A Quick Binomial Interlude

I just thought I would quickly share with you, what I consider to be a lovely way to demonstrate what polynomials actually represent, geometrically.

For some reason, until I saw this image, I never really fully understood how it all fits together, as you approach higher powers.  I have lifted this image from Wikipedia, which you can find here.



I think it's wonderful.

David Brannan Versus M208

I have now ploughed through the first two units of Analysis A; which is part of M208 Pure Mathematics, with the Open University.  Now that I am off work sick, awaiting back surgery, I have managed to consume an entire Unit in two days (should take about 16hrs per unit); so things are going okay at the moment.

As I was studying, I picked up my wonderful copy of David Brannan's 'A First Course in Mathematical Analysis'.  Now, the uninitiated may not actually know, but approximately 95% of the Analysis section of M208, is a direct lift from the pages of Brannan.  Shock, horror, you say!  I think that they have just left it mainly unaltered, because of the simple, sequential way that he expands on series and continuity, before launching into differentiation and integration, making the subject easier to follow.

This is not dissimilar to the way that Spivak, approaches the subject in his book, 'Calculus', the title of which may throw one a little, as the book actually covers much more than the simple mechanical aspects of calculus; as it weaves through series, limits, vectors, conics, planetary motion and plenty of rigour to whet the appetite of any purist.  I have the third edition, and I thoroughly recommend owning it, even if it's for nothing else other than the beautiful way he explains the basic theories of numbers in chapter one; or the way he makes one chuckle, by entitling the seventh chapter 'Three Hard Theorems'.  You certainly know what to expect from that chapter!

Now, one thing that struck me, about the M208 Units, whilst I was leafing through Brannan, is that each M208 unit is an A4 size booklet, of, say 40-60 pages of text.  Each one, sometimes, feels, like a 'pamphlet', an easy read option, almost like lecture notes, purposely written to give a student an easy life.  Yet, they are actually very deceiving!

It only took a few moments to realise, that just 8 of these 'pamphlets' actually cover the entire Brannan Analysis Textbook.  I found this surprising, as when you look at Brannan, it looks pretty scary in size.  So, maybe M208 isn't all that light on reading, after all.  The only problem with that assertion, is that one of the tutors at day school (an insanely brilliant mathematician), recently described M208, as Pure Maths 'lite'.  Well, it feels pretty 'full-fat' to me.  Time will tell, as to whether it will provide enough background to cope with any future studies.

An Errata?

I, like some of my esteemed peers and colleagues in the Mathematics world, have a penchant for pedantry (never thought i'd ever use those two words in the same sentence).

I don't do it for the sake of just 'doing it' (if so, is it still pedantry?); rather, I find that sometimes a little error, can actually throw you off and prevent some of the more subtle understanding that is needed in mathematics; especially in pure mathematics.

I was reading Unit AA1 of M208, Pure Mathematics, and spotted what I believe might be an error; but I am second guessing myself, as to whether it is just my limited understanding of the subject.

The possible errata I refer to, is on page 28, Example 4.3.  In the solution it describes the inequality \[1 - {\textstyle{1 \over {{n^2}}}} \le 1\] for n = 1,2,....

However, I would have thought that as n tends to infinity, that \[1 - {\textstyle{1 \over {{n^2}}}}\]  would approach 1 but never actually reach it.  So, therefore, that inequality should be just \[1 - {\textstyle{1 \over {{n^2}}}} < 1\]

Do I have that right?  I know it is a small thing, but I really must make sure, that my understanding is solid.

ps: if the latex above is not displaying, do let me know.  I am testing out Mathtype 6.7.

Blind-sided by LA5, Pure Mathematics.

I thought I would just quickly comment on the difficulty of the later units of the M208 section on Linear Algebra.  It was a real shock, how quickly it all became so abstract and a little bit counter intuitive in some parts (matrix arithmetic when diagonalising).

The whole Linear Algebra offering, was a really good example of a time where you, absolutely, must master the first 3/4 of the material, before even hoping to understand the last 1/4.

I did enjoy it, but I think it may take me the next year or so, to re-read, think about, and then finally absorb some of the finer subtleties of this subject.

I hope it doesn't take too long though; as I will probably need to apply it all, next year, if I finally register for Quantum Mechanics (The Quantum World) with the O.U in October 2013.

I am already on the look out for any interesting / classic Linear Algebra reference texts, to add to my creaking bookshelves.  I think, as with most of these subjects; that enrichment by reading widely, without necessarily increasing the difficulty of the material, is the key to understanding.

TMA03, Away...

Finally, after an extremely difficult week, I have managed to get TMA03 finished and in the post.  It was a little late, with a 3 day extension, but I am glad to say that I was able to finish late on Friday, post it Monday morning and make headway with the new shining, wonderful, solid and oh so delicious, Analysis Units for M208.

I don't know why, but I love Analysis.  Having said that, I have already pre-studied Brannan's, 'A First Course In Analysis', which the O.U Analysis Units, are broadly based on.  I found it exciting and elegant.  Much better than Group Theory at the beginning of the year (I still have nightmares).

I have also taken some good advice from my friend Chris, who suggested that I print off the audio transcripts for the course, rather than go through the pain of listening to some stranger, attempting to lead me through the material.  It's working well, so far, as I have nearly finished Unit AA1 and will start AA2 on Thursday.

Anyway, I sort of have some bad news that could also be slightly fortunate, in a strange way.  I am just about to go under the knife, for a herniated spinal disc.  Bottom line?  I will be off work for at least 2 months.  What's good about that?  I'll have about 12hrs a day to read mathematics, finish off Tolstoy's War and Peace, get my Rubik cube times down; and last but not least, catch up on some exciting enrichment material in Calculus (Spivak).

Summer, here we come!

Normal Service to Resume Shortly

Just a quick post.  I am working hard towards completing TM03 of M208, and now just have the last question to answer, which is all about working out standard form quadrics.  I am a bit behind on my reading (still half of Unit LA5 to read; as, due to a family bereavement, I have been quite distracted this week.

Anyway, I will complete TMA03 and start the Analysis section next week, on time; after a short period of solitary reflection.

Audio Sections; Contributing to Confusion.

I am 3/4 of the way through writing TMA03, using Mathtype for word.  I have to say, that having spent approximately 20hrs using Mathtype, now;  it is becoming very much a breeze, to use.  The results of using it are nicely typed, automatically well formatted maths equations.

With regards to the content of TMA03, I did find it a real mixed bag of difficulty.  I found most of it relatively straight forward, except for the question on orthogonality.  I found myself in a real twist for about 6hrs, before working out what I needed to do, in order to find an orthogonal basis for a subset, that they had given.

I think what really threw me, was the increasingly annoying way that the O.U secrete an 'audio' section into the units, with cryptic accompanying notes, to guide your path.   I am not a purist, by any means.  I do enjoy occasionally learning in ways that are not connected with reading dusty old text books; but these audio sections are driving me insane.

I could understand the use of them, if they were needed to describe intricately subtle ideas, that wouldn't be apparent from just reading a text book; but my experience of the O.U's treatment of this format, leads me to the conclusion that they are just used to read out what is already written on the page of notes.  Hardly helpful!

They are an unnecessary distraction, in my view.  I believe that the use of an audio section in the 'Orthogonality of subspaces' section, has contributed to my confusion on TMA03; and I fear that there are more audio sections, to come.