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.