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December 28, 2009

Popular Maths

Christmas is over, and a lot of us are presumably already enjoying our new presents. For me, no Christmas is complete without getting at least a couple of books, a present I always welcome. Lest I should sound like a total dweeb, I'd like to point out that I also wished for, and received, more standard presents. Still, a geek is a geek, and if you have followed my blog thus far -- or if you've had a peek at the 'About Me' section -- you may not be surprised to hear that I'm often given books discussing mathematical topics.

This year, however, I explicitly stated that I did not wish for any such books.

The reasons are twofold: First, it is a lot easier for me to decide which books are interesting and at the right level, and which books are too dull or too trivial. Additionally that means I can buy them second-hand, which saves money as well as paper. It is the second reason, though, that I want to focus on in this post. It may sound odd, but the majority of popular maths books that could potentially be a Christmas present, do not in fact target mathematicians like myself. Rather, they are meant to be enjoyed as gentle introductions to certain maths-related topics, to people whose main area of expertise is not maths. An appetizer, if you will. So whenever Iget one of these books, I always feel a little sad that whoever bought the book for me, did not buy it for him- or herself. 

Popular science books (books on popular science), such as Richard Dawkins' 'The Selfish Gene' and Bill Bryson's 'A Short History of Nearly Everything, are cropping up everywhere. Likewise, there are also plenty of popular maths books out there, some of which have received considerable attention and praise from the general public. Ian Stewart is worth mentioning in this context due to the popularity and success of his popular maths books, ranging from the serious, but still accessible, works ('Does God Play Dice', 'Letters to a Young Mathematician') to the more playful ones ('Professor Stewart's Cabinet of Mathematical Curiosities', 'Math Hysteria'). Then there are books which explore presents one concept in detail, but in layman terms (like 'Imagining numbers', 'Fermat's Last Theorem'), books which explore a wide range of topics on a superficial level (like "Why Do Buses Come In Threes?' or 'How to Cut A Cake'), books which give a brief introduction to maths in general ('Introduction to Mathematics'), and so on.  Not to mention the somewhat childish yet extremely enjoyable 'Murderous Maths' series. I have read many of these, and always find them agreeable to read, but when I'm confronted for the 55th time with a detailed presentation of the golden ratio, I can't help thinking: I shouldn't be the one reading this.

Stephen Hawking writes in 'A Brief History of Time':

Someone told me that each equation I included in the book would halve the sales

Most popular maths books are therefore written in an informal style and contains as few equations as possible, so as not to scare the reader. In fact, this is the main feature that distinguishes a popular maths book from a standard maths book. The author usually takes care not to lose the reader in his reasoning, and it is these clear explanations that sometimes make me realise: someone else should read this. I think this is where the problem lies. People think that any kind of maths is beyond them, and that they will never be able to understand or appreciate it. They think that what lies between the covers of those books is an inaccessible world, when in fact the content of such books is not the 'real' maths that university students are being taught, but a modified version of it, specifically designed to be understood by the non-specialist. My Algebra lecture notes are an example of a text that requires a certain degree of mathematical ability to read; 'How To Cut A Cake' isn't.

Please, come visit Mount Maths. It's a little lonely up here. But the view is incredible.


October 27, 2009

Native Counting

I've had the immense pleasure of being contacted by a fellow anosmic who happened to stumble upon my blog post on the inability to smell. She proceeded to read the rest of the blog (which is one of the greatest rewards any reader can give me) and says that she "especially like[s] [my] posts on language". So here's to you, Catriona, a post about counting in a foreign language.

