# Marvelous Maths

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?

Amazing, huh?

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.

Like, wow.

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.

Cromercrox is a recovering palaeontologist, author and editor who lists his recreations as writing, beachcombing, playing hard rock organ, supporting Norwich City FC and falling asleep.
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### 24 Responses to Marvelous Maths

1. Steve Caplan says:

If I recall correctly, on one of your blogs some time ago, you had mentioned enjoying Borges’ “Labyrinths?” Or did I remember incorrectly? In any case, I read some of the stories in “Labyrinths” eons ago, and recently decided to reread the collection. Incidentally (and this is how the tie in with mathematics comes), the sponsor of my recent art/science seminar at Purdue is an expert in S. American history and is extremely interested in the role/significance of mathematics in the works of Borges. He actually sent me an English translation of a scholarly work on the matter, which I’d be happy to share if you are interested.

Note: I too am reliving my mathematical experiences through my kids. Both are extremely gifted, and one is an absolute numerophile who can, among other distinctions, recall 116 numbers after the decimal point for PI…

• cromercrox says:

Thanks Steve. Indeed, Labyrinths is my favourite book, the one I’d take with me to my Desert Island.

I’d be interested in the grimoire you mention. Please send to the usual address: Third Park Bench On The Left, The Esplanade, Cromer.

Kids are amazing. Especially their memories. It’s all I can do to remember where I put my spectacles.

2. Heather says:

Yes! I love being surpassed vastly by my math-inclined children – at their age, I hadn’t been so shabby, but it makes everyone feel good to patronize Mom a bit. Dad does it, too – he actually remembers things and for the moment, slightly better than the older teenager, which keeps the natural order straight. As of next year, that will probably be all finished.

Anyhow, that feeling of wonder and awe is a keeper. I get it relative to harmonics, and looking at the stars, and at beating cardiomyocytes disconnected from the rest of the body.

Isn’t that feeling related to the desire we all seem to have, to enjoy vistas – uninterrupted views except for the horizon, such as that featuring in the back of this blog? I was thinking so the other day, how much more peaceful it is to come home and not have the vis-à-vis not only of another apartment building across the street, but of any building at all, than formerly.

I was amused that Facebook chose as one of the highlights of my seven years of its use, my photo of the delivery of your book “The Accidental Species”.

• cromercrox says:

It could be that education is wasted on the young. Soon Crox Minor will be starting her A-level course, and it’s at that point that I think I’ll have to do some actual revision, rather than relying on the wily cunning that middle-aged people pass off as superior knowledge and experience. I expect I’ll find much to be amazed by, things that I simply found ornery back when I was sixteen.

I am thrilled that FB rates the delivery of The Accidental Species as a highlight. The publisher told me last night that it’s sold about 4,000 hardbacks and 600 eBooks worldwide so far, and will soon be into a second printing – which is rather good, apparently.

3. stephenemoss says:

My son (16) is also in the final preparations for his GCSEs, so we too are enjoying revisiting the wonderful world of numbers. He’s a decent mathematician (A* in both his mock maths papers), but biology is what really excites him. That and Arsenal FC.

• cromercrox says:

Arsenal Schmarsenal. Support Norwich. Now, there’s suffering.

4. Owen Dunn says:

That’s quite nifty. Folk say GCSEs are getting easier, and they may or may not be, but I don’t remember any questions of the form “Prove blah” back in my day…

• Mike says:

I’m pretty sure the basic methods (solving an equation/simultaneous equations) were part of the curriculum when I did the equivalent (early 1990′s]), but the language used to describe the problem above does seem to differ from my recollection of how these things were presented.

• cromercrox says:

I did my GCSE maths in 1978, before Mrs Thatcher was Prime Minister. Back then the exams were known as ‘O’ levels. I really can’t remember what we did, but some of today’s questions do seem quite sophisticated. According to Crox Minor, when you read the paper for the first time under exam conditions, some of them look like this:

You have nine apples. You give five to your friend Paul. Now, calculate the mass of the Sun.

But seriously, there was one recent question in GCSE Physics that involved calculating the mass lost by a star in certain circumstances. I certainly don’t remember doing astrophysics at O-level. (Crox Minor got an A*, natch.)

5. Mike says:

It also works if n=a.

Have you read Simon Singh’s “Fermat’s Last Theorem” (pronounced Tehorem on the interwebz)?
If not, get your irreducibly complex eyes around it. It’s a beaut.

