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March 30, 2009 | by  | in Opinion | [ssba]

The Basics of Counting and other Tricky Maths


Do you know someone who thinks university is a huge waste of time and money? Maybe a grandfather or uncle, who left school to work when he was 14, and was supporting a wife and a couple of kids when he was your age, you bloody lazy smart-arse. Are you sadistic enough to want to irritate that person just a little bit more than usual? Then get a copy of the first set of notes from Math214 this semester, and let them casually slip out of your bag at an appropriate moment. Because written right across the top of those notes in big bold letters is the topic of those pages and pages of notes: the basics of counting.

And that’s one reason that mathematics is so damn cool. Those letters are just sitting there all secure, saying to anyone who looks upon them: “Oh hi! You thought you learned all about counting? You think you’ve mastered it? Oh no, grasshopper. You have so much to learn we’re gonna start right here with the basics of counting. You won’t be ready for the hard stuff for a long long time.”

You probably learnt 1+1= 2 pretty early on in your schooling too, right? Early last century Alfred North Whitehead and Betrand Russell co-wrote the Principia Mathematica. The aim of Principia was to create a framework in which all mathematical truths can be derived. By page 378 there’s a proof that 1+1=2. Well, Whitehead and Russell don’t give a proof, but show that, once they’ve defined addition, 1+1=2 will follow. Russell also wrote a book called In Praise of Idleness, so he trumps the ‘basics of counting’ notes by a factor of about a billion. (In case you’re wondering about the goal of Principia, after had published it, a mathematician called Kurt Gödel came up with his incompleteness theorems which basically meant Whitehead and Russell had just wasted a lot of their time … which was probably karma seeing as Russell did pretty much the exact same thing to Gottlob Frege a few years earlier).

But mathematics isn’t just about proving what Uncle “get a job” Bruce considers to be blindingly obvious. If you’re more into contemplating the universe and infinity and stuff while watching the Wizard of Oz with the sound turned down and listening to the Dark Side of the Moon, then maths is totally for you. There are theorems that conclude that there are infinitely many sizes of infinity, there’s the mathematics of knots, and fractal mathematics (if you’re the type of person who listens to the Dark Side of the Moon while watching the Wizard of Oz, you’ve probably got posters of fractals up on your wall!). Mathematics has applications in the mind-bending (think string theory, chaos theory), and the mundane (calculating compound interest … which I guess might actually be exciting to people who are even sadder than I am).

Prime numbers fall firmly in the mind-bending category. A prime number is a number that has precisely two distinct divisors – one and itself (2, 3, 5, 7, 11 …). When it comes to maths and mathematical objects, one of the first things people do is look for patterns. But prime numbers seem to defy patterns: in the counting numbers they occur in bursts of seeming regularity, then disappear for such a long time they seem to have finished, but then one occurs, and then maybe another… People have been arranging prime numbers to look for patterns for thousands of years, in columns, in spirals; but a pattern still evades them. There’s even a prize offered for coming up with the next prime number. The biggest one found so far is 2^43,112,609 − 1.

Even though we can’t count them all, we know that there are infinitely many prime numbers, thanks to a Greek mathematician, Euclid, who was the first to come up with a proof of the “infinitude of prime numbers”, some time around 300BC. His proof relies on an earlier proof of his that all non-prime numbers are made up of a set of prime numbers that multiply together to the non-prime one. Euclid’s proof of the infinitude of primes is roughly: if there is a finite number of prime numbers, multiply them all together and add one to get a number. Call that number P. We can’t divide P by any of the prime numbers, because this would give a remainder of one. So either P is a prime number, or it isn’t and there are one or more primes that P can be divided up into that aren’t in our original list. Either way, we get the required contradiction – there is no finite list of prime numbers; given any such list, there is always at least one more prime. In other words, even though we can’t count them all, we know with certainty that the prime numbers go on and on for infinity.


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