Anyone who has ever properly attempted to learn a foreign language, will know that the most efficient way to learn, is to spend some time in the country in question. Learning French in a classroom with a teacher and some books is always good -- even necessary -- but the effects of spending a holiday in France are considerable, especially if you are on your own and thus forced to speak French all the time. Eventually, the language will almost "infiltrate" your brain and your sentences will no longer be direct translations of your native language, but instead they will be constructed and structured directly in French in your brain, without going through the intermediate step of English, or Danish, or whatever language you feel the most comfortable in. This may naturally take more than a week, but given enough time such a change is bound to happen. I remember waking up one morning during my first long school trip to France (the whole purpose was to expose us to the French language) and realising that my dream had been in French. It's a peculiar feeling, a mixture between pride that your language skills have allowed you to reach that stage, and shock due to the unfamiliarity of the experience.

I have spent the majority of last year in England, each of the three academic terms being 10 weeks long. 30 weeks of speaking all but English, with the only exception being my Danish conversations with my parents over Skype. All my dreams are in English, I swear in English, my brain constructs my sentences directly in English, and even my internal dialogue is in English. In fact, when talking to my parents or writing them emails, I often find myself searching for a word or a phrase in Danish that I would have no trouble expressing in English. I sometimes use my English-Danish dictionary in cases like these, effectively creating a new, unforeseen application of the book. So it would seem that my brain has somehow adapted to the environment, and is performing all tasks in a foreign language. And yet... Every time I'm playing cards with my friends, and I need to quickly evaluate the number of cards in my hand or in the deck, I find myself doing it in Danish!

I count in Danish. Be it cards, or people, or papers, it is the one function that my mind still does better in my native tongue. I am of course perfectly able to do it in English, but I can do it significantly faster if I allow myself to switch back to Danish. Consequently, I always count in my head, in order not to frighten or confuse people. From what I have heard, it is in fact quite normal. And rightly so; one would expect the brain to perform better in most tasks when using a more familiar language, so the surprise is that I would rather think in English for most other tasks (at least during term-time). The question is, what is so pecial about counting that makes it tedious in a foreign language?

I believe the answer lies in the fact that counting is something you learn at a very young age, and that you learn to do automatically. It is like walking or riding a bicycle: the process have become something so mechanic and automatic that very little brain-power is actually required to perform it. The only difference is that counting is linked to a language. When you count, a sequence of words, or sounds, will be going through your head, but the significance linked to each number will have disappeared. When you reach "twelve" in your head, you don't stop to think that about its quality as a number, its properties, or even the fact that 12+1=13. "Thirteen" is just the sound that follows naturally on from "Twelve". Only when you reach the last card in your deck, do you stop to interpret the sound "Fifty-one". This now becomes a number, with its usual properties and connotations, and you realise that you're one card short.
So the reason I still count in Danish, is because the sequence I have learnt to recite as a kid starts with "Entotrefirefem...", and changing that automatic process would be, if not impossible, then a very difficult and time-consuming task, requiring a lot of practise. All of which would be pointless in the end, since I might as well just stick to counting in Danish.

This, however, raises another question: What other tasks will also automatically be performed in one's mother tongue? The first answer that comes to mind, is reciting the alphabet, since this is essentially the same as counting, except with letters. Indeed, when looking up words in my afore-mentioned dictionary, I will silently be going through the alphabet in Danish. Also, it would seem like a logical step to assume that if counting is performed in one's native language, then so is basic mathematical operations, like addition or multiplication. Here, I will disagree. While it is true that I can only recite my 7-times table in Danish, I don't use any language when adding, subtracting, multiplying or dividing. If I write "6 x 7" on a paper, the symbol "42" will automatically form itself in my mind. I may then add words to it, which could theoretically be in any language. If I'm multiplying them orally, I will still see the mathematical symbols in my mind, and from there "translate" it into whatever language I am speaking. I think the reason this is so different from counting, is because in this case, the numbers are actually perceived as numbers, mathematical objects, rather than a string of letters or sounds.

I am unaware of other tasks that would be unnatural to perform in a foreign language, but I'm sure there must be others. And I dare anyone to show me how you ride a bicycle in English.


September 23, 2009

Choosing 2nd Year Modules

Inspired by this post, I decided to talk about the different modules I plan to take this year. If you're not a maths person, you can take pleasure in knowing that you will never have to know about this stuff. Also, it's a good way to get an idea of how many different areas Mathematics cover. It's not just one big 'Maths' module.

