Tag Archives: Maths

Critique, Maths, Proofs, Text Books

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I received the marks back for my first monster assignment! Did quite well as it turns out! But this blog isn't about spouting about my success, it's about the learning process! So here's some of the things I screwed up...

First off, my algebra is clearly rusty as fuck. In one instance put a minus sign in the wrong place AND mysteriously lost a factor of 2 in the progress of my working. In future I really need to re-read my working really carefully (three or four times over it seems), both the hand-written and the full typed-up LaTeX...

Something else I lost marks for was the apparently simple task of graph sketching, either where I hadn't considered asymptotes or had not considered the limits of the domain. Overall I clearly need to be a lot more mindful of whether I'm dealing with \leq or <. When I read those symbols I see them both so often, I frequently gloss over them without properly considering their usage. Again, pretty basic stuff.

With complex numbers I apparently need to be more explicit with my declaration of forms. My polar form was implicit in the answer, but there wasn't anywhere I actually stated it. Silly boy.

I fell down on a proof of symmetry for an equivalence relation. I just wasn't mindful whilst answering this. It is assumed that x-3y=4n. This can be rearranged in terms of y as y=\frac{x}{3}-\frac{4n}{3}. So substituting y, in the symmetrical y-3x results in: 4 \left(-\frac{2x}{3}-\frac{2n}{3}\right). Of course, at this point, proving that what's inside the brackets is an integer is pretty difficult. But that's where I left it. A bit more play would've shown that I could easily have arranged the first equation in terms of x instead which would've resulted in 4\left( -2y-3k\right), which is rather obviously an integer given the initial variables. More exploration required in future...

Lastly, in my last post I mentioned how there was a distinct lack of symbolic existential or universal quantifiers in all this new material. After Velleman, I was so used to seeing them, and working with them appropriately but because they're now not around, I got totally burnt by assuming I had to prove "there exists" instead of "for all" for one question. I suppose I'll be able to get around this with making sure my notes explicitly state whatever quantifier we're actually talking about. Damned English language... Symbols are much more concise! 🙂

General, Maths, Proofs

Large Intro

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Finally submitted my first assignment. It was monstrous. Just over 23 pages of mathematics and sketches of graphs. All of it typed up in LaTeX. Skipping ahead to look at the rest of the assignments, it looks as if this first assignment may very well be the biggest of the whole lot. This is a very good thing as I really don't think I could churn out that much work of a high quality every month.

Glad to say that most of this introduction section I was familiar with. Only really new topic was equivalence relations, which caused some problems initially.

Overall though, what I've found difficult is the apparent lack of logical notation. After reading "How To Prove It" I've become half-decent at making sense of and rearranging logical notation to solve a problem. The difficulty comes in looking at the plain-English description of something in the texts and then having to translate it into logical notation to allow my fussy brain to think about them logically.

Perfect example of this is the definition of a function being "onto". In the text, the definition reads:

"A function f: A \longrightarrow B is onto if f(A)=B".

Which is fine, but the Wikipedia definition reads:

"\forall y \in Y, \exists x \in X such that y = f(x)"

Which for me, gives me a much better idea about how to go about proving if a function is onto. Why leave out the quantifiers? The Wikipedia definition tells me so much more. I suppose translating English into logical notation is just something I'll have to get good at!

Though even after this long intro section, I really feel I need more practise with proofs... I guess this may have to wait until revision time... Next up is the first section on group theory, with an assignment due on November 24th. Onward.

Bertrand Russell, Books, Proofs, Text Books

End of the Quantifiers

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A month and a week, and I've just come to the end of the second chapter. Reasonably happy with the progress, but I could be going a bit quicker... Mind you, over just two chapters I've now created 46 pages of A4 of exercises. So there has been a LOT of material to go through. Frankly, just these first two chapters have worked wonders for my understanding of logic and what proofs are founded upon.

This second chapter mainly introduced quantifiers. The concept of "for all x" and "there exists at least one x...", but quickly branched off into more involved set theory.

