The Art Of The Infinite - our lost language of numbers.

Robert Kaplan and Ellen Kaplan Allen Lane (Penguin Books) 2003

A typical hardback. 160 x 240 x 30 320 pages.

Thick pages. An old fashioned style of book.

Good binding but that doesn't help much. It won't like open. Pages too stiff with that thick paper. Which is a shame because it is really a text book or reference book and leaving it open would be what you'd often want to do.

It teaches mathematics in a novel way, to me at least. By visualising numbers as collections of dots. i.e. remember three, think of three, as a pyramid of three dots.

Right at the beginning it starts out by showing you how visualising 1,2,3,4 etc, as dots and imagining (or writing) them a written below each other to make a pyramid - 1 dot at the top, then 2 beneath it and then 3 beneath that and so on you can come to many surprising understandings.

Like take one such pyramid and stand it on it's side instead of on it's base. So that the base becomes the hypotenuse of a triangle thus formed.

Then it is easy to see that you can take another such rotated pyramid - of the same size - and stick it onto that one to make a rectangle out of the two of them.

Try it. You'll see it won't be a square, it'll be a rectangle.

Say your pyramid had 7 rows, i.e. the numbers 1 - 7 expressed as rows of dots one beneath the other. Well stick two of them together in this way and you'll have a rectangle of 8 x 7 instead of a square 7 x 7.

Well the number of dots in that will be 7 x 8 = 56, right?

Well that's not a square, it is a rectangle, but we can clearly see that it falls into two halves. We made it out of two halves.

And maybe we're interested in the half, what we started with. Would like to know the number of dots. Which is like the total of the series 1 + 2 + 3.... to 7.

Well it has to be half of the rectangle. It is half of the rectangle. But the rectangle is the number of rows multiplied by the number of rows + 1 and all divided by two. n . (n+1) / 2

And you've derived a formula for the sum of a any series - just by looking at some dots.

And he goes on like that. Demonstrating amazing mathematical realities derived from looking at representations of numbers.

Fascinating.

Not too hard to understand. Not too simple to be useful and interesting. Very much not too simple... it moves on to quite involved stuff I'd never get my head around I'm sure, but can't be certain because I didn't get that far.

So it not really 'teaching mathematics' as I said it was - rather it is doing what it suggested in the title - giving an intro to a lost art, a lost language of numbers.

It doesn't have chapters for 'algebra', 'set theory', 'geometry' or whatever. It has chapters such as 'Designs on a locked chest', 'Skipping Stones' and so on. Quite a different language, quite a different approach.

The text is quite conversational and easy to read. It ranges across the ages and talks of figures from the past, problems and problem solvers from the past....

I'd say it is a great book. Not just a good one but a great one.

Sort of thing you'd want laying open in your study all the time....

## The Art Of The Infinite

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