First, the compulsory ones (Core modules):

Second Year Essay: I have a friend who studies Politics (at Warwick). She always has at least two essays she's working on. We Maths students, on the contrary, only have yearly essays, which is of course totally sweet. And as if that weren't enough, we also get to choose the topic! I haven't decided on that one yet, though I'm wavering between Mathematics-of-Card-Shuffling and Something-To-Do-With-Chaos-Theory. The maths behind the Rubik's Cube could also be fun to have a look at, but I wonder if my tutor would accept that.

Differentiation: You'd think that by the end of our first year Maths course we would at least have covered differentiation. But oh no, it has only just started. From what I've gathered, it's about differentiating several functions of several variables. A generalised notion of a derivative, basically.

Vector Analysis: I've heard it's a bit like Geometry and Motion, with paths and trajectories and areas and surfaces and volumes and change of coordinates and all that.

Analysis III: I know what this is all about: formally defining integrals. Judge all you want, but I actually liked Analysis I and II. Learn and understand definitions of intuitive notions ("increasing", "tending to a limit", "continuous" ...), and rigorously work from there to prove complex theorems that often seems dead obvious when you think about them, that's what I like. Also the overall direction was very clear. I think I'll like this.

Algebra I: Also known as Advanced Linear Algebra. I wasn't too keen on Linear Algebra last year; it went from being mind-numbingly boring to over-your-head difficult. But I've made peace with eigenvalues and eigenvectors over the summer, so I think it'll be all right in the end. It'd better be; it's compulsory.

Algebra II: This is basically Group Theory, as far as I know. We touched a bit of Group Theory in college, and since then it has had a few cameo appearances in lectures. I don't understand what all the fuss is about, the definition of a Group seems rather straightforward to me, albeit a bit pointless. I hear Group Theory is a crucial concept in Mathematics, though.

Now we get to the optional ones, of which I still have to take a certain amount:

Metric Spaces: There's something magical about this. I've read ahead on this topic, and every time I read something new, I get this tingling feeling of delight in my stomach. There's just something neat about visualising metric spaces in your head. Maybe it's because I know that this is what leads to Topology, that it gets me so excited. Metric Spaces, please don't disappoint me.
This isn't strictly speaking a compulsory module, but one must take either that or...

Partial Differential Equations. And I've chosen both. PDEs don't appeal to me in the same way Metric Spaces do, but I've been told it's a useful tool to have. Although I know exactly what the module is about -- it's differential equations, but using partial derivatives instead of normal ones (duh) -- I have no idea of the difficulty, the concepts, the scope or whether I'm going to like it or not. We shall see.

Geometry: I want to do Geometry. Proper, formal geometry. Yes.

Mathematics of Random Events: The title sounds tantalising, but I guess the content is what matters. From what I can tell from the description on the Maths Department's website, this is something of a mixture between Analysis and Probability. While I adored Analysis, I abhorred Probability, so this is going to be an interesting one. But come on, "This module aims to provide an introduction to the mathematical ideas and language underlying the notion of randomness, which permeates through much of modern mathematics, as well as statistics and probability theory." I mean, who can resist that?

Stochastic Processes: Another module linked to probabilities. As much as I dislike probability, it is an important area in the mathematical world, and I know I can't try to work my way around it. So I might as well meet it face on, with my head high and a positive attitude. Besides, we did a bit of stochastics in college, and that wasn;t too bad. Also, I like the idea behind random walks, and that's one of the topics that will be covered, I believe.

Mathematical Economics A: Last year I did Introduction to Quantitative Economics, which was essentially Economics from a mathematical point of view. The one aspect I really enjoyed about the module was Game Theory. Game Theory is, in a nutshell, a mathematical study of what happens when two people play a game but they don't know what move the other person is going to do. Rock-paper-scissors style. Now, Mathematical Economics A is all about Game Theory, and nothing else it would seem. And I think it's fun.

