Oh my! I have spent the last three weeks, using up every available moment of my time, studying my geometry and number theory courses. I hadn't realized just how much new stuff there is to learn; and not only learn, but manipulate and mould, into something that can be used in my TMA questions.
It has sunk in that number theory, is really, really hard. I mean, not just hard, but insanely hard. I'll try and explain a little. I am currently staring at my TMA sheet, in which it has taken me 20hrs, to answer 3.5 questions out of 10. I have looked at two of the questions and not only can I not work out an answer for them; I can't even recognize which bit of the course, they are supposed to be testing.
This isn't just some changing of symbols used, that is throwing me; this is a case of me tackling this stuff without a full and comprehensive background in mathematics, at A level and 1st year university level maths. These questions are calling on stock pieces of mathematical knowledge, that I am shaky on, combined with high level abstraction.
All I have been able to do, is work, work and work some more. I have kept an accurate note of my time spent studying this month; and I am about on track with my studies and TMA work; but it has taken me an average of 30hrs per week, to get there.
And the worrying thing is, I still feel like I don't really know the material all that well.
All I can do is maintain this level of study for the entire course; and I won't give up. I can't really say more than that, at this point.
This is the hardest that I have ever worked, in my life; and I am going to make it count for something.
Tuesday, 29 October 2013
Saturday, 5 October 2013
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.
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.
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