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400 words in 30 minutes on why I took maths further

By Peter Rowlett. Posted February 20, 2011

Goal: 400 words researched and written in half and hour. For me, for practice. Corrections welcome in the comments.

This time I try to answer a question from Ruby Childs: “why did you study mathematics?” Recently we recorded a Math/Maths Podcast with Ruby as a special guest and we discussed issues around this question. Why do some people like maths when others don’t? Why do people take maths further?

This is a bit of a cheat for 400words, because it’s a topic I didn’t need to research. It’s all personal. I could have just written this as a standard blog post but I’ve set myself up to try to do this piece of constrained writing and, hey, I’m busy this week so two birds with one stone is attractive!

Ruby would like more answers to this question. You can contact her via Twitter as @RubyChilds or send her a message via the Ask me anything feature on her tumblr blog.

∞

When I was nine, we moved house and school. I wasn’t particularly good at or keen on mathematics. Nine years later, I went to university to read mathematics. Recently I have been asked: Why did I choose to take maths further?
I recall moving into a small village primary school with the two boys in my year taking classes with the year above. The dozen or so girls in my year didn’t have this privilege. I was placed with the boys even though I would sit in the corner doing different work. However it sounds, I don’t remember this being a bad experience – it was simply that I hadn’t, at my previous school, covered the same material.
The headmistress, Miss Crabtree, was very keen on mathematics and though I don’t remember specific incidents I can only assume my enthusiasm for the subject from that age was from her. I seem have caught up well enough and, when we finished the primary school maths curriculum months before the end of school, Miss Crabtree brought out other “interesting material”. There was mathematics in these few months that we didn’t cover until the third year of secondary school.
At secondary school I found maths easy and enjoyed learning new topics. At GCSE I took three separate sciences. I was interested in science, too, but taking these as separate subjects unlocked a new mathematical GCSE: statistics. At A Level three of my choices were obvious – Maths, Further Maths and Physics. I chose Chemistry as my fourth.
Then came the careers questionnaires. I answered questions and the answer came back: Chemical Engineer. I did work experience in a local cement works and the university chemical engineering labs. Although it was interesting enough, I didn’t feel a proper enthusiasm for the subject. I decided to take physics instead, but still there was a bit too much messy lab work for my liking. I liked the logical structure and reasoning, which fit my way of thinking about the world. Eventually, and I don’t remember how, I came to understand mathematics was a valid option for university and signed up.
So why did I choose maths? I liked it, was good at it and the structure of it fit my way of thinking well. Did it fit my way of thinking because of early training from Miss Crabtree? Perhaps. In any case, this confused mixture of interest in physics and ability at logical thinking led me to prefer applied maths but do better at pure, and ultimately to excel at programming and go into a computing Masters degree. So that’s what led me to mathematics and what led me away again. Now, when something goes wrong or a problem needs solving, I can feel my brain starting a logical breakdown and I relax into mathematician mode, and I still like it.

∞

Time: 27 minutes. 473 words. Performance: too many words. I feel it would have been easy enough to take the last few minutes to shorten it by amalgamating paragraphs two and three to a new paragraph of half their overall length, but I decided not to because I am trying to give as full as possible an answer to Ruby’s question. This doesn’t quite fit the “400words” idea, but it’s my game and I’ll break the rules if I want to.

Explanation. More 400 words.

400 words in 30 minutes on sexy primes

By Peter Rowlett. Posted February 13, 2011

Goal: 400 words researched and written in half and hour. For me, for practice. Corrections welcome in the comments.

∞

With Valentine’s Day upon us a seemingly appropriate mathematical topic is sexy primes. These numbers surely have some attraction to those in the mood for love.
In fact, the “sexy” in “sexy primes” is a reference to the Latin word for the number six: “sex”.
Sexy primes are prime numbers separated by six places from another prime number. You may remember that a prime number is a number that can only be divided by 1 and itself. So 5 is prime because you can only divide it by 1 (to give 5) or 5 (to give 1). All other numbers divide into 5 to give answers which are not whole numbers. Try it: 5 divided by 2 gives 2.5 – not a whole number.
In fact, 5 is not just prime, it’s sexy too. That’s because 11 is also a prime and is six places from 5.
The next sexy prime pair? 7 and 13. Then, 11 and 17. Actually, that’s the second time we’ve seen 11, isn’t it? It’s six places above 5 and six below 17, and all three are sexy primes. This means we’ve found our first sexy prime triplet: three primes with common difference six. The second of these is 7, 13 and 19. You can get sexy quadruplets too and one sexy quintuplet, but only that one.
A natural question to wonder at this point is: what is the largest sexy prime? The largest we know of has over 11 thousand digits, but is it the largest? Well, we know there are infinitely many prime numbers – Euclid proved this in ancient Greece – but the higher up the list of numbers you get, the rarer prime numbers are. The question of whether there are infinitely many sexy primes depends on the distribution of primes, and this is something we don’t know. In fact, the question of the largest sexy prime leads us neatly to an unsolved problem called Polignac’s conjecture. Alphonse de Polignac, a French mathematician in the 19th Century, made a conjecture in 1849 that, if true, would mean there are infinitely many sexy primes. However, 162 years later, this has neither been proved or refuted.
That mention of the year brings up another connection: 2011 is a sexy prime. That’s because 2017 is also prime. Mathematics is an active field of research. Perhaps by 2017 we’ll have our answer.