Mathematical Methods for Physicists II: I'm no physicist, but this module was recommended to me by an older student, because it provides a nice introduction to something called Fourier Analysis. I have no clue what that is, but it comes up in later years and is a big thing. It should also be noted that while the word 'Physicists' is in the title of the module and while the exercises will probably be Physics-oriented, the actual content of the module is (apparently) purely mathematical. Which is a good thing.

Quantum Mechanics and its Applications: Last year I did Quantum Phenomena which was okay, if not a bit dull. Quantum Mechanics should provide a more mathematical and abstract presentation of Quantum Physics, which is just the way I want it. I am, however, taking this module tentatively, because I've been told it's a heavy load and it might not be so fun in the long run. But I want to give it a chance.

C programming: For the uninitiated, 'C' is the name of a programming language. I really like programming, although I would still qualify myself as a beginner. It's a kind of hobby for me, except I ought to spend more time doing it if I want it to become a serious pastime. Last year's module about Java programming was a good introduction to Object-Oriented Programming, but according to my friend and local Computer Scientist, Sarah, C is a much better language for programming games, a skill I would love to develop and perfect. After all, why else would you want to program?

Finally, there's Russian for Scientists. I definitely won't be doing this for credit, but I'm considering doing it for fun anyway, because I love languages and because I have a Russian-speaking friend. Plus, being able to say that you know a bit of Russian sounds awesome. Doing it for no credit also means I can easily drop it if it becomes too much of a burden.

All that brings me to a total of 173 CATS. The minimum is 120, maximum 180, and recommended maximum 150, the most sensible thing to do for me is to drop one or several of these modules as soon as I know which ones displease me the most.

Little side note explanation here: non-UK people often get confused when I talk about CATS, the Credit Accumulation Transfer Scheme, since they are used to ECTS, the European Credit Transfer System. ECTS is part of the Bologna process to make educational systems in Europe more comparable, which is why most European countries now use ECTS -- except England. The systems can still be compared, though, as 1 ECTS point = 2 CATS points. The irony is that "ECTS" is an English abbreviation.


July 02, 2009

Covered Clocks

The Warwick Arts Centre has these four clocks:

Clocks in the Arts Centre

(Picture taken from Dilip Mutum's blog)

My initial reaction upon seeing them for the first time was a feeling of bemusement. The function of a clock is to tell the time, so what is the point in hiding half of it? Also, why are the four clocks clustered in one place, rather than spread out nicely in the entire Arts Centre? I eventually realised: it is a work of art. And a clever one, in my opinion. One never needs the entire disc of the clock to be visible in order to tell the time; it is enough to know the position of the two hands. And by placing the clocks together in this fashion, the "artist" made sure that most of the time, it would be enough to look at one clock.

"Most of the time" is the key phrase here. A little thought reveals that sometimes, no single clock shows both hands. If one is only concerned with the hour and the minutes, it is also obvious that suffices to look at two clocks to see both hands. This leads to the question: how often does does one have to look at two clocks to know the time?

As a mathematician, I felt it was my duty to solve this. It turns out that the probability that the probability that two clocks are needed to tell the time, is 0.25. The following argument should make it clear:

A clock is divided into four quarters, or "zones". On every clock, two of these are visible, and two are hidden. By "adjacent zones" I mean zones that are next to each other; by "opposite zones" I mean zones that are not.

For any given hour, the hour-hand is in one of the four zones. If the minute-hand is in the same zone, or in one of the two adjacent zones, there will be a clock showing both hands. Only if the minute-hand is in the opposite zone, then no clock will show both hands. So during every hour, there will be a 15 minute time lapse (when the minute-hand is in the "opposite" zone) during which one has to look at two different clocks. It follows that 1/4 of the time, two clocks are needed. Hence the probability of 0.25.

We have made a few implicit assumptions along the way:

  • One can only tell the time by looking at the long ends of the two hands. Seeing a bit of the short end of a the hour-hand on a clock on whcih the hour-hand is hidden by the black area, is cheating.

