Archive for the ‘general math’ Category
There is a post over at Mapleprimes called ‘The feeling of Power‘ which contains an extract from a short story by Isaac Asimov written in 1958. The story is also called ‘The feeling of power’ and is set in the far future at a time when humans completely rely on computers to do all of their calculations for them. In this vision of the future even the keenest human minds are incapable of performing simple calculations such as ‘nine multiplied by seven’ without the aid of a machine.
Until reading the Mapleprimes post I had never heard of this story and I wanted to read more. A quick google search gave me the full text (it is very short and well worth a read). I am not sure if putting the full text online in this manner is completely legal but I doubt that Asimov would have minded – especially since it seems to be currently out of print.
The message of the piece struck a chord with me since I spend a lot of my working and personal life working with various mathematical packages such as Mathematica, MathCad, Matlab, Octave, the NAG libraries and so on (as you may have guessed from the sort of things I write about in this blog) – all of which give humans various ways of doing Mathematics on a computers.
You need a PhD in order to do percentages in your head you know?
Thanks to all of these technological aids, it is certainly not difficult to imagine a time in the very near future when those of us who can compute 9*7 in our heads are in the minority. In fact, it is quite possible that this is already the case. Some time last year I bought a new pair of spectacles that had a marked price of exactly 300 pounds. When it was time to pay for them, the nice lady at the till told me that it was my lucky day because they had just started a 15% off promotion. She then started hunting for her calculator so she could let me know how much I needed to pay as she didn’t know how to get her fancy till to do it for her.
I patiently waited…and waited…and waited some more – her desk was rather untidy you see. Eventually I said – “The discount is 45 pounds so I need to pay 255.” I didn’t want to appear arrogant so I added “I think” to the end of the sentence. She smiled but continued hunting for her calculator. Eventually she found it – punched in the numbers and was utterly amazed that I had worked this out in my head.
This bothered me – why should such a simple calculation be considered amazing by someone who was well educated? When I recounted this story to my wife she rolled her eyes and said (and I quote) “Do get off your high-horse Mike – not everyone has a PhD like you do. You think you are so clever just because you can do a bit of maths.” This upset me even more and off I went to sulk for a bit. If I tell the same story to a mathematician or physicist then they usually respond with a knowing nod and then something along the lines of “That’s nothing – the other day I saw a student get out his calculator to multiply 32 by 10”.
So is this a bad thing? Should we be worried that many people today would struggle to do these basic calculations in their head? Alternatively, to put it in a more domestic setting, who is right – me or my wife?
Strong opinions
A lot of people have some very strong opinions on this – check out “Will it rot my students’ brains if they use Mathematica?” by Jerry Glynn and Theodore Gray for example – where they argue that skills such as mental arithmetic or being able to evaluate integrals by hand are no longer useful in today’s society. Actually they put their point a little more forcefully than this when they say
“If you are worried that your child will suffer by not learning to solve a polynomial by hand, I would suggest worrying more about not learning how to skin a rabbit, or how to start a stalled car. Of all the failures of education likely to get your child into trouble, manual polynomial solving is not high on the list.”
My own opinion is a little less polarised than that – I personally feel that maths education in the future should contain both hand calculations and the use of computers. Basic mental arithmetic, for example, gives one a certain level of intuition about numbers that would be lost otherwise (I am reminded of the story about Richard Feynman and the abacus here) but if I had a large list of numbers to add up then I would use a computer to do it.
Similarly I think that is important that calculus students learn manual techniques such as integration by parts or partial fractions and be able to apply them to reasonably complicated functions and not rely on packages such as Mathematica all the time. On the other hand, they should also know how to evaluate integrals using a computer (both numerically and symbolically) and check that the result is plausible (Even Mathematica gets it wrong sometimes you know). Ideally they would also be able to knock up a simple trapezium-integration routine in a language such as python, estimate the errors, and talk about how you might improve it.
Enough of what I think – feel free to read the Asimov story and the other articles linked to in this post and post a comment letting me know what you think.
While tidying up the attic I came across my copy of the 2nd edition of The Penguin Dictionary of Curious and Interesting Numbers
“1980-0891=1089 which is one of only 5 patterns in which subtracting a 4 digit number from its reversal leaves the digits rearranged.”
The next entry was 2025 which implies that we have a lot of rather dull years ahead – (only numerically speaking I hope). Of course, as everyone knows, there is no such thing as an uninteresting number (check out the wikipedia page for The Interesting Number Paradox to find a “proof”) and so I embarked on a search to find out why the number 2008 was interesting (to me at least).
I failed! Here is the best I could do (thanks to this site):
2008 is a Kaprekar constant in base 3
So there you have it – 2008 is a boring number and to make things worse my attic is still untidy. Someone out there please try and prove me wrong – find something interesting about this number and let me know in the comments. If it passes my own personal test of ‘interestingness’ (loosely defined as anything that makes me say “thats cool”) then I will update this page with the results
Update
I have had a few great responses to this post most of which can be found in the comments section. My favourites are:
2008 is a Happy Number
There are exactly 1000 numbers less than 2008 that share no factors in common with 2008. That is, Euler’s totient function phi(2008) = 1000 (check here.) Would it be correct to say that 2008 is the beginning of the next Euler totient millenium then?
2008 is the prime number 251 multiplied by the sum its own digits.
Update 2 (Jan 8th 2008)
I have just discovered another blog post that has some fun facts about 2008 over at 360 – my favorite of which is
2008 can be written as the sum of of 16 consecutive positive integers: 118+119+120+…+132+133
The latest carnival of mathematics, hosted by Wild About Math, has been posted and contains 16 different articles – well worth a look.
I was recently playing around with version 14 of Mathcad and discovered that it had a problem with calculating the following indefinite integral symbolically.
