One Student’s Story
by Joe Noss
(Continued)
Don’t get me wrong here; I liked it. I liked the thrill of bringing a complex jumble of letters and numbers, down to a seemingly simplistic, innocent looking answer. Then came the fun of pulling a large wedge of paper over, and glancing nervously through rows of answers; and bingo: finding I’d got it right.
So why am I writing in the past tense? To the larger extent, my maths in school still goes on in this way. But in the last 12 months a maths project based in Boston, (along with the miracle of Bill Gates’ computing empire), has allowed me to begin to see mathematics in a rather different light, and take some tentative steps away from standard school maths.
Living in England, where extracurricular mathematics comes with the same degree of abundance as good weather, it was a new concept for me. Largely by accident, I fell into communication over email with a mentor, and we began discussing a number of maths problems and topics. Sometimes these were connected with the work I was completing in school, and at other times, more recently, we have been concentrating on the problems available on the web.
Consider this: What’s the probability that an integer chosen at random is prime? Sounds intimidating doesn’t it? At least, that’s what I thought as I looked over a problem on what I now know to be "probabilistic number theory" on Saturday morning. This was way outside the realms of anything I’d done in school, and of course, there was no recipe! In the beginning I was stuck. Then in a memorable message, my mentor asked, "Is there a difference between an event carrying probability zero, and an event which is impossible?" At the time this meant very little to me.
The "prime problem" burnt on in my head, and I read that the density of primes surrounding a given integer, "n", was equal to the natural logarithm of that integer. I saw the implication this had on the problem: If one in every "n" numbers is prime, then the probability of a given number being prime is the reciprocal of "n". But as "n" tended towards infinity, its natural logarithm became infinite, and the probability zero. The problem went round my head once more. Could this be the answer? It seemed so silly  I knew there existed prime numbers  how could the chances of hitting one be zero?
Through further emails my attention was drawn back to that message
in my inbox. "If you specify a point on a dartboard," came
another message, "what is the probability of me striking that point
with a dart?" Well zero, I supposed. The point had no area. But
was it impossible? Definitely not.
So it turns out that the probability of a randomly chosen number being prime does equal zero. But obviously it’s not impossible for a number to be prime. It came as a shock to me. As it would probably have done to my maths teacher, had I told her.
Our discussion has moved on to other topics now  but the intrigue of what maths is really about hasn’t left me. Things in school are largely still as they were, though the "recipes" have become more complicated, and I still enjoy carrying them out.
My experience in school has been that enjoying maths makes you "weird", and many a conversation with my peers has been thwarted by me announcing which is my favourite class. But if you’re a student somewhere in the world who’s interested in what maths is really about  dare to try this. I did.
I’m currently a 6.1 student at a school near London, England. I’m taking Maths, Further Maths, Physics, and English Literature at "A"level. I would be happy to discuss anything I’ve written here with anyone interested.
Drop me a line at: joenoss@ioe.ac.uk.