  • A hand is never in two zones simultaneously, e.g. a hand cannot be in zone A and zone B in the same time. It will always tend a little to one side. Strictly speaking, this isn't totally true; when the time is, for example, exactly 3 o'clock, the hour-hand will be in zone A and B, while the minute hand will be in zone A and D. However, just one second later, the hour-hand is technically in zone A, and the minute-hand in zone B. My point is, the probabilities of the hands being in two zones are so small (1/3600 for the hour-hand and 1/60 for the minute hand) that we can neglect them. Also, if we view time as something continuous, these probabilities do in fact become zero, but I won't go into that, since it would only serve to confuse everyone.

  • The time on the clocks is uniformly random. In other words, we are equally likely to view the clocks at any time of the day. This isn't realistic. The probability that we are looking at these clocks in the Arts Centre at 3:14 in the middle of the night, is close to nil. But heck, we're setting up an abstract model to figure out this probability, and assuming the time is uniformly random is the most sensible thing to do.

  • The hand showing the seconds isn't needed to figure out what the time is. This assumption is reasonable, and it makes things easier. But it isn't necessary. If one includes the second-hand, we get a new situation which can also be analysed. In this case, the probability that one has to look at (i) one clock, is 0.4375, (ii) two clocks, is 0.5625. Can anyone show this?

But then, this year, something happened. The names of four important cities (New York, Moscow, Beijing and Coventry) were painted underneath the clock, and the time on each clock was then adjusted accordingly.

Clocks in Arts Centre - with names!


I'm sure the individual who made this happen felt that he or she had just shown the signs of a pure genius, and turned 3 redundant clocks into something more practical and business-like. In reality, he or she simply messed up [EDIT: I was wrong about this; see Sarah's comment for more details].  Before the names were painted, telling time was never too problematic, as we've seen. Now, if the hour-hand is not visible in the Coventry clock, one must know the time difference between Coventry and, say, Beijing. Even worse, there are times when the hour hand is visible only on a single clock. Indeed, the visibility of the our-hand goes as follows:

12.00-3.00: Visible only on the Beijing clock
3.00-5.00: Beijing, Coventry
5.00-6.00: New York, Coventry
6.00-9.00: Moscow, New York, Coventry
9.00-11.00: Moscow, New York
11.00-12.00: Beijing, Moscow

(This is during summer time, for the record)

This new layout leads to some new interesting questions:

  1. What is the new probability that one has to look at two clocks in order to see both hour-hand and minute-hand? What if we include second-hand?
  2. If time zones are chosen at random, what is the probability that, sometime during the day, no clocks show the hour-hand? What is the probability that one has to look at two clocks now?
  3. If the each clocks is set at random to any time of the day (so that minutes and seconds don't necessarily match each other on the four clocks), what is the probability that one has to look at two clocks? What if we include the second-hand?
  4. What if, instead of three hands, we have four? To generalise even more, what happens with N hands?

I leave these as an exercise for the reader.

***

Someone must have realised that people were having trouble deriving the actual time from the four clocks. So, as you can see above, each clock was set back to English time. All the clocks say 15:54 in the picture. The funny part is that they have to leave the names of the cities, unless they want to repaint the entire wall white. As a consequence, most people (or at least the ones I've asked) haven't noticed that the clock are all showing the same time.

I guess nowadays most people would just take their mobile phones out to figure out what time it is. I'm old-fashoned in the sense that I still wear a wristwatch.


May 23, 2009

Student Quantum Uncertainty

Compared to the activity from a few weeks ago, my blog is a bit silent at the moment. And (sorry Vincent) it will remain so until the 9th of June, the day I finish my exams. Right now, I need to do some serious revision instead of spending my time in the blogosphere. After that, I should have enough time to post regularly, as long as I don't run out of ideas.

