Todoist, so that should generate more posts.

Have you ever looked at someone who is walking down the street listening to an MP3 player and said, “Gee, I wonder if that person is listening to a rap song about physics!”¹

Because that’s what I spent most of Monday and yesterday doing. Seriously.

Today marks the first circulation of particle beams through the Large Hadron Collider. This is the largest particle accelerator ever built—27 km in circumference! Soon scientists will begin high-speed particle collisions, and thousands of scientists from around the world will analyze the results of these experiments to help us better comprehend the universe.

I love physics. It interests me almost as much as math does. I‘m also one of those people who believe that science, especially physics, doesn’t need to be inaccessible to laypeople. While you may not be able to grasp the more esoteric mathematics behind the theories, it is possible to distill it down to the most basic points. Katherine McAlpine managed to do just that with her Large Hadron Rap. If you want to know what the LHC does, but you just don’t get all those explanations on Wikipedia or other news sites, watch this:

Yes, it’s a rap video about physics!! There’s also an MP3 available for download (so I can listen to it elsewhere). I think it does a remarkable job at explaining the LHC—the diagrams in the video assist the lyrics, especially for those of us who don’t speak rap. The video reminds me of the music videos at the end of Bill Nye: The Science Guy episodes. Oh, those were the days….

Those of you who just came for the video can go now; anyone who’s going to stay for some science babble may continue reading.

The LHC is a remarkable achievement because it will give scientists a glimpse at subatomic reactions with the fidelity we’ve never had before. It’s like HDTV for physics, only not only can we see better, but we can actually create more types of collisions.

What’s the big deal about particle accelerators anyway? I mean, it’s a couple of protons zipping around so quickly that they’ve completed a circuit faster than your brain can tell your eyes to blink. But by smashing protons into each other, and observing the results—i.e., what sort of particles and energy gets emitted—we can verify theories about how the universe works.

Particle physics and particle accelerators do have real-world applications. These scientists aren’t just spending billions of dollars because they are bored. Thanks to nuclear reactors, we have the ability to treat (alas, not cure) cancer. One form of cancer treatment requires radioisotopes, and the nuclear reactor in Chalk River, Ontario produces eighty-five per cent of the world’s supply of medical isotopes (primarily Cobalt-60, I believe). We live in a society where we have the ability to split the atom not just to destroy, but to create substances to save lives. Science is wonderful.

Particle accelerators have medical applications too. The LHC is kind of large for that purpose, but its smaller, more linear cousins (called “linear accelerators”, unsurprisingly) are an alternative form of therapy for cancer. Rather than creating the isotopes in a nuclear reactor and then storing and transporting them, the particle accelerator collides ions during the treatment, emitting the radiation that kills the tumour. Since storing radioactive isotopes is rather dangerous and expensive, producing radiation only as needed is safer and more efficient.

CERN has slightly larger aspirations for the LHC. Scientists are hoping to test numerous theories, as explained in the Large Hadron rap. We want to know where all the antimatter went; we’re trying to figure out what exactly composes dark matter; and failing all else, the LHC will at least help us verify or disprove the Standard Model of physics. All of this is built upon the work of those who came before, and the results of these experiments will in turn contribute to improvements in science and society in the future.

• [ 1 ] If the answer is yes, and you haven’t heard of the Large Hadron Rap, then you may be a closet physicist. Don’t worry, there’s support groups for those now.

I got to miss the entire day of school for a math competition at the university.

The individual competition was in the morning, so I sat in the lecture theatre with a hundred or so other kids and answered 15 multiple choice and 5 full solution questions (or attempted to answer, I guess). It was pretty hard, but not as bad as last year. I did well on the multiple choice, I think, although not so well on the full solution.

Then we got a free lunch, which is always a nice break. Pasta, chicken wings. Nanaimo bars for dessert. Mmm. After lunch those of us from Westgate tossed a Frisbee around for a little, since it was so nice outside. They gave away free Frisbees, coffee mugs, and of course, the annual t-shirt.

Then there was the team competition. That went better than the individual one. I was partnered with two other fairly smart kids, so we did fine.

I‘m not sure why I love this math competition so much, except that I do. Well, it was cool to get a day off school and a free lunch just for doing math. And the t-shirt is nice.

Not that I’m materialistic or anything…

Note that f(x) must be the same over this interval. Silly me.

(My only consolation is that not only did my two classmates fail to spot this, but the student teacher was the one who tried to do this with different functions and led us into the incorrect solution. )

Today I participated in the TD Canada Trust Math Competition at LU. I went last year too, although it was at the college then because of a strike at the university.

I don’t think I did too well at the individual portion. Part of the reason was that some of the questions involved stuff I simply haven’t learned yet. As for the team competition…I think we did reasonably well. Again, I felt a bit useless because I was on a team with two grade 12s (the grade above me) and I didn’t know half as much as they did. Plus, there was nothing on conic sections! That was my “thing”, since I had just finished learning it, whereas the grade 12s had forgotten it. Oh, there were sequences and series all right (another grade 11 subject that I haven’t learned), but no conics whatsoever!

So yeah…we did get down an answer for every question, but we used the horrendous method of guess-and-check for nearly every one, and our proofs were quite flimsy. The answers will be posted online sometime this weekend, so I’ll get a link up to them later.

Lunch was quite good. There was pasta, caesar salad, buns, and lots of little pieces of cakes and sundaes for dessert. I enjoyed it muchly. Oh, and I got a t-shirt.

So it wasn’t a bad day; I wish I could have done better, but I did get the day off school and I think I acquitted myself as well as I could have. I’m tired now and feel burdened by all the stuff I have to do. None of it is marked “urgent”, so I don’t do it. I’m so lazy.

I did type this up, so I guess that’s today’s accomplishment.