Powered by MathJax From GCSE Maths, to Rocket Scientist...: April 2011

Saturday 30 April 2011

Physics and Maths, Studied this Week.

It has been a short week, on account of a two day family break from studying, as part of the royal wedding celebrations.  However, I have still managed to break the back of some decent work, including some deeper understanding of vector use in static force problems.

Anyway, here is this week's completed study:

MST121


Analysing force problems (vectors)
Integration and area problems


The Teaching Company: Calculus lectures

The area problem and the definite integral


Other reading:  Calculus made easy, Silvanus 


Integration as the reverse of differentiating


Other reading: Dr Riemanns Zero's, Karl Sabbagh


Chapters 1 - 3


Total time studying this week: 10hrs

Next week, I will be making a start on the next piece of coursework for MST121, called TMA03.

Tuesday 26 April 2011

The Genius of Gauss

With the royal wedding approaching on Friday, it seems fitting to quickly mention a man who was known as the 'Princeps mathematicorum', or the 'Prince of Mathematics'.  He also often referred to mathematics, as the 'Queen of Sciences'.

Carl Freidrich Gauss was an early 19th century mathematician and I have recently had the good fortune, to encounter one of his elegant mathematical feats.  I am referring to the anecdotal story of when he was a child at school.  His mathematics teacher asked the class to add up all of the integers from 1 to 100.  This task, by hand, would have taken some time, but Gauss is alleged to have pondered the problem momentarily - and then come up with an ingenious solution, to allow the sum to be done in a few seconds, using a simple formula.

He approached it, thus:

If you take the sum of the  numbers from 1 to 100 i.e.

1 + 2 + 3+ 4...+ 100

and make these equal to X

and if you then take the same numbers 1 to 100, but this time, you reverse them so you get:

100 + 99 + 98... + 1

These numbers added up, would also equal X.

Therefore, if you added together both of these series, they would equal 2X.

Easy enough so far.  But here is the really clever bit.  What Gauss did, was to add together, the two series of numbers in the order shown above.

For example he added:

(1+100) + (2+99) + (3+98).... + (100+1)

Now, all of these little sums such as 1+100, all add up to the same number, which is (101).  And if we say that in  our series from 1 to 100, that the number 100, is equal to (n)

Gauss realised that 101, is equal, in general terms, to (n+1), because each of the above terms such as (1+100), (2+99), is equal to (100 +1), or  (n+1).

He also realised that (n+1) must be multiplied by how many terms, he was adding up.  In this case, the terms from 1 to 100, equal 100 or (n) terms.

Therefore, he was then able to produce a general formula, to add together the sum of any sequence of consecutive numbers, which was:

n(n + 1) = 2X

This is simply, the sum of the two sequences of 1 to 100 and 100 to 1.  They are the same sequence, just in the opposite order.  So, they both still add up to 2X.

Therefore X = n(n+1)/2,  where X = the sum of the series of (n) consecutive integers.

So, in our above problem; Gauss wanted to add up the numbers from 1 to 100.  So, in this example, n=100.

Putting this into our equation, we find:

X = 100(100+1) / 2
X= 5050

If you do it longhand, and add the numbers from 1 to 100, it does indeed equal 5050.

Now, all this may seem a little complex; and, for a young child in a maths lesson, it was.  Yet Gauss, worked it out, in a very short period of time, and demonstrated his prowess and precocious nature, that was to have him crowned the prince of maths, studying the queen of sciences.

And not a single line of Union Flag bunting, in sight!

Thursday 21 April 2011

Physics and Maths, Studied this Week.

Another week done!  I am finally back in my stride following my months of sickness and this summer is shaping up to become a very interesting period of study.  I have started the 'official' calculus part of my University studies.  It is not surprising, that the Open University have a compulsory postgraduate course entitled, Calculus of variations and advanced calculus (M820), which must be studied, before they allow you to take any further postgraduate courses with them.


I can certainly see the merits in doing so.  I mean, most real world processes, that need modelling in a precise and predictive way, tend to use calculus as a large part of the calculations.  I think that calculus is becoming a personal passion and I find it intuitively simple, to understand, contrary to my pre-ideas about the subject, before I ever encountered it.


Anyway, here is this week's completed study.  I finished S196 and submitted that coursework.  I have now almost completed my draft version of TMA02 for MST121 and I am now headlong into calculus.  The time spent studying this week, has been 16hrs:




MST121


Recap on Vectors
TMA02 draft version completed
Lots of practice, differentiating polynomials!