• cromercrox says:

I have certainly read ‘Fermat’s Last Theorem’, and it is indeed marvelous. It takes a writer of rare skill to be able to explain higher mathematics to those of us who have trouble with our bank statements.

I once attended a lecture by Singh in which he explained how hard it was to pitch an hour-long TV special about maths to the BBC, that is, the Horizon episode to which the book Fermat’s Last Theorem related. You could tell, he said, that the commissioners would have preferred a program called Fermat’s Last Dinosaur or Fermat’s Last Volcano.

The Taniyama-Shimura Conjecture notwithstanding, Crox Minor hasn’t come across calculus yet. I explain to her that differentiation is fun – it’s like breaking eggs to make an omelette. Integration, however, is hard, like trying to turn the omelette back into eggs.

• Mike says:

For me, they key to understanding integration was to realise that it’s the same thing as summation (only it’s a continuous, rather than a discrete process). Unfortunately, I didn’t make this connection until many years after my last mathematics exam.

• cromercrox says:

Just two words will send me quivering fetally behind the sofa. No, not ‘Dalek Invasion’ – but ‘Partial Fractions’.

6. Laurence Cox says:

It reminds me of the story that Feynman told about himself and Hans Bethe at Los Alamos. It is repeated in “The Pleasure of Finding Things Out”. Feynman was squaring numbers for use in a formula using a mechanical calculator, when Bethe came in and was giving him the results by mental arithmetic faster than he could get them on the calculator. Naturally, Feynman asked him how he did it and he explained it in terms like those above. Bethe was really sharp like this, he could even remember logarithms of numbers, which came in handy when you had to calculate functions like cube roots for which there is no simple mechanical procedure.

One consequence of our reliance on computers is that almost everyone has forgotten these mental arithmetic short-cuts.

• cromercrox says:

These days children are taught how to get close to a result by first approximation, to get some idea of what you are up against. There are all sorts of simple tricks, such as when multiplying by 5, it’s actually quicker and easier in one’s head to multiply by ten and then halve the result, a bit like counting all the cows and dividing by 4. Or not. (Need Coffee.)

7. I loved loved loved algebra at school! I would do algebra puzzles for fun.

3D trigonometry, on the other hand, broke my brain, no matter how hard I tried to see the solutions. But my carpenter husband can do it in his sleep.

• cromercrox says:

I never did 3D trig. My favorite part of maths is number theory (see above) incited mainly in later life by reading popular books on the subject. (Note to self – read Brian Clegg’s book on Infinity).

8. My sprogs are a lot younger, of course, so thus far my maths (not up to that much, though I do have an A level in it from long ago) has been equal to the task. When it gets complicated c. GCSE level I am planning to turn all maths queries over to ‘Er Indoors, who tells me maths was her best subject at school, and that she might have done a degree in it if she hadn’t chosen medicine.

Re. the difficulty or otherwise of modern maths syllabuses, my impression is that the main change from 30 to 40-odd years back is that there is a lot more varied stuff in them now. Thinking of what my brother’s kids (who are older than mine) have done in the way of GCSE maths, I’ve seen (e.g.) really quite sophisticated probability stuff that we never did in O or even A level. Conversely, it’s pretty clear they do less classical calculus (differentiation / integration etc) than we did, simply because there are more different topics to cover.

• cromercrox says:

Indeed – this makes the annual complaints about ‘dumbing down’ hard to believe. The exams are not only just as hard, but they cover all sorts of different things.

9. KJHaxton says:

I loved this kind of proof at school. So elegant and straight forward ( compared with other subjects and *people*). I’ve never figured out long division though…

• cromercrox says:

Quite. As for people, I much prefer golden retrievers.

• Well, Tolkien famously said he preferred trees to people, so you’re in august company.

10. That’s a lovely little proof.

I confess I have a liking of that kind of geometry that lets you draw various angles, make circles through sets of points, and construct perpendicular lines, all with nothing but a compass and a pencil. I love that stuff. It’s not exactly mathematics, but it’s still fun.

A set of skills I have that nearly nobody cares about. Kind of like pouring 14-by-17-inch polyacrylamide slab gels for DNA sequencing. Archaic, but something I was very, very good at.

• Update – and a straight edge. To go with the compass and pencil.