Wolfram Alpha

I was flicking through my Mensa magazine, at the weekend; when I read an article written about a chap called Dr Stephen Wolfram.  He is a brilliant scientist and mathematician and has created a new search engine, that actually gives you specific answers to questions, rather than just producing a list of results, as is common with Google et al.  The engine is called: Wolfram Alpha.

I would advise any mathematician, to check it out immediately.  It is, quite simply, stunning.  You can ask it any sort of mathematical question and it will punt out answers and variations on the answers, that are just brilliant and very complete.

I tried to trick the engine, into giving me a stupid answer, but I struggled to do so.

For example, I asked it:  What is the meaning of life, the Universe and everything?

I was expecting an error message, but it simply returned just one answer:

42

For anyone who knows Douglas Adam's, Hitchhikers Guide, they will know that this is the only correct answer! (see below video)







I also asked it:

Give me all the English words, which contain the letters 'M,A,T,H,S'

It gave me 476 words in total, of which, a selection are below:

abashment  |  accomplishment  |  accomplishments  |  achievements  |  admonishment  |  admonishments  |  aestheticism  |  aftermaths  |  alchemist  |  alchemists  |  algorithms  |  amaranths  |  amateurish  |  amateurishly  |  amateurishness  |  amethyst  |  amethystine  |  amethysts  |  Amherst  |  amphetamines  |  amphitheaters  |  anathemas  |  anathematizes  |  antechambers 

Physics and Maths, Studied this Week.

Another week of studying complete, and I just absolutely love my current course, MST121, Using Mathematics, with the Open University.

I am deep into calculus, which just seems to be such a beautiful part of mathematics.  It is elegant and so very useful.  I am very tempted to take the OU, MSc in Maths, just so I can study one of their very advanced calculus methods courses.  I'll have to see how that fits in with my overall plan to finally study a PhD in Mathematical Physics or a related subject.

So here is what I have studied this week:


MST121


Revision of Product rule, Quotient rule, Composite rule
Revision of advanced derivations

Basic Integration
Integration by inspection
Further integration: Using algebraic manipulation and the composite rule


General maths exam practise


Sequences
Lines and Circles
Functions
Modelling with Sequences
Modelling with Vectors


Coursework completion


Questions 1 and 2 of TMA03, draft completed.


S197: How the Universe Works


Questions 1,2,3,4,5,6 of Final End of Module Assessment


Further Reading




Well, a very busy week completed and I am well into my stride now.  Although I don't have an exam this year, for MST121, I am still revising the course, as if there was to be an exam in October.  The reason for this, is simply one of practise. From January 2012, I am studying MST209, a more advanced course in mathematical physics and modelling, which is the same length as MST121, but has twice the study material and also a 3hr exam, at the end of it.  So, I am treating MST121 as a mock version, to prepare for next year's course.


Total time spent studying this week:  20hrs

Maths Study Tips I

This is my 50'th blog post! so as a way of celebration, I thought I would share an old Jedi Mind Trick.

A quick war story first:

I took a professional, 3 hour exam about 8yrs ago, in which I needed to learn, one-thousand pages of law, in great detail. 

I took the exam twice.  The first time I took it, I just read the material over and over again, and felt I had a good grasp of it.  I failed the exam on that occasion by 2%.

So, the following year, I re-sat the exam, but this time, I radically altered my approach and used the technique, that I describe below. 

This time, I read the books through and then started to put together practise exam questions, based on the text.  But I didn't just do the questions once, or twice, or even ten times.  I completed each practise questions, 50+ times.

By the time I had finished, I could even close my eyes and see the page in front of me, almost word for word.  It was like when you look at a bright light, close your eyes and still see the image.  I have no special skills (apart from Jedi Mind Tricks), It was just through practise,  nothing more and nothing less.

I scored 100% in that second exam.  I later learnt that I am the only person, to date, in the country, to have taken that exam and scored 100% in it.  That is out of over 100,000 people that have taken it, in the last 8yrs.

Why am I boring you with this?

Well, I believe that my exam revision technique works and, what’s more, it is perfectly suited to maths.

I noticed that when I first started studying maths with the OU, that I fell into a pattern of study, without too much thought.  It was roughly balanced as below:

50% - reading new material

45% - writing Coursework (TMA's)

5% - doing practise questions

Based on my Jedi exam prep technique, I quickly amended it to:

30% - reading new material

30% - writing TMA's

40% - doing practise questions

But, you may ask, where can I find all of those practise question, to test my knowledge?  Well, I have a stock of questions that I have collected from the OU texts and exercise books.  Also, I have several good text books such as Spivak et al.  I have written out, all of the practise questions, that best reflect the syllabus and all of the pit falls, for each area of maths.  I have just chosen one carefully selected question, for each learning point.

 [This selection of questions takes time, but it is probably the most productive use of time, in my whole year, without exception].

I then repeatedly do those sums, over and over and over, until I can do them in my sleep (not an exaggeration, I actually dream about this stuff). 

It doesn't matter that you are using the same sums repeatedly. The structure of how to answer these practise problems will be so ingrained into the subconscious that they will never, ever leave you; a bit like your times-tables.

So, when a similar exam problem comes along, but it contains different variables, constants, notations or even a different structure; your amazing brain won't panic, it will simply adapt.  It will solve the problem, based on your solid knowledge, built up through good old fashioned 'elbow grease'.

If I can find the time, I am currently considering starting a comprehensive didactic compendium of several hundred / thousand study questions and solutions, covering my entire maths journey, from 1+1= 2, right through to Post Grad MSc work; along with an explanation of how to use the questions and get the best out of them.  The practise-question method, works for me; so maybe I could share it, in a structured way, with other students, in the long-term. 

If I can find a willing computer geek, I might even try and build some software / smartphone app, that alters the parameters of each question and provides meaningful feedback after each study session.  My computer knowledge isn't that good, so it will definitely take collaboration of some sort. As a very wise man once said, 'I, have a cunning plan', almost.

Maths Revision

Maths, as an academic subject, is very difficult to learn, if you don't practise doing the sums.  That might sound like an obvious statement; but I know several people who have tried maths modules in the past, and failed them.  Not because they weren't bright enough, but because they read the maths, understood it, but then didn't apply what they had learnt, straight away. 

I have personally found that I can read a text and fully understand the beautiful concepts and elegant methods involved.  I then go to bed, sleep, get up, have my morning coffee, take the kids to school etc... To then find, that my recollection of the material, is not just rusty, but actually non-existent.

This did initially come as a bit of a shock to me, as I have studied with the Open University, before.  I did some art-history and French modules, a few years ago, and found that I could just read the texts, write a few essays, cram like crazy and pass with a good grade.

However, I have discovered that Maths is totally different.  It is both accumulative and technique based.  You need to learn and remember each and every concept, theorem or technique; as most of the more advanced maths work, is based on these earlier, elementary steps.

About 5yrs ago, as a confidence booster, I took part in the Kumon Maths programme.  It involves doing up to 236 basic maths sums, per day, 365 days a year.  Each worksheet is just a little bit more advanced, than the previous one.  And, although it is rather repetitive; it does make those skills and techniques stick.

My next post, gives some detail of the techniques I use, to study and learn, (really learn), University maths and other important bits of information.

 

Physics and Maths, Studied this Week.

Okay,

I have just come back off holiday, so this will be a short post of work carried out whilst I was sunning myself, in the pouring rain!


MST121


The Chain Rule
Exercise book C (calculus)


Misc Maths


Differentiation practise
Trigonometry identities and their differentials


Misc reading


Astro-photography (Sky at Night mag).

Calculus Made Easy, Thompson, Gardner.


The Teaching Company Lectures: Calculus


The Chain Rule



So, total time spent studying was 12hrs.

There has been a pretty amazing announcement in the cosmology community this week,  which appears to have been largely bypassed by the mainstream media, regarding some experimental evidence, for the existence of dark energy.

I personally hate the idea of dark mass and energy, as I find the whole thing, a bit of a cop out.  I am hoping, that in future years a more complete and satisfying explanation of physics, other than the standard model, is discovered.  In the meantime, I hope I don't spend the next 5yrs, becoming an expert on the standard model, only for it to be replaced by something more simple and intuitive ;-)

If it is disproved, I hope it happens sooner than that.  Maybe the LHC in Switzerland, will finally let us know, either way.  I just hope they hurry up, before I fill my head with one theory, only to be replaced by another.

But then, I suppose that is just the nature of science.  It just isn't exact; if you pardon the pun.

We just keep trying to understand 'stuff', using our limited knowledge, to date.  And, for now,  I am happy to be one of those who keep trying.

Physics and Maths, Studied this Week and TMA02 results

Just a quick post today; as I am packing, ready for a few days away with the family.  So, here is this week's study:

MST121


Gradients of curves
Gradients of quadratic graphs
Differentiation of a function re-cap.
Differentiating polynomials
Features of graphs
Optimisation


Other maths


Advanced factorisation practise

Total time spent studying: 14hrs

I also received back my course work (TMA02) for Open University course MST121, yesterday.
I scored 94%, which I am very pleased with.  I actually left the back page of my TMA, in the photocopier, so I lost 2% because of this silly error.  Other than that, I am now preparing for TMA03, which will be all about calculus!  It's due in July, so I'd better get cracking soon.