The biggest issue I had towards the end of the second chapter was that on a couple of occasions, I don't think I thought carefully enough about the kind of answer the questions required. ie: in this context, whether the answer was required in logical notation, or whether it was required in set theory notation. Translating between the two is something I certainly found tricky. As such, I decided to write my own definitions of notation in the form of a list (thanks Lara Alcock!). Though the lack of lists of definitions could be considered a slight shortfall of the book, I think I benefited from constructing my own notes and definitions.

I found that towards the end of the questions (because of the more lengthy logical notation) I was concentrating more on the definitions than what the notation actually meant. Not convinced this is so good for the learning process, but at least I'm mindful of it now.

Last little question the second chapter covered was Russell's Paradox, as discovered by Bertrand Russell in 1901. The fact that I'm being introduced to stuff like this in the second chapter is pretty cool. Very enjoyable!

Next up, proof technique!

Maths, Proofs

Reading Mathematics

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Still working through my book on Analysis (which I will be until July 2016, so I should probably stop mentioning it...), but I recently came across another proof that I had difficulty understanding. I had to reach out to the mathematics forums in the end, but after getting a reply and working through some further steps that they mentioned, everything fell into place.

I felt really good that I'd finally understood the proof, but I also tried to work out what I could've done to push for that answer myself (so hopefully next time, I won't have to post a question to the forum).

I realised that I might not be reading through the mathematics effectively enough. After reading through Lara Alcock's book I realised how important it is to make effective notes whilst reading through "all the symbols".  When reading through proofs I have this nasty habit of reading them like a novel, keeping this story of logic in my head... and then very quickly becoming confused because I didn't see how you could logically progress from one sentence to another.

This sounds really simple, and almost obvious, but I think all it really takes is to sit down with a pen and go through the mathematics of the proof in gritty detail, liberally re-arranging things as you go. It's worth mentioning that this is perhaps quite a different act than just "taking notes".

Being completely confounded by something only to solve the issue entirely on your own is enormously satisfying. The hope is that if I stay mindful and remain aware as to when to write the right kinds of notes, not only will I be able to solve more complicated problems on my own, but also in time these seemingly large logical steps will become second nature.

Anyway, doing independent reading of mathematics is proving, generally, to be really satisfying. Let's see if I can become a more effective reader...

 

 

Bertrand Russell, Mathematicians, Pop Books

Fermat's Last Theorem

One of the first "Popular Maths" books I decided to pick up was Fermat's Last Theorem by Simon Singh. In the book, he expertly tells the story of Andrew Wiles and how he (practically) single-handedly solved an age-old mathematics problem. I won't go into any more detail than that because you can read a lot about the story on the web (but really don't, just buy the book).

fermatsLastTheorem

Part of what I really love about this book is that it goes into a lot of background detail, a lot of history. It builds up your understanding of why mathematics is the way it is today, and exactly why Fermat's Last Theorem was such a prestigious problem to solve.

One section that really sticks in my mind features a certain man called Bertrand Russell. Now, the more I read about modern mathematics the more his name crops up. Right at the beginning of the 20th Century, Bertrand Russell's research in logic appeared to show that mathematics was flawed. Again, I won't write down the details, I'd only end up copying Simon Singh's own words (buy the book!!!).

Although one passage did stand out in particular. Simon Singh mentioned that mathematicians obviously questioned Russell's work, and then goes on to quote Russell's response:

'But,' you might say, 'none of this shakes my belief that 2 and 2 are 4.' You are quite right, except in marginal cases - and it is only in marginal cases that you are doubtful whether a certain animal is a dog or a certain length is less than a metre. Two must be two of something, and the proposition '2 and 2 are 4' is useless unless it can be applied. Two dogs and two dogs are certainly four dogs, but cases arise in which you are doubtful whether two of them are dogs. 'Well at any rate there are four animals,' you might say. But there are microorganisms concerning which it is doubtful whether they are animals or plants. 'Well, then living organisms,' you say. But there are things of which it is doubtful whether they are living or not. You will be driven into say: 'Two entities and two entities are four entities.' When you have told me what you mean by 'entity', we will resume the argument.

A brilliant read (buy the book).

Books, Proofs, Text Books

How To Study For A Mathematics Degree

After my third module, completed in June 2005, I realised that although I enjoyed the module immensely, I still wasn't entirely happy with my ideas behind how to present proofs. Apparently this is a common problem. The shift from normal arithmetic to proofs is a jump that a lot of students find difficult.

I was so bothered by it I decided to post a query on the Open University forums about further reading. Part of my post read:

"...[I] consistently struggle with all kinds of proofs. I either start off completely incorrectly, or have no idea where to even begin."

The response was amazing, everyone (including more advanced students) agreed with me on the difficulty proofs and the shift of mindset required, and I now have a sizeable reading list to work through!

One of the books suggested in the forum was How To Study For A Mathematics Degree by Lara Alcock. I've tagged this post as being about a text book, but only because it's certainly more of a text book than a pop(ular) book. Although pitched at students that have just finished their A-levels and are about to start at university for the first time, I thought I may benefit from picking it up.

howToStudyForAMathematicsDegree

I came away with some good notes as to how to adjust my learning slightly, and she talked a lot about the basics of proofs and how to acclimatise yourself to them. One of the parts that I feel was key was the realisation that I can try to construct premises and conclusions in terms of definitions. "Starting with the definitions" seems so obvious in retrospect, but when shown something so unfamiliar it can be difficult to align your brain so as to give yourself a solid starting point. Outlining all the definitions you might need gives you that solid starting point. I'll be keeping all this in mind as I start through my first text book...

Another section I benefited from was a section on how to reading mathematics. Or rather, how to benefit whilst reading mathematics. After spending three years of reading mathematics and feeling I've so far done a pretty good job, but Lara introduced some thoughtful comments on note-taking and comprehension. Again... I'm looking forward to applying this when sifting through my next text book.

As good as the book was, I must say I didn't benefit much from the second part on "Study Skills", but this section was only about 80 pages long. The section covers what "university life" is like, and how best to be a student in such an environment. If you've already been to university it'll be of limited use, but if you're straight out of your A-levels it could make for quite an insightful read.

All in all I feel I took away from it what I wanted to take away; that is a base understand of how to approach proofs in a more structured way. At the very least, I now have a starting point!

Analysis, Books, Maths, Text Books

The Maths Club

After my first year, I had a look at the clubs and societies you could join as a member of the OU. I thought it would help me feel a little less like I was learning so remotely. Sure enough, there was a mathematics club, M500.

A nice little bonus that you get for being a member is a discount for the bi-annual revision sessions that are held by the mathematics faculty at the OU. So when I started my last module, I signed up for a revision weekend that was arranged to be held about a month before my exam.

It was great meeting other students. With all that remote learning, it was great to be reminded that you weren't alone. We even had nightly trips to the pub! Just like a physical university!!!

Towards the end of the weekend, one of my fellow students, studying the same module as me, mentioned a book that he'd recently ordered.  It was written by David Brannan, a man apparently partially responsible for the syllabus of our first double-credit module in our second stage, and wrote the book to reflect the content of that module. My classmate thought he buy it to give him a bit of an edge when he finally started the monster-module. In our class he gave everyone the details. The book is called A First Course in Mathematical Analysis, and at over 450 pages, it looks like one serious introduction!

aFirstCourseInMathematicalAnalysis

Finding out about this book couldn't have been any more perfect. It means I can start my easy introductory module this year, and read through my new book on analysis so I'm ahead of the game for when my double-credit module starts in 2016! Perfecto.

Maths

Choices, choices.

It's only a few weeks now until my next module starts! The beginning of my fourth year looms...

This module choice proved to be an interesting one. My degree consists of three stages (think of them as proper university academic years), and this module will be the last of my first stage.

The last module choice is actually very broad (some of the modules are foreign languages!), but of all the modules, there is only one that has 100% mathematics content called "MU123". The only other one I would want to take has about 30% mathematics content.

The problem with MU123 is that is an introductory mathematics module. -one that I'm bound to find far too easy. Worried by this, and wondering whether I could just skip straight to the next stage I called the OU support center. They agreed that I would indeed find the module easy, but did suggest something that other students have done in the past. Previous students have taken MU123, AND the first module for the Stage 2 together and done them in the same year. The problem for me there is that the first module of Stage 2 is actually a double credit module, meaning twice the usual workload. So instead of doing one modules worth of work in a year, I'd be attempting to do three!

