When maths screws you over
Reasoning is the language of mathematics
There’s an interesting pattern involving points on a circle. We start with a single point on the circumference.
Now let’s add a second point and join them up. We’ve split the circle into two regions:
Add a third point and repeat.
Four regions. Any ideas what will happen when we take four points? That’s right, we end up with 8 regions.
Have you found the pattern? Of course you have; the number of regions is doubling each time. That seems to make sense. If you’re still in doubt, try the next one along:
Excellent, we get 16 regions as expected.
By now you’ll know what to expect for 6 points — double 16, or 32. Are you a betting person? How much would you gamble on the next circle returning 32 regions? Here goes — count them:
Did you do a retake? Me too; I couldn’t believe it either. Surely I’d missed one out — there’s no way it ended at 31 regions. Rest assured, your eyes have not betrayed you. Your mathematical reasoning, on the other hand…
Why did we expect 32 regions? Most likely because it continued a pattern we’d stumbled upon. Our minds are attuned to patterns; we instinctively seek them out. Each new circle strengthened our belief system, exponentially.
But that’s all it was; a belief. There was no rigorous argument, no solid reasoning that ordained the regions to double. The beauty of maths is in its patterns. But therein also lies its danger. Maths is more than pattern recognition. It’s also about understanding why those patterns hold.
That’s where reasoning comes in. We can only celebrate the irrefutable truths of mathematics once we’ve paid our dues and convinced ourselves, through flawless logic, that they are so. Maths will reward your diligence, but it will just as well screw you over if you abandon reason.
Is there another pattern to the circle regions, one that is more difficult to describe than simply doubling? Possibly; you may not care enough at this stage. Perhaps you’ve been wounded enough. Still, if your curiosity is biting, you may want to venture on, drawing the next circles along as you develop and test new hypotheses. Maybe a subtle pattern is lurking after all.
Now consider the following:
n² + n + 41
I reckon this expression returns a prime number whenever n is a non-negative integer. Don’t believe me? Try it out.
When n=0, it gives 41. Good start.
For n=1 we get 43 and for n=2 we get 47. Both prime.
My claim looks promising. You have every right to be sceptical; after the torment of the circle regions I’d expect nothing less. And if you know anything of primes, it is that they are a mysterious species. Surely we can’t breed an infinite supply of them so easily.
But how will you disprove my claim? You have to find a counterexample; an example of an integer n for which the expression n² + n + 41 is not prime. If you elect the systematic approach of trying each integer in turn, you could be here a while. In fact, the expression will return primes until n=40.
We may have intuited a counterexample by reflecting a little more on the expression. Can you see why it can’t be prime for n=41? One reason is that each of the terms in the expression is then divisible by 41, which means 41 is a factor of the whole expression. It certainly can’t be prime if that’s the case! My claim is exposed; it’s only remarkable it held up until n=40.
The brute-force method of checking each integer has nothing on the elegance of picking out n=41 from the expression. Reasoning is beauty.
Reasoning is getting more airtime in the school curriculum, and quite rightly so. It is too often seen as a self-contained topic, kept separate from all the others. A little something extra for Friday, once the ‘core’ material has been covered. Or something to challenge those so-called gifted students with.
Here’s the rub: reasoning is the language of mathematics. It is how we can be sure of anything. The what and the how of mathematics are important, but both are meaningless without the why. The most important question any maths student can ask is why? There is nothing more empowering than being able to rigorously justify your mathematical arguments. It gives you ownership of that knowledge; not even the most hardened among us can begrudge sound logic. Reasoning must be suffused in the learning and teaching of every concept.
Maths is the subject that always gives. But when we take it for granted it will just as easily lead us astray. We’ll stay on the righteous path, and guide our students there, so long as we retain our right to ask why?
Images sourced from this article on the circle regions problem.
I am a research mathematician turned educator working at the nexus of mathematics, education and innovation.
Come say hello on Twitter or LinkedIn.
If you liked this article you might want to check out my following pieces:
