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.