Faced with the absurd workload, I decided to just sign up for MU123 on it's own, have a nice easy year and get the credits I require to move on to the second stage.

Having said that, my fourth year will not be all plain sailing... more to come in a future post!

Group Theory, Maths, Proofs

Year Three

My third module, completed last June, took me by surprise. As I'd started my statistics In February 2014, and completing in September 2014, I was hoping for a nice break before starting this third module.

The module I needed to take was MS221, rather innocently called "Exploring Mathematics". However, it transpired that the class that would be starting in September 2014 was to be the last ever of MS221. Normally this wouldn't be a problem, but MS221 followed on directly from material presented in my first module, and I was told I had to take this course. If I waited to take the replacement course I would face a ton of catch-up material .

So in summary, due to the slight course re-structure, I ended up not stopping to have a break between my second module and my third. This meant I was studying non-stop from February 2014 to July 2015. Exhausting.

I did manage to take a long holiday at the beginning of 2015, but tried to make sure I was several weeks ahead of my study before doing so.

Although "Exploring Mathematics" sounds rather unassuming, it covered some rather in-depth topics. On the one hand I was presented with proof by mathematical induction (which only proved itself to be baffling) but on the other I was also presented with group theory which describes a really nice formal form of symmetries. Seriously cannot wait to learn more about group theory...

Once July arrived, and I had taken my final exam I was once again a free man. Finally free to enjoy a Summer.

So that brings us up to date! I'm now enjoying my first proper break away from mathematics in 17 months, and the materials for my next module are due to arrive in under a month!

Maths, Statistics

Year Two

After a long break from my first module, I decided to start my next module in February 2014, half-way though the academic year.

My second module was to be Statistics. I was looking forward to this, and wasn't disappointed. The module covered a lot about survey results, and how to conduct surveys effectively and make sense of the data. It also had a biology experiment thrown in there for good measure, the results of which were meant to be used in an assignment.

One of the things that really bothered me though, was their description of statistical variance. To my mind, one measure of variance would be the sum of the difference between the population mean and the sample mean, all divided by the sample count.

In fact, that's not the case. It's not divided by the sample count, it's divided by the sample count minus one. This seemed totally counter-intuitive, and apparently the full explanation was outside of the course material!!! Frustrated I scoured the web looking for an explanation... Eventually I came across a decent youtube vid. Thanks youtube!

To summarise, this video proves that sample variation s^2 is an unbiased estimator of the population variation \sigma^2 as shown below:

    \[<span class="ql-right-eqno"> </span><span class="ql-left-eqno"> </span><img src="http://adrianbell.me/wp-content/ql-cache/quicklatex.com-8309d8e21d5c74b4a21588f648103d5a_l3.png" height="45" width="261" class="ql-img-displayed-equation quicklatex-auto-format" alt="\begin{align*} E(s^2) =& E \left( \frac{ \sum_{i=1}^n( x_i - \bar{x} )^2 }{n-1}\right) = \sigma^2 \end{align*}" title="Rendered by QuickLaTeX.com"/>\]

See that pesky "n-1" divisor? Check out the full explanation here.

So after my study starting in February and ending in September, I felt I'd had a decent introduction to statistics.

Although, something that does surprise me is that looking ahead at my possible future module choices... statistics doesn't really crop up again for the rest of my maths degree. With the growing importance of statistics in modern society (let alone mathematics itself!) I would have expected a lot more of those modules on offer. Having said that, there is a separate BSc (hons) Mathematics and Statistics that the OU offers, but this appears to almost be purely statistics and doesn't give much variety. -certainly wouldn't be good for me.

In fact, there are  only two real interesting modules on this Maths and Stats degree course, one of which is the final year module "Mathematical Statistics" (M347) that introduces Markov Chain Monte Carlo, which is an area of interest. I suppose after my ten-year degree, if I'm still desperate for more, I can sign up for it as a stand-alone module. 🙂