Mathematics for Dragons

C'est magnifique mais ce n'est pas la mathematique

2010-04-14T23:18:00.000-07:00

I got my first MST221 TMA back yesterday, 95% which is a cause for much tail-wagging in the dragon household ( we do not sing when happy - despite what that film might have said - but our tails do tend to wag which is unfortunate for any small animals or trees in the vicinity; do not stand behind a happy dragon ).

However there was one or two questions where I got full marks, and I don't think I should have done because I really didn't do anything to deserve them.

There's a question in TMA02 as well that I guarantee I'm going to absolutely ace and all I did was press a few buttons on a computer, something I do for a living most days, and fire off a few printouts. Question one took an hour of fiddling with some rather torturous algebra, question two took 10 minutes of keying in a formula into a pre-built spreadsheet and then setting a couple of parameters. Yes its quite wonderful but it's not maths.

In this green scaly creature's opinion using Mathcad in the course is fine enough but it should be used to illustrate and expand on the subject at hand - it's good at that, especially when it shows 2-cycles breaking down to 4, 8 and then into the chaotic region. What it should not form is part of the assessment. All testing me on Mathcad does is prove I can amend a pre-existing mathcad sheet and get some different numbers to come out.

Oh and Mathcad is a sucky, bug ridden piece of 1990's vintage software as well. I have to run it on a virtual XP machine on my main PC otherwise the second I touch the scroll wheel it freezes.

A classroom on a wet wednesday evening

2010-02-19T00:09:00.000-08:00

Wonderful thing Open University tutorials. Number of people in the tutorial group, twenty seven; attendees at the tutorial, three.

The best one I ever went to was when I was the only bugger there but it did mean that I got two hours personal tuition on vectors and matrices from a research fellow of a Cambridge college.

To be fair the tutorial earlier in the week was not too badly attended with about 8 people showing up on a wet and cold winter evening in a sixth form school in Cambridge*. I'd already covered the subject of the tutorial but I tend to get more out of tutorials that way and I find them handy just for picking up tips on how best to do things rather than for ab-initio instruction.

This time the big take away was "how not to show a proof".

Lets say you have the focus-directrix property of a non-degenerate** conic Pd = ePF. Now what you don't do is start off by saying "As Pd = ePF then... blah blah blah ...see they are"

Rather you go

"Pd = some formula... rearrange rearrange rearrange... = x"

and

"ePF = some other formula ... rearrange rearrange rearrange... = x"

"so Pd = ePF for this conic"

If you don't do it that way then you fall victim to an argument by circular reasoning which, as a rabid athiest used to duffing up religious types when they use such arguments I should have realised.

So even if you've got down pat the subject your OU tutorial is about it's always worth going along IMHO because you're bound to pick up something.

And what else would you be doing on a wet wednesday evening anyway.

* Which follows in the noble tradition of all educational establishments by selling undrinkable coffee that tastes like hot dishwater with grit in it.

** I have a picure in my head of a "degenerate conic" as a parabola that hangs about smoking behind the bike sheds and committing acts of petty vandalism.

Madonna's Bra

2010-02-15T10:30:00.002-08:00

I was studying chapter A2 of MST221 this weekend, working away in the office I use when I work from home. As is usual the dogs were in there with me to ~~give me a hand with the algebra~~ sleep on the sofa but at one point Mrs Dracunculus came in with a cup of tea and asked what I was up to.

Being all enthusiastic about my new knowledge of parabolas and hyperbolas I started to scribble some cones on my whiteboard and show what happens when you cut them up.

"Oh," said Mrs Dracunculus, " so how come you have to use Madonnas' bra in mathematics?"

She's not very mathematical, Mrs Dracunculus.

0,1,1,2,3,5,8...