Elegant Algebra

I have been doing lots of practise in manipulating polynomials, as it is strangely addictive, once you get going.  It is really just about picking out patterns in an equation and using tools on those patterns, to simplify what is in front of you.  One of my favourites is below:

x²y² - x² - y² + 1 =

(x²y² - x²)  - (y²- 1) =

x²(y² - 1) – (y² - 1) =

(y² - 1)(x² - 1) =

(y + 1)(y – 1)(x + 1)(x – 1)

Lovely!

Can Mathematicians Add up?

For some reason, whenever I speak to any maths academics or math teachers, (I know a few), I keep hearing the same anecdote about the fact that mathematicians can’t count.

To expand a little, I believe that what they mean is that many academic mathematicians don’t have a masterful and fast grasp, of basic calculation and mental arithmetic skills, such as using fractions, doing multiplications and divisions of large numbers, or deft ability in the handling of factorisation or expansion of polynomials.

I have been looking around the internet, looking for some evidence to support this urban myth / anecdote.  And, although I haven’t located any specific evidence yet, about general arithmetic skills of maths academics; I have notice some subtleties in the subject of maths, that probably lend themselves to the conclusion, that mathematicians don’t need to count, all that much.

I bought a book this week entitled Calculus, 3^rd Ed by Michael Spivak.  It is a classic text of analysis and calculus.  In his book, Spivak states that he sees mathematics as the act of ‘thinking’ about mathematical questions.  He goes on to discuss how the aim of his book, was to present mathematics, not as merely a collection of tools; but as a framework of ideas on which to form a view of mathematics as a holistic subject, interlinked in all ways.

I agree with this hypothesis, and that led me to postulate that perhaps, as mathematicians develop in their field; they become more detached from ‘numbers’ and more involved with ‘ideas’.  Hence, perhaps, the anecdote ‘those academics can’t count’.  Perhaps they don’t actually need to?

My only problem with this approach is that having studied maths recently, and as part of an attempt to apply it to physics; I have discovered that each mathematical skill or tool, builds on the ones that preceded it.  By mastering the basics I believe that you can do them on automatic, allowing one to concentrate on the ‘ideas’ that Spivak describes so beautifully.

An example:

Want to fly your private jet across the globe?

Need to think about continuous vector forces…

Want to work out those vectors?

Gotta know calculus…

Want to use calculus?

Gotta know differentials, integrals, functions and limits…

Want to work out limits?

Gotta know how to simplify, factorise or expand…

Want to factorise?

Gotta know your algebra…

Want to be an algebra God?

Gotta practice, lots…

Want to practice algebraic manipulation?

Gotta know the fundamental properties of numbers…

Want to know those properties?

You need to know ‘four operations’, fractions and factors…

As Newton stated in his ’Principa’;     ‘ 1+1=2, is a very important sum’.

You MUST be able to count!

There are so many rules and methods that are used in manipulating numbers, to arrive at a model such as the one you would need to plot your aircraft’s path around the world; that mastering as many of the early steps as possible, must make it easier to concentrate on the bigger questions such as, ‘What colour should my private jet , be’?  Or ‘what wine should I serve, as we fly over Paris?’

Of course, you could always use Mathcad or Maple, computer packages.  But even then, as good as they are, you still need to know that the answer that they provide is the right one.

Physics and Maths, Studied this Week.

This week, I have been starting to expand my studies again, to include not only what is required in my current university course (MST121); but to take in some fun and interesting side issues.  It is a great way to hold my interest, if studying gets a little tough.

To that end, this week, I have been looking into some cosmology.  It never ceases to amaze, just how many new discoveries are being made, every single week, in cosmology.  This week, a team of scientists were able to prove, via experimental evidence, that the earth causes time and space to warp around it.  Also, they proved that the earth actually drags space and time, around with it, as it rotates and moves in orbit around the sun.

The satellite that was built to achieve these results, had an experimental accuracy which was the equivalent of being able to detect a hair's breadth difference, at 10 miles distance.  Pretty amazing,!  And, for those instruments to survive a violent launch into space at thousands of miles an hour, on the back of an Arian rocket, is equally impressive.