The Teaching Company: Calculus lectures


Infinite limits and limits at infinity
The derivative and the tangent line
Basic differentiation rules


S196
Completed EMA (coursework)
Finished course. (result due approx September 2011)


I can't wait to do some more calculus this week and I may even dip into the Integration and modelling books early, to keep my perspective broad, as I assault the differentiation material.

Wednesday 20 April 2011

MST121 Study materials have arrived.

The second half of my Open University course books for MST121 Using Mathematics, have finally arrived today.

This is where the maths really picks up and I can't wait to get started on it.  It consists of introductions to some of the core subjects, that I will be learning, in detail, as I progress towards becoming a theoretical physicist.  The subjects are all at the heart of quantum processes and are the mathematical tools needed for most physics applications, whenever you are dealing with particles and nuclear processes.

The course-books cover the following subjects:

Differentiation and Modelling
Integration and modelling
Differential Equations and modelling
Probability studies
Modelling distributions and deviations
Statistical Investigations

These subjects will be studied over a period of 4 months and will be tested by two pieces of coursework and a final marked assignment.

S196, Finished!

I have finally, after much reading and study time, finished my Open University course, S196 Planets: An Introduction.

It was very interesting to learn about the planets; however, trying to write the coursework was a real challenge.  The coursework asked many questions of almost forensic detail, about the planets or their many, many satellites.  Now, I don't mind a bit of detail.  In fact, the detail is exactly what I enjoy when I am learning maths.  But the detail that I needed to learn for S196, about individual craters, scarps, rotation periods, ad nauseam; was a little bit like train spotting.  I felt like I was just having to read a load of boring facts about planets, and then regurgitate them for the coursework.

Some people may like the parrot fashion, learning of a subject, but I don't.  I prefer solving problems, forming abstract concepts and using them in application on important issues.

Well, it's all done now.  I submitted my coursework this afternoon.  But I have to say, that if I never have to see another Trans-Neptunian object again; it will be far too soon!

Monday 18 April 2011

Rounding Errors

One thing that is causing me a major headache currently, is having to handle answers that need to be rounded  to (n) amount of decimal places or significant figures.  The act of rounding these numbers is simplistic its self; however, the trouble comes, when you then need to start using the answers from those calculations, in later parts of the same question.

For example, recently, in my MST121 coursework, there was a question involving vector analysis, to determine the speed and direction of an object that has multiple forces, acting upon it.

The question involves approximately 14 different steps which are formulaic once you know how; however, you need to provide the answers to (n) decimal places.  However, you need to keep the answers in their full unadulterated form, for use in later calculations, as you move through varying component forms of those vectors.

Now, having dyslexia, has caused me a few problems with vectors already; and these rounding issues are no exception.  I do find myself getting lost in the middle of a calculation, due to the constant act of rounding up or down of answers, to fit what is required in the final answer, by the Open University Tutors.

Another silly little problem with vectors, is that they use the notation of either (i) or (j) to denote the two different components of the vectors.  Being dyslexic, I have always found that I mix up the letters 'i' and 'j'.  So, you can see, that this is a recipe for disaster when handling vectors.

I have now started to use Mathcad to check my written answers and catch those rounding errors or those little mishaps with 'i' and 'j', which does help in the long run; even if it does add to my workload slightly.

Friday 15 April 2011

Physics and Maths, Studied this Week.

Okay,

This week I have mostly been studying MST121, Vectors.  I felt that I had glossed over the intricacies of vector manipulation in the various forms.  So, as well as continuing my TMA02 preparation and also Finalising my coursework for my O.U Planets course; I have found a nice settling feeling, that comes with a second glance at work that I have previously studied.  It has provided a period of reflection and has been a great confidence booster, as I now feel very familiar with the material.

Well, here is my study material for this week.  Next week I will continue to write TMA02 and write the coursework for S197, How The Universe Works.


MST121


Vectors in column form
Vectors in geometric form
Vectors in component form
Sine rule
cosine rule
Three force modelling problems (static)
Modelling friction

S196
Jupiter
Saturn
Uranus
Neptune
Kuiper belt objects
Exoplanets

Completion of Part A coursework, Q's: 10 - 17


Book of the week


Imagining Numbers  by Barry Mazur


Total study time = 15hrs

Monday 11 April 2011

Warming to Mathcad

I had a problem this week, with multiplying two matrices.  They were a 2x3 and a 3x2 matrix, and I sometimes get muddled in which numbers need to be added to which others.

Mathcad to the rescue!  I punched in the numbers, and out popped the result.  I did the maths using pencil, paper and grey matter, and then used Mathcad to verify.

In situations like this, it has started to become a bit of a life saver as it stops all of those silly errors in counting.