2010-02-07T23:04:00.000-08:00

Just finished Chapter A1 of MST221 in which we get to poke around that set of numbers up there, the Fibonacci sequence, an example of a second order recurrence system. All quite interesting stuff and I didn't have too many problems getting my head around the sums including the derivation of a closed form for such sequences - which given the struggle I had with wrapping my poor dragony brain around the same derivation for first order sequences last year I was rather happy with.

I did have a bit of an issue with the algebra when doing the exercise to derive a closed form for a Cassini identity as the OU have started truncating their solutions and I am assuming just expect you to be able to follow this stuff by now.... bit like this...

Anyway a quick email to the tutor (and a same day response - great service!) gave me a poke in the right direction with something that is bleeding obvious once you've been shown it, which is.


 n-1                          n       n+1
a     is a common factor of  a   and a

so 2 is a common factor of 4 and 8, 3 is a common factor of 9 and 27, and so on.

I was even happier when Question 2 of the TMA was on just such an identity.

So, onto conics

2010-01-26T00:07:00.000-08:00

Here we go again

Woot! Large brown cardboard box arrived yesterday which means it can only be Open University time again.

Sure enough the first half of MST221 has arrived along with the usual assortment of DVDs and funny coloured bits of paper. I now have three copies of the "Enrichment Material" thanks to previously doing MU120 and MST121 but oddly enough the didn't send another copy of Mathcad and presumably assume you did MST121 and have the box handy somewhere (I did, but spent ten minutes rummaging through funny coloured bits of paper to find the sodding password).

On the subject of Mathcad... has ever a more crappy piece of software ever been designed? I mean it's hideous ugly, buggy as hell (touch the scroll wheel of your mouse at your peril) and I think I could do the maths faster sometimes its that slow. I really do not like using it at all and I really wist the OU could sort out a deal with Wolfram so we could use Mathematica - you know software people in the real world use?

Rather amusingly I notice that my tutor is the same one I had when I was doing MU120, less amusingly one of my tutorials is in Ipswitch which is quite a flapping distance from the lair.

Anyway time to dust of MST121 for a bit of revision on sequences and then I'll get an early start on unit A - I find keeping two or three weeks ahead of the study timetable works best for me and allows for a bit of wiggle room should things get tricky away from sums.

Aquiring a rubber fetish

2010-01-11T10:51:00.000-08:00

No, I haven't been getting kinky (well no more than normal) but I have been getting into LaTeX which as you may or may not know, is a bunch of software and a mark-up language for typesetting documents, particularly maths ones.

I've been sort of dimly aware of it for years but a few weeks ago me and everyone else in the East of England region doing MST221 this year were invited on a day-long "Intro to LaTeX" session in Cambridge given by one of the OU math team.

Well I had a rummage in the documentation online and it all looked quite interesting and actually not that horrendous to use. I'd done all of MU120 using Word 2007 which has a half-way decent equation and math editor nowadays (certainly way better than earlier Word versions) but I must admit after the first TMA for MST121 the amount and complexity of equations and formulae became a real chore to word process so I went back to the mk-1 pencil and paper method; far from ideal as (a) my handwriting is crap and (b) I get writers cramp really quickly and had to stop to stomp up and down and curse after every page.

I've also been around the block when it comes to word processors, the first "real" word processor I used - although you would be hard-pushed to call it that nowadays - was Wordstar running on a 286 in DOS so dicking around with formatting codes was not new, and having paid my dues in days gone by as a developer hacking together old-school ASP pages I was no stranger to markup and the whole concept of "What You See Is What You Mean" (rather than "Get") was familiar.

However I was surprised by just how powerful LaTeX is and what lovely results it produces and how quickly you can get it to do nice stuff.

Over the years I've kind of fallen out of love with the whole Linux/Open Source ideal - I am pretty sure it's because of my job as I am under lots of pressure just to get the code working and onto the trader's desktops so I really want to shove a couple of DVDs into the PC, install the development environment and off I go. I just don't have the time to fanny around with hand crafting config files, rummaging in HKEY_LOCAL_TOUCH_THIS_AND_YOUR_PC_WILL_NEVER_WORK_AGAIN and all the usual fluff that goes with open source. So with that said at first LaTeX was pressing all my "oh dear" buttons.

