I love maths. My adoration of maths has, however, been largely unrequited. Having long since abandoned my one-way devotion as hopeless, I have recently been forced into a number of startling episodes of nostalgia, now that Crox Minor (15) is studying hard for her GCSE exams at school. Notwithstanding inasmuch as which, her efforts bring back for me those long nights toiling over my homework, with all the attendant agony and impotent frustration.
But that was then. After ploughing through the fetid undergrowth of the Foundation paper (which is largely sums, sums and more sums) Crox Minor has pushed through to the sunlit uplands of the Higher paper, which offers more in the way of conceptual clout.
Now she, too, has fallen in love with maths and the power of number.
Recently, while we were walking together on the beach, she confessed to me her wonderment that a simple ratio such as pi can crop up repeatedly in the natural world. How can something that seems so demonstrably a product of human ingenuity find its expression in nature?
I came hard up against such wonderment when Crox Minor showed me a problem in a GCSE past paper. It went like this:
Prove that the sum of two consecutive integers is equal to the difference between their squares.
It’s amazing to me that the proof of something so staggeringly fundamental is within the grasp of GCSE maths. (I’m also amazed I’d never heard of this.)
It’s really, really easy.
If your integer is n, then the next consecutive integer is n+1. The sum of these is n+n+1 = 2n+1. Put that one aside, and let’s get squaring. The square of n is n2, and the square of n+1 is (n+1)(n+1) = n2+2n+1. The difference between these squares is n2+2n+1-n2 = 2n+1 … which is the same as the sum of the two integers. QED.
I don’t know about you, but I can’t help but be astonished by this. Take any two consecutive integers, and they add up to the difference between their squares.
It makes me start thinking about integers as concepts, and what it might be that makes them distinctive, and a whole lot of philosophical stuff I really haven’t the ability to articulate. I don’t think, however, that I should feel belittled, intellectually, in my sense of inexpressible awe. After all, such longheads as Whitehead and Russell needed three whole volumes to summarise the basics of the properties of numbers, and I believe (I haven’t read Principia Mathematica myself, you understand) it took them a hundred pages to show that 1+1=2.
As for me, I am just hopelessly lost in the wonder and awe of it.