The reason I am nevertheless writing a post now, is that I had a sudden flash of inspiration a moment ago, when looking for my pencil. I think I have discovered an important truth that should be incorporated in the general theory of Quantum Physics. I claim the following:

If pressure is applied to a student, he cannot simultaneously know where his pencil and his keys are.

Quantum is next Thursday. 9 exams in 12 days. And one is on a Saturday, on top of that. At least most maths exams are 1-2 hours long, unlike some other courses, in which the exams all seem to be 3 hours long. First exam on the list: Linear Algebra.

Linear Algebra pun.

(Maths joke. Never mind)

Wish me luck.


April 24, 2009

CBD

Okay, here's something I wanted to write about a while ago, and then completely forgot about...

*MATHS WARNING: If you are allergic to Mathematics, please stop reading now.*

Around two months ago, the 27th of February to be precise, it was my birthday. On one of my birthday cards, the date had been written: 27.02.90. Upon seeing it, I paused for a moment, becuase it was as if there was something familiar, yet not quite right about this sequence of numbers. Then it dawned on me: it was a simple permutation of the daate of my birth, 27.02.90, a sequence I happen to see quite often. And that got me thinking... how many permutations of my birth date exist?

Now, mathematically speaking, there are 720 ways to permute a set with 6 distinct elements (digits in this case). In this case, two pairs of digits are the same, the 2's and the 0's, so in total there is 180 distinct permutations. But there's more to it. Indeed, not all of these 180 sequences are valid dates. For instance, 79.20.02 is not good, because there's is no 79th day in the Month of Octodecember. More generally, the days must be between 1 and 30/31, the months between 1 and 12, etc. Formally, the following restrictions are imposed on the sequence D1-D2-M1-M2-Y1-Y2:

-D1 and D2 cannot both be 0
-M1 and M2 cannot both be 0
-D1 cannot exceed 3                                                            [Less than 40 days in a month]
-If D1 is 3, then D2 cannot exceed 1                                    [At most 31 days in a month]
-If M1 is 0, M2 is 4,6 or 9, and D1 is 3, then D2 must be 0   [30 days in April, June and September]
-If M1 is 1, M2 is 1, and D1 is 3, then D2 must be 0             [30 days in November]
-If M1 is 0, and M2 is 2, then D1 cannot exceed 2                [Less than 30 days in February]
-If M1 is 0, M2 is 2, and D1 is 2, then D2 cannot exceed 8   [28 days in February]
-M1 cannot exceed 1                                                            [Less than 20 months in a year]
-If M1 is 1, then M2 cannot exceed 2                                    [12 months in a year]

Notice first that I have ignored leap years. Those can be taken care of "manually".

Now, with those rules in mind, we can get a computer to work out how many permutations of my birth date exist. Henceforth, I shall call these Co-Bithdates, or CBDs. An example would be 29.07.02. Note that the year, "02" could mean anything, like the year 2, or 1902, or 2502. So to make sure that each CBD points to a unique date, we'll take it to be the first date after my birth. In other words, all my CBDs are going to lie between the 27th of February 1990 and the 27 of February 2090.

The only thing left to do is to write the program that is going to work out my CBDs. Which I did. In Python (I'm always a bit ashamed when writing "simple" linear programs in Python, since it has so much potential as an object-oriented programming language that I feel like I'm abusing it, but it's such a handy tool when it comes to these kind of I-need-a-small-and-simple-program-and-I-don't-want-it-to-take-ages-to write-situation). If there the least interest from people, I'll gladly send the source code to you, print it in another blog post, or just tell you when your Co-Birthdays lie if you're not that much into programming.

The conclusion of it all, is that I have 26 CBDs, not including the day I was born (My potential leap CBDs are the 29.02.07 and 29.02.70, but none of these are leap years). I have already missed 9 of them, which means I've got 17 left in my entire life, assuming that I live for 89 years or more, thus experiencing my last one on the 20th February 2079. The next one is on the 22nd July this year. I'll try and make something special out of every one of them from now on.


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