Need to install more than one program to get it working - CHECK
Various bits of code scattered all over the web - CHECK
1419 obscure tiny files, one of which allows me to typeset Ancient Nabbatean - CHECK
Irritating cliquey in-joke in that you need to write the name of the PrOgrAm in a miXeD CaSe and then it's not even pronounced "latex" but "lay-tec" - CHECK
Cutesy animal mascot for the application...

- CHECK

Not looking promising so far and indeed although I got up and running pretty quickly with an .iso distro off the net (for a given value of "running") there was a lot of fiddling around to do to get it working right until I realised that it hadn't actually installed everything I needed to make things work right so I needed to fiddle with the innards of a package manager (eventually giving up and saying "Yes, you win, install every sodding package including the ones that let me typeset Elder Futhark Runes, Astrological Birth Charts and even bloody Hieroglyphs")

So - not exactly software I would expect your average Joe to be able to install but then that's the point I guess, this stuff was written by geeks for geeks so your average geek (like me) with a bit of perseverance and consultations with Professor Google can get it working.

And when it is working... blimey it's powerful. The syntax is not too tricky but there's a lot of it and all those little files just extend the basic facilities by orders of magnitude and wow does it produce good looking, very readable output. Here's an example...

and you produce that using this markup:


\section*{Question 2}
\subsection*{(a)(i)}
The sum of the first $n$ integers is given by
\[\sum_{i=1}^n = \frac{1}{2}n(n+1)
\]
So the sum of the first 28 integers is $\frac{1}{2}28(29) = 378$ 
and the sum of the first 107 integers is $\frac{1}{2}107(108)=5778$ 
so the sum of the integers from 29 to 107 is $5778-378=5400$
\subsection*{(a)(ii)}
The sum $\displaystyle\sum_{i=29}^{107}(4+3_i)$ can be rearranged as 
\begin{equation}
4(107-29+1)+3\sum_{i=29}^{107}i
\end{equation}
and from the answer obtained in \textbf{(a)(1)} above we can write (1) as 
\begin{equation*}
4(107-29+1)+3(5400)=16516
\end{equation*}

I was hooked within minutes - one of those 3am "are you coming to bed yet love" times (sorry Mrs Dracunculus if you're reading this). Within a couple of days I'd already started work on a template for submitting TMAs just so I could learn how "fancyhdr" worked and that was before the class last saturday.

The day didn't start well with me totally failing to get into the building and walking round twice before finding the goods bay and a doorbell. Once inside I spotted the tutor giving the day school - wayward hair, yellow sleeveless jumper, blue bow tie, brogues... yep, you're committing every style crime in the book so you must be a maths professor and indeed he was. His enthusiasm for the subject was plain to see and quite infectious and like everyone on the OU tutoring staff he really knew his stuff. I was pleased I'd already had a play around with the software before the day but I got a whole load out of the day, fixed my install on the lappy to something better and came away with a different template for TMAs which I didn't like a lot of so I stole some of the ideas out of it and added them to mine.

So I'm a happy LaTeX convert... expect more witterings on the subject as MST221 progresses. Now where did I put my gimp suit.

I passed MST121

2009-12-30T23:20:00.000-08:00

Sorry, not really been keeping up to date with this blog (bad dragon! naughty dragon!) I will try harder during MST221 I promise.

Anyway over Xmas I got a little letter saying I passed MST121 with a rather credible 93% which I was pleased with. I was especially pleased to see I got 95% on the final end of course assessment which was a surprise as I thought I had royally screwed up on the calculus question and I wasn't too happy with my answer to the crocodile modelling question either - turns out they were my best 2!

Anyway more to come soon.

One down, lots to go.

