An excursion into integers, rationals and `real' numbers (Part 2)


Dad showed us some consequences of the irrationality of the square root of two, which I found startling but which gave me a deeper insight into what it meant to be irrational. The side of a square and its diagonal cannot both be measured exactly with the same ruler no matter how fine its markings. Can you believe that? A fancy way of putting this is to say that the side and diagonal of a square are incommensurable. If you have a ruler which measures the side exactly (meaning that the two endpoints of the side coincide with two markings of the ruler), then that same ruler when placed along the diagonal so that one of its markings coincides with the initial point of the diagonal will have the endpoint of the diagonal lying between two of its markings. Always.

And if the ruler measures the diagonal exactly it will fail to measure the side exactly. If you had at your disposal a new ruler with ultra close markings, or even an infinity of such rulers with every conceivable separation between the markings, it would make no difference. This geometrical fact gave me more respect for the square root of two - it held some interest.

Dad gave a two-dimensional example of a rectangle in which the lengths of the longer and shorter sides are in the ratio the square root of 2:1. This rectangle cannot be tiled with square tiles, no matter how small the tiles.

We have all seen rectangles which are approximately this shape. If you have an A4-pad of paper close by, look at its cover on which it says that the pad measures 297mm x 210mm. Now, the last fraction in the sequence mentioned above is 99/ 70. Multiply it above and below by three to get 297/210.

The A4 sheet is supposed to have the length of its longer to its shorter side in the exact ratio of the square root of 2:1 for a very practical reason. In this case (and in this case only, as a little algebra easily shows), the sheet can be folded along its longer side to give two smaller rectangles, for each of which the ratio of the longer to the shorter side is again the square root of 2:1.

Thus each of the new sheets thus formed has the same property as the parent. Two A4 sheets are the offspring of a single A3, and in turn, two A3s come from one A2. The first in the line of the A-series is an A0 sheet whose dimensions are 1,189mm x 841 mm. This is approximately a square metre in area.

Of the other three numbers, e