The result Mathcad gives is
Which looks pretty scary. If I bypass the symbolic engine and calculate it numerically then I get a result of about 1.089 which is equal to the first term of Mathcad’s symbolic solution:
So if MathCAD has got the symbolic solution correct then the scary looking term containing the limit should cancel out the imaginary term. I have never come across the polylog function before so I tried to look it up in Mathcad’s help system but there was no reference to it there – a bit of an oversight in my opinion considering it seems to be a built in function. (A google search later and polylog turns out to be short for polylogarithm – the entry on Wolfram’s mathworld looks like a good place to start learning about it – yet another thing on my list of things to learn.)
Clearly Mathcad’s symbolic engine wasn’t capable of evaluating this limit so I had to try something else. The thought of trying to evaluate it by hand briefly crossed my mind but I didn’t fancy my chances much since I didn’t have a clue about the polylog function. My next thought was to fire up Matlab (2007a) and try its symbolic toolbox. First of all – Would it succeed to evaluate the integral where Mathcad had failed?
>> syms x
>> int(asin(x)/x,x,0,1)
ans =1/8*i*pi^2+1/4*pi*log(2)+1/2*pi*log(1-i)
>> simplify(ans)
ans =1/2*pi*log(2)
No problems there then but could it help with the polylog limit that would determine if I had found a bug in Mathcad? At first, the answer seemed to be ‘no’ since I couldn’t find a reference to polylog in Matlab either but, after a bit more googling, I discovered that polylog(2,x) is sometimes called the dilogarithm function and this is implemented in matlab as dilog(x) according to the help browser. Unfortunately it is not implemented for symbolic variables so I was stumped there as well.
Since I am a fortunate soul and have access to pretty much every computer algebra system that is worth having I then turned to Mathematica. Just like Matlab, Mathematica had no problem with solving the integral that stumped Mathcad:
Integrate[ArcSin[x].{x,0,1}]
gave
1/2*Pi*Log[2]
It also fully implemented the polylog function so I could try the limit:
-I/2*Limit[PolyLog[2, 1 – 2 x^2 + 2 x Sqrt[1 – x^2]], x -> 1]
gave
-I Pi^2/24
Which confirmed that Mathcad had got the correct result but just couldn’t simplify it fully like its two rivals could.
Finally, I would like to set a challenge to anyone who stumbles across this little part of the web – could you evaluate this integral by hand? I can’t but would love to know how it might be done. If anyone finds a solution please let me know.
While trawling around some of my favourite maths blogs I came across one from Good Math, Bad Math that left me slack jawed with wonder!
The British national lottery operator, Camelot, recently put out a scratch card game called “Cool Cash”. The rules are simple enough – there is a target temperature on the card and in order to win you have to uncover a temperature that is LOWER than the target.
This has been covered in several other blogs and the Manchester Evening News so I will not discuss it much further but will leave you to ponder the following quote from one confused customer.
“On one of my cards it said I had to find temperatures lower than -8. The numbers I uncovered were -6 and -7 so I thought I had won, and so did the woman in the shop. But when she scanned the card the machine said I hadn’t.
“I phoned Camelot and they fobbed me off with some story that -6 is higher – not lower – than -8 but I’m not having it.”
Perhaps this is why so many people are in debt here in the UK – they see a balance of -1000 pounds and assume that they are better off than last month when it was only -500.
Are you geeky enough to have a favourite formula? Because I am and it turns out that I am not the only one either. I have just googled ‘favourite formula’ and the second result was a link to the aply named ‘Mathematics Weblog‘ where the author described an equation usually known as Euler’s formula.
The Nobel prize winning physicist, Richard Feynman, once referred to this equation as ‘the most remarkable formula in mathematics’ and he really knew his stuff. Now I agree with Feynman, it really is an amazing formula since it connects some of the most important constants of mathematics in a small and elegant package, but it’s not my favourite. Elegant it may be but it’s not particularly useful.
My personal favourite is the general case of the above equation. Also called Euler’s equation, it is written as
Set 
to 
and out pops Feynman’s favourite but we can do so much more with it than that and in this post I am going to focus on using it to obtain the trigonometric addition identities.
At the moment I am writing an introductory Mathematica course and was recently looking for inspiration for potential exercises. One website I came across (I have lost the link unfortunately) suggested that you get something interesting looking if you plot the following equation over the region -3<x<3, -5<y<5. It also suggested that you should only plot the z values in the range 0<z<0.001.
Suitably intrigued, I issued the required Mathematica commands and got the plot below which spoke to me in a way that no equation ever has before.
So now I have a question – What other messages could one find hidden inside equations like this? For example, is it possible to generate a three letter word with a relatively simple equation such as the one above? Of course if you were allowed to use very complex equations (and make use of Fourier transforms maybe) then I guess you could spell out whatever you choose but that’s no fun.
If anyone finds other such messages in simple(ish) equations then please let me know.
While sat in the pub on a damp and grey Friday evening enjoying a few drinks with my friendly neighborhood web-guru, I had the idea of starting a blog with some maths in it. After all, I thought, surely there are not very many blogs all about maths (yes I know that THIS blog has been more about computers than maths so far – but there will be a lot of maths…eventually). I was wrong! Very wrong!
There are LOTS of blogs all about Mathematics and associated subjects and a wonderful tradition has emerged from them all – The Carnivals of Mathematics. The idea is quite simple – a blogger is chosen to be the host of a carnival and then he/she accepts submissions from other bloggers. The host then writes up a blog post which includes a summary of the accepted submissions. The carnivals are a great way of discovering new mathematics and mathematics based blogs and on 21st September the 17th carnival was published over on MathNotations
If the 17 links on this month’s carnival are not enough for you then you can find the locations of all previous carnivals over at carnivalofmathematics.wordpress.com/