Also, the last space shuttle launch ever, is being dogged by delays, but is due to launch at some point this week.  I don't know why, but I just seem to hold my breath each time they launch one of those things.  It just seems like 1970's engineering, being patched up to work in 2011.  Lets hope Endeavour has a safe flight and that the particle physics experiment that it is carrying, delivers some good results.

Anyway, study this week was a total of 15hrs:


MST121


Vector and static forces modelling.
Completed final draft of TMA02 coursework.
Differential practise.
Expanding polynomials practise.
Factorising polynomials practise.


Teaching company lectures: Cosmology


Using sound as Cosmic Diagnostic.


Misc maths work

Long division and multiplication practise.

Next week, I will be getting to grips with some serious calculus work.

Lightning-Fast Mental Arithmetic

To complement my university maths studies; I have decided to do some real back to basics, study, to improve my mental arithmetic and calculation speed for long multiplication and division.  I have found a system, that is used by those maths magicians, who seem to calculate huge numbers, in an instant, without writing down any intermediate steps.

The system that I found, is called the Trachtenberg Speed Math System.  It was devised in the 1930's by a Russian engineer who was detained in a Nazi war camp.  During this time, he devised a system for arithmetic, that could be done without paper and pencil or calculator.

I have been studying the system for a few days, and already can see its benefits.  One thing I am planning to do, is to teach my children the system, to compliment their school work.

Just to give you an example of how easy the system is; we will try, for example,  to multiply a large number by a double digit number:

Such as     452413 x 11 =

Now such a number would probably take anyone a few minutes to calculate in a long hand fashion; but with the Trachtenberg system, it can be done mentally, in an instant, without writing down any intermediate steps.  How?  I'll demonstrate:

Just as normal, you write down the answer from right, to left.

In this example, when multiplying by 11, the first digit of the answer, is simply the furthest right hand digit, of 452413 i.e.  3.

For the next digit of the answer, each successive figure of 452413, is added to its right hand neighbour.  In this case, you add  1+3 = 4.

You then repeat this adding.  Thus, next 4+1 = 5.  Then  2+4= 6.  Then 5+2=7.  Then 4+5= 9.  When you only have one number leftover, on the far left, you just write it down.

And that is it.  So 452413 x 11 = 4976543

After practising this for 2 minutes, I managed to do these a in a few seconds.  There are other 'rules' that are slightly different for some multiplications; but the principles are all the same.  When it comes into its own, is where you have a sum such as 3857856 / 843.  By anyone's standards, this sum would take a while to work out.  But with this system, it can be done mentally, in about 10 seconds.

There is a caveat with the system; the rules used for arithmetic are very simple and don't ever require you to add up to more than 11; however, it does take some practise to get it slick.

Plus, it's great fun and a useful diversion from calculus!

Big Bang Acoustics

I have been looking into the sound-waves created by the big bang and also how those waves, that still exist today, can be used to interpret the size and structure of the universe, since the big bang.

It seems amazing, to think of sound-waves existing in space, after all, we often think of space as cold dark and silent, apart from the odd star and planet populating its regions.  However, that is only half the story.   At the time of the big bang, for about the first 40,000 years; the universe was a hot dense 'fog' of materials, including photons, sub-atomic particles in a photon-baryon gas and also dark mattter.  This 'fog' transmitted sound waves through it, and these waves, still exist today, although the 'fog' of materials, is now more of a fine mist! caused by the expanding universe, reducing its density.

From these waves, and using what we already know about how waves behave, as they travel through different materials and densities, at different frequencies and wavelengths; we can compare the waves created by the universe, as it developed over the last 15 billion years, with these 'benchmarks' of known physics / wave properties.  This provides us with a sound spectrum, that can be crunched in a computer programme, giving us a very accurate indication of what the universe is made from.

What has been discovered, is that the sound-waves of the universe indicate that its composition consists of 4% atomic matter and 23% dark matter.  No other combination of matter, holds true, for these waves.

These type of investigations, show that cosmologists are really just technical-historians.  They are always dealing in events that happened in the past.  For example, the light from the nearest star, our sun, takes 8 minutes to reach us, so we are always viewing what has happened in the past, and not what is happening in the present.  This is caused by the fact that the speed of light is fixed, and that nothing in the universe can travel as fast as that speed.

Therefore, information from anything that is very far away from us, such as a galaxy or a primordial sound wave, take years or even centuries, to reach us.  Until then, we can know nothing about the secrets, that are waiting to be discovered.

However, it is this precise property of light, and the universe, that allows us to see the first moments of the big bang, as some of  that information is only just reaching us from the outer parts of the expanding universe.  That means that some of the light reaching us now, is billions of years old and has been travelling through the universe for a very long time.