2009-08-17T09:32:00.000-07:00

Well I got my first Open University result back, this one from "Easy Maths Even You Can Do" or "MU120" as they call it. To be honest I probably could have skipped that course as it became pretty obvious early on that I could easily handle the maths involved but it was useful for revision, especially some of the later on trig as that was rather rusty and some of the stats stuff like boxplots was new to me (they didn't teach those back when I was a hatchling) and most of all it was useful for getting into doing things the OU way so I was not stressing about the maths and the admin.

I actually completed the course well ahead of the alloted time too and so it was a bit of a surprise when the results actually turned up and a very nice surprise to find I'd got 96% which is a first (or "distinction" as they call them) in anyones language except MU120 is a straight pass/fail.

It was even more surprising that what I call the "wibble" questions I got full marks for. The "wibble" questions are the non-mathematical ones that ask you to pick a bt of the course and say what you found hard, how you coped and what your "learning outcomes" were. I guess having lived with a social worker for the last 15 years helps here as I just came up with a stream of wibbly nonsense with the right kind of buzzwords in it and bingo, 16 marks out of 16. I don't mind this sort of question and I can see that in a low level course with people just starting out it makes sense but I don't agree with them being a marked part of the course and final exam work as, well, it's not really maths is it?

Still, bumped my marks up!

Integration.

2009-06-04T10:33:00.000-07:00

MST121 has moved onto integration now. Integration conceptually seems pretty simple, it's like differentiation only seen from the back.

It's not too hard to do either: if you remember down there when I differentiated X^2 and ended up with 2x you can do a reverse operation and it turns out the formula, yielded by some relatively straightforward arseing about with algebra or geometry, to integrate a simple function of the form


      n
    ax


   a  1   n+1
     --- x
     n+1

which happily yields x^2

However the devil in the detail is that the result of integration needs to have an arbitrary constant added on as differentiating x^2 + a (where a is any constant) gives 2x so the reverse operation has to add this "constant of integration" back in. Apparently this makes these "Indefinite Integrals" but luckily for me you can get them to be a bit more definite when you subtract one from the other.

And subtracting the results of integrating a function for different values of the dependent variable is the one bit of calculus I vaguely remember being shown at school (I never did A-levels as I loathed the two maths teachers with a passion so I'm guessing this got touched on somewhere in AO level¹ pure maths) probably for finding areas under curves.

Oh and I got a book on Calculus as well (see - still a swotty dragon, reading around the course now!) called "Calculus made Simple (for scaly green firebreathing creatures like you)²" written by some guy in the early 1900's and apparently still the definitive text on the subject today. I will let you know how I get on

¹ This was an old UK exam that I think got dropped sometime in the 80's. I, being a swotty hatchling, was in an "advanced" stream at secondary school and we took our O levels (called GCSEs now) a year early at 15 and we also did a couple of these "midway between an O level and A level". I had a google and although I couldn't find a AO level paper I did come across a marking scheme for one here: www.edexcel-international.org/VirtualContent/49349/7362_PURE_MATHEMATICS_F.pdf and it does have a little bit of calculus in it.

² I think I might have imagined the subtitle.

Zero as a limit

2009-05-10T23:04:00.000-07:00

Well MST121, which is the Open University maths course I'm doing, has now reached the dreaded "Chapter C" in which we get to study "The Calculus".

Quite why calculus gets a definite article is a mystery to me, I mean I never studied "The Geometry" or "The Algebra" that I can remember.

Well the first thing that's presented is a demonstration of differentiation which, in essence, is finding out about the rate of how things change.

A nice way to think about this and one that helped me get my head around it is to think of a graph like this which shows distance travelled (on the y axis) versus time taken (on the x axis).

In this example the distance travelled in metres is the square of time taken in seconds so in maths-speak we say that distance (d) is a function of time (t) and that in this case the function is t squared; which you write like this

d = f(t) = t^2

OK so from this you can tell how far you have gone in how many seconds. But how fast are you going at any time? Well as speed is distance divided by time you can work out an average speed between two times, t and t + h by joining the points on the curve at t and t+h and dividing the distance covered by the time taken.

Now the time taken is (t+h) - t which is just h and the distance covered is the difference between the values of distance d at times t and t + h; but as we know that d = t^2 we can express these distances as (t+h)^2 and t^2.

So that gives us


      (t+h)^2 - t^2
      -------------
            h

which simplifies to

 
      t^2 + 2th + h^2
      ---------------
            h

and more simply:


      2th
      ---
       h

Now what we do is start to reduce h and keep reducing it to a limit of 0

Ah, you can't divide by zero though, we went through that on the last post didn't we, you get a black hole if you do.

"Yes", says my OU tutor who has a brain so large there's nothing left for dress sense, but we don't go to zero, we TEND to the LIMIT of zero so think of h getting infinitessimally small, so small it doesn't count. So small you can just get rid of them leaving:

2t

And there you go, the first derivative of t^2. So after 5 seconds you've travelled 25 metres and you're going at 10m/s

Wel it does work but there's something inside of me that doesn't quite like it. I'm just a bit concerned that it's a bit of a fiddle throwing away these very tiny numbers. Apart from that at least I'm feeling pretty comfy with differentiation so far.

By the way if you type "Zero as a limit" into Google you get a load of song lyrics to a Human League tune from the 80's. Was the floppy haired one a mathematician I wonder?

Divide By Zero

2009-04-28T23:22:00.000-07:00

Despite what you might have read on teh interwebs you don't get a black hole if you divide by zero. But what do you get?

If you try it on a calculator you get an error or "can't divide by zero" message, although oddly if you are writing a computer program like this one...

double DivByZero(double n)
{
return n/0;
}

...you may well get a special value passed back which is either "Positive Infinity" or "Negative Infinity" which is a plausable number, but mathematically speaking wrong.

Lets look at the infinity business first. Imagine you divide 1 by a half (0.5), you get 2. Now divide 1 by a tenth (0.1), you get 10, now divide 1 by a hundredth (0.01), you get 100. So as the number you divide by gets progressively smaller and smaller the result gets larger and larger and this is true regardless of the number that is being divided.

In maths-speak what they say is that as the divisor (the number on the bottom) tends to zero then the result tends to infinity. Of course if the number being divided is negative then the result tends to a larger and larger negative value, "negative infinity".

The important thing to notice here is the "tends to zero" bit. The number is approaching zero, becoming exceedingly small but it is never actually zero. This works fine for computers where the rules say that all floating point operations have to have a defined result but we still have not actually divided by zero.

Why you can't actually divide by a real zero is that it makes maths itself break.

To see why you need to look at what division is. Basically put it is multiplication backwards. If you take a number, say 6 and divide it by 2 you get 3, now if you multiply it by the same number 2 you get back to 6. We say that division is the inverse function of multiplication.

Right, so let's for now pretend that you can divide by zero and you get this number "infinity" as a result: so 1 divided by 0 is infinity and infinity multiplied by 0 must be 1. Except it isn't as anything multiplied by zero is zero.

You could do the same thing with 2 divided by zero, three divided by zero, in fact any number divided by zero Now the same calculation can't have different results as the only way this could happen is if the result of a divide by zero as being every possible number simultaneously; so the correct answer to what do you get when you divide by zero is "the result is undefined".

"Simples" as that irritating meerkat on the telly says.

Mathematics For Dragons

2009-04-28T23:14:00.000-07:00

Hello everyone (all two of you who follow my witterings anyway).

This is going to be an offshoot of my normal blog Grumpy Dragon (Warning, adult language, swearing, cursing and stuff) where I'm going to make the occasional post about adding up as I wander through my Open University maths degree.

If you're at the Open University, a student or just interested in sums you might like to have a read. The rest of you will probably be bored witless.