Monthly Archives: April 2012

A pigeon for every hole, and then one (sort of)

There is a certain pattern to learning mathematics that I got used to in primary and secondary school. It starts like this: first, there are only positive numbers. We have 3 apples, or 2 apples, or maybe 0 apples, and that’s that. Sometime after realizing that 100 apples is a lot of apples (I’m sure that’s how my 6 year old self would have thought of it), we learn that we might have a negative number. That’s how I learned that they don’t always tell us everything, and that sometimes the things that they do tell us have silly names.

We know how the story goes – for a while, there aren’t remainders in division. Imaginary numbers don’t exist. Under no circumstance can we divide or multiply by infinity, or divide by zero. And this doesn’t go away: in my calculus courses (and the ones I’ve helped instruct), almost every function is continuous (at least mostly) and continuity is equivalent to ‘being able to draw it without lifting a pencil.’ It would be absolutely impossible to conceive of a function that’s continuous precisely at the irrationals, for instance (and let’s not talk about $latex G_\delta$ or $latex F_\sigma$). And so the pattern goes on.

So when I hit my first class where I learned and used the pigeon-hole principle regularly (which I think was my combinatorics class? Michelle – if you’re reading this, perhaps you remember), I thought the name “pigeon-hole” was another one of those names that will get tossed. And I was wrong.

I was in a seminar today, listening to someone talk about improving results related to equidistribution theorems, approximating reals by rationals, and… the Dirichlet Box Principle. And there was much talking of pigeons and their holes (albeit a bit stranger, and far more ergodic-sounding than what I first learned on).

Not knowing much ergodic theory (or any at all, really), I began to think about a related problem. A standard application of pigeonholing is to show that any real number can be approximated to arbitrary accuracy by a rational $latex \frac{p}{q}$. What if we restricted our $latex p,q$ to be prime? I.e., are prime ratios dense in (say) $latex \mathbb{R}^+$?

More after the fold –

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Posted in Expository, Math.NT, Mathematics | Tagged , , , , , , , | 2 Comments

From the Exchange: Is it unheard of to like math but hate proofs?

A flurry of activity at Math.Stackexchange was just enough to rouse me from my blogging slumber. Last week, the following question was posted.

I have enjoyed math throughout my years of education (now a first year math student in a post-secondary institute) and have done well–relative to the amount of work I put in–and concepts learned were applicable and straight-to-the-point. I understand that to major in any subject past high school means to dive deeper into the unknown void of knowledge and learn the “in’s and out’s” of said major, but I really hate proofs–to say the least.

I can do and understand Calculus, for one reason is because I find it interesting how you can take a physics problem and solve it mathematically. It’s applications to real life are unlimited and the simplicity of that idea strikes a burning curiosity inside, so I have come to realize that I will take my Calculus knowledge to it’s extent. Additionally, I find Linear Algebra to be a little more tedious and “Alien-like”, contrary to popular belief, but still do-able nonetheless. Computer Programming and Statistics are also interesting enough to enjoy the work and succeed to my own desire. Finally, Problems, Proofs and Conjectures–that class is absolutely dreadful.

Before I touch upon my struggle in this course, let me briefly establish my understanding of life thus far in my journey and my future plans: not everything in life is sought after, sometimes you come across small sections in your current chapter in which you must conquer in order to accomplish the greater goal. I intend to complete my undergraduate degree and become a math teacher at a high school. This career path is a smart choice, I think, seeing as how math teachers are in demand, and all the elder math teachers just put the students to sleep (might as well bring warm milk and cookies too). Now on that notion and humour aside, let us return to Problems, Proofs and Conjectures class.

Believe me, I am not trying to butcher pure math in any way, because it definitely requires a skill to be successful without ripping your hair out. Maybe my brain is wired to see things differently (most likely the case), but I just do not understand the importance of learning these tools and techniques for proving theorems, and propositions or lemmas, or whatever they are formally labelled as, and how they will be beneficial to us in real life. For example, when will I ever need to break out a white board and formally write the proof to show the N x N is countable? I mean, let’s face it, I doubt the job market is in dire need for pure mathematicians to sit down and prove more theorems (I’m sure most of them have already been proven anyways). The only real aspiring career path of a pure mathematician, in my opinion, is to obtain a PHd and earn title of Professor (which would be mighty cool), but you really have to want it to get it–not for me.

Before I get caught up in this rant, to sum everything up, I find it very difficult to study and understand proofs because I do not understand it’s importance. It would really bring peace and definitely decrease my stress levels if one much more wise than myself would elaborate on the importance of proofs in mathematics as a post-secondary education. More simply, do we really need to learn it? Should my decision to pursue math be revised? Perhaps the answer will motivate me to embrace this struggle.

I happened to be the first to respond (the original question and answer can be found here, and I’m a bit fond of my answer. So I reproduce it below.

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Posted in Mathematics, SE | Tagged , , , , , | 2 Comments