In a future post, I will give an evaluation of the Mathcad functions, both good and not so good.  It is starting to become a useful tool, as it is helping in the checking process, as part of an overall maths problem solving strategy.

New Particle? Probably...

This week, the U.S Tevatron accelerator, had some unusual readings in one of their experimental data streams.  The readings, if confirmed, could hint at a new particle.  The existence of this particle would then tend to suggest, that there is a new fundamental force in nature, previously unknown to science.

What does this mean?  Well, it could possibly require a revision of the standard model of particle physics and may lead to a deeper understanding of matter and, (it wouldn't surprise me), if the theory of dark matter / energy, was revised or discarded completely, for a new fundamental theory.  I have personally never accepted the dark mater theory which, to date, has never been fully proved.

But, this new particle discovery, is far from confirmed or certain.  Scientists have stated that, because of rounding errors and with the probability factors; that the experimental data is of a 3 sigma level.  Just to explain, most science is not absolute.  Because of minor inaccuracies in the construction of the experimental equipment, calculation errors carried forward and even down to the uncertainty surrounding quantum processes themselves; you can create spurious data, that can mimic that which would be created, by a new particle or force being discovered.

3 Sigma is accurate to about 1 in 1000, whereas a 5 sigma  level of certainty, is required for this new particle to be hailed as a discovery.  There is already data in existence, from particle accelerators around the world, that can prove or disprove this theory; it will just take time to go through all the lines of code etc, to check these results.

Friday 8 April 2011

Physics and Maths, Studied this Week.

Another week done and I am slowly working back up to full speed.  At the end of May, there is a substantial piece of coursework due, for MST121 and also an extensive marked assignment for my course S196, Planets, An Introduction.

So most of my study, has been around answering the questions on those papers.  Once those are out of the way, I then have 8 weeks of Calculus, as part of MST121.  I can't wait and this is the part of the whole course, that I am looking forward to.

At the end of MST121, there is a section on probability.  I initially thought that perhaps that particular subject, wouldn't hold much interest for me.  However, I have been looking into Quantum Mechanics this week and, in particular De Broglie Probability Waves.  It is an elegant and facinating topic, which introduces the Heisenberg Uncertainty Principal.  Key to both areas of study, are the maths of probability and also wave functions.  This has whetted my appetite and is another motivator, for studying those topics at the end of this year.

Anyway, here is my study list for the week just gone.  I have concentrated on my core work this week, so that I can ensure a good pass, on both pieces of coursework.  Total study time = 8hrs

MST121


Questions one and two from TMA02

Sequences and limits - Reciprocal and constant rules.
Sigma notation
Vectors - converting from component to geometric and back.

S196


Earth
Moon
Jupiter

Questions 5-10 of Final assessment coursework, answered.

Another week done.  This week I will hopefully complete the bulk of the coursework and then start to reintroduce some video lectures on Calculus, Cosmology and some Stanford University lectures on Quantum mechanics.

Saturday 2 April 2011

Writing Mathematics

As an Open University student, I was surprised to discover, that for many of the OU Maths courses; it is a requirement, that all maths course work be handwritten, and not done electronically or on word processed documents.

Before I started studying, I assumed that maths work would be done electronically.  However, apart from the odd use of Mathcad to complete some modelling work; the OU insist on the old pencil and paper methods.

I can only assume, that they don't want to cause confusion for adult learners, who may be less computer literate, than those younger University students who have grown up in the last 15yrs, with computers being used for everything.

I know, at the age of 35yrs, that my computer skills have nearly all been learnt, since 1997, when I first started using the internet.  It wasn't until 2002, that I started using email at work.  From previous experience of humanities study with the OU; I know that many of the courses only started to move from using tape cassettes and VHS videos, in about 2004; which is many years after the digital revolution, began in earnest.

The OU is a forward thinking institution and they are involved in some rather ground-breaking research and innovation, especially in astrophysics and planetary science; so they are certainly not stuck in their ways!

This perhaps suggests, that the reason for a lag in the use of computers, to complete science or maths based coursework; is a symptom of them having to cater for a largely adult student population, who are trying to play catch up with technology.

Alternatively, many distance learners generally need to be able to scribble down their answers, in short bursts of study to fit around their family or professional lives.  I know that much of my coursework is done on my lunch breaks at work, or in short bursts of 10mins, when I get a chance to sit down.  So as an adult distance learner, I know that if I needed to fire up the laptop, every time I wanted to write some maths; then I may be less inclined to study the high volume that I currently do.

Perhaps by not using computers to write maths, the OU are allowing more freedom for busy people, who are not full time students that would be otherwise too busy, to follow a course of University study.