∞

Time: 31 minutes. 397 words. Performance: not bad. I discovered my text editor was counting words wrong, so that “we’ve” was counted as two words. Sigh.

N.B.: @stecks suggested that we cover ‘Valentine’s maths’ for today’s episode of the Math/Maths Podcast. @axiomsofchoice suggested sexy primes. So this piece owes a debt to them for the suggestion.

Explanation. More 400 words.

400 words in 30 minutes on chaos and the weather

By Peter Rowlett. Posted January 15, 2011

Goal: 400 words researched and written in half and hour. For me, for practice. Corrections welcome in the comments.

∞

Why does the weather forecast only go a few days into the future, and why are they so often wrong about what will happen in a couple of days time? If they know where the weather systems – clouds and so on – are right now, and they have a good computer model for how things change over time, why can’t they predict the weather a fortnight from now, or a year?

The answer is that weather is a chaotic system. In everyday language, chaos means complete disorder without pattern but in mathematics the same word chaos refers to a particular phenomenon. Chaos theory studies the behaviour of systems that are very sensitive to small changes in initial conditions. This means slight differences in the starting point of a system can give very different results in the long term.

Because the system gives very different results when you run it twice with similar starting conditions, you might think that it involves some random elements that change each time you run the process. However, this mathematical chaos does not involve random elements. If you run the process twice with exactly the same starting conditions, it will give the same results. This is not like a random process – each time you run it you get a different outcome. We call such a process, in which running the system in the same way gives the same results every time, deterministic, because if we know everything about it we can determine in advance what will happen.

What makes chaotic systems interesting is that they are deterministic but so sensitive to initial conditions that they aren’t predictable. A slight difference in the initial conditions can give massively different outcomes.

For the weather forecast, the computer model is fed by initial data from weather monitoring stations. If some of that data is slightly off, or if the computer model doesn’t correctly guess what is happening between stations, the model can give radically different answers to what will actually happen. These differences mount up – if you got tomorrow’s weather forecast slightly wrong and used it as the initial conditions to forecast the next day, then that could be very wrong indeed.

This is why weather forecasts are continually updated and why forecasts are only made reliably for a short period in the future.

∞

Time: 39 minutes. 388 words. Performance: took too long. I started without a clear idea and started writing as I was researching. At 25 minutes I gave up on one idea and started anew. Bad practice! I should try to remember to take some time to process before I start writing.

Explanation. More 400 words.

400 words in 30 minutes on zero

By Peter Rowlett. Posted November 28, 2010

I am bad at writing concisely. I am not great at writing to a deadline. In order to practice both skills I make this pledge: 400 words researched and written in half an hour on a mathematical topic, weekly. I may not keep this up but we will see. It is for my benefit but if you also enjoy reading it that is good. I make no promises about the content except that it will be related to mathematics, so expect some variety and no pattern. I will think of a topic outside of the half hour, and sometimes I may have some prior knowledge of it, but generally I will research and write each piece very quickly. Corrections welcome in the comments. For the first entry, let’s begin where things begin, from zero.

∞

When you don’t have any apples you have zero apples, right? Zero is a natural concept and easy to understand. However, zero is a relative newcomer to the mathematical world and its introduction caused a shake-up in what mathematics can do.
In ancient Babylonian mathematics, the lack of a numeral in expressing a number was indicated at first by a blank space and later by a punctuation mark between numerals. This would be like us writing 2011 as “2\11”. However, this was not our zero because it was not used alone.
All over the world – notably in Greece, China and South America – different cultures used symbols for zero. Some, like the Babylonians, used this as a placeholder but others began to understand zero as a number itself.
In 9th century India, decimal zero was used as we might use it today in practical calculations. The symbol for zero was incorporated in Hindu-Arabic numeral system, which comes to us through the Islamic mathematician al-Khwārizmī. This system of numbers was brought to Europe in the 12th century, particularly by Fibonacci, and so zero was brought into modern mathematics.
But why is zero important?
Zero plays one role in helping to distinguish numbers. Take the number 101: this means one hundred, zero tens and one unit. Without the zero, we would struggle to use this shorthand to represent 101, 1001 and 11.
Apart from this, zero serves as a number in its own right. If you subtract any number from itself, you get zero. This opens up all sorts of possibilities. For example, if rather than just having nothing you can consider yourself to have zero of an item then this opens up the possibility of being owed items as debt. If you allow zero you then get on to negative numbers, and the idea that you can have, say, -3 apples. This means that if someone gives you 3 apples, you will return to the default state of having zero apples again. This idea is very important in trade and economics.
In mathematics, zero plays an important role as an identity element for addition. This means that when you add zero to any number, you get the number itself. This fact is important in group theory, which underpins much of mathematics and must be left for another time.

∞

Performance: not great first time out. 393 words in 38 minutes. At 30 minutes I was about 70 words over.