Division

Last year, I completed a T.M.A for my pure mathematics course with the Open University, in which one of the questions asked for a simple proof of a number theory example.  I forget exactly what the example was, but it involved proving that a certain expression was divisible by 12, or some other integer.

At the time, I had a bit of a brain freeze and struggled with the question for some days.  It consumed my days and nights until I finally gave in and rang my tutor, begging for salvation.  I though, at that point, that I wasn't cut out for number theory.

It wasn't until he asked some very probing questions, that we finally got to the bottom of where my difficulties lay.  It was the simple fact, that I didn't quite understand what was meant by 'divisible by'.

What?  Nonsense, I hear you say!  Every six year old knows how to divide two numbers.  But the simplicity hid a rather elegant idea, in that, to divide two integers, and get another integer, is quite difficult to prove or understand, from axiomatic principles.

This was born out when I delved into my first book of number theory, last week, and came across... you guessed it... a proof for the Division Algorithm (...how to divide two numbers).

Its not a short proof, but it is elegant, none-the-less.  If I had my LaTex editor working, I would write it below.  I might do that next month.

The division algorithm is really quite lovely and it is one of the few gems in this new course, including my favourite so far, the proof of general mathematical induction; and my least favourite, the rather puzzling, second principle of mathematical induction.  I say it's puzzling, as I haven't quite managed to figure out why, and when, you need to apply it to a problem, rather than just using the first principle.

I'm working on that one.

First Impressions

Well,  I think I like my new courses.  Number Theory isn't as heavy on content, as I first thought, and I have been able to manage to stay on track for my first 1.5 weeks of study.  The main thrust of the material, is the mastery of Mathematical Induction as a way of providing simple proofs for integer problems.

I struggled with induction, when I studied M208 Pure Maths; but I think this was because I didn't allow enough time to complete practice problems, on the subject.  However, I have taken head of the warning in the material for the number theory course, which states that Induction must be mastered, asap.

So, I have spent hours doing problems and redoing them, until the method started to stick.  Having said that, I have pretty much struggled with the algebraic step in the last part of all of these proofs.  Now, I considered myself to be at an advanced intermediate level, when it comes to algebraic manipulation; but these examples in the texts, make huge jumps in the algebraic reasoning, that I have struggled to follow or replicate (even by punching the expressions into Wolfram Alpha).

I definitely need to ask my tutor for some advice, about that.  I do hope that I have not reached a natural ability ceiling, that hinders my progress on the course.

The groups course has been quite heavy on material volume, in the first book.  You only get 1.5 weeks per book, where as, you get 2 weeks for a similar amount of material in number theory.  Most of the material introduced, was basic concepts and language, that will be needed for the course; however, there was a cheeky little chapter on Affine geometry, which I started off hating, but then when I got all the examples right, without referring to the answers; I started to love it.  How fickle!

Also, I have to say, that I find it soothing, rather than boring, to manipulate tiling's.  I know that people who previously took the course, felt that it was all a bit dull.  I find that they are all so geometrically neat and satisfyingly symmetrical; just the right amount of order needed, after a torturous and chaotic day at work!

I am adopting a new strategy for my studies this year.  That is, I have decided to complete all of the practice examples in the books, having a good go at them, before referring to the answers.  In my last few courses, I had started to get into the habit of skimming the examples, kidding myself that I was saving time, or leaving myself extra practise examples for exam time.  I think it was a false economy and harmed my understanding of some of the more difficult concepts in analysis and linear algebra.

Lets see if it does me any good, this year.

Go...

They have arrived.  I now have the first few month's worth of study materials for both Number theory and The O.U Groups course.

The Groups course is full of groovy plastic overlays and lots of different handbooks and equation sheets; yikes!

I have already started both courses, completing my first hour of study, on each of them yesterday.  It's always difficult to say which will be the more challenging course; however, going solely on first impressions - I would say that the Groups course looks a little easier than the Number theory course.  Time will tell, as to whether this first impression, lasts.

On Your Marks... Get Set...

After a lovely summer break which included a month off work and a trip to France and then Barcelona; the kids are back at school (my sanity is returning), and I am now sitting by the front door mat, like an expectant dog, waiting to chew the hand of the post man, as he delivers my fresh, new, Open University course materials, for this year.

The last time I blogged, I mentioned some of the prep that I would do over the summer, including some of M337, in preparation for next year.  I didn't manage to do that, as I made a decision to start giving myself a 'proper' break between periods of study, so that I can maintain a higher level of intensity, when the courses begins.

For the last few years, I have studied pretty-well none stop, in some form or another, which while admirable in keeping up a study habit, has probably harmed some of final TMA marks, as I find my attention waning, after non stop studying for 24+ months.

After my summer break, I have picked up my French and Spanish Assimil language courses, that I did a few years back; and I have resolved to spend an hour a day (in my work lunch break and during my commute), to listening to these audio tracks, with a hope of reaching a better level of conversational proficiency, in the next few years.

I had found previously, when I took my OU Art History module's in 2004-2005, that doing a short period of language study (at that time French with the O.U course L192) each day, somehow made it feel easier to learn the Art History from all of those dusty old books.  I don't know the reasons for this.  Whether it was just some novelty and a different part of the brain being exercised (auditory and language centres), I will never know.  I will report back as to whether I perceive any difference in my Maths study, as I listen to a small bit of languages, alongside my formal studies, this year.

Anyway, I am booked onto Number Theory and Logic, along with Groups and Geometry.  I've read Duncan's posts on these topics, and it all sounds very interesting.  I just hope that it isn't too stressful.  I've almost forgotten how many times, I sat up until 2am, during 12hr marathons to get my TMA's completed for M208.  Only now, I have two courses to contend with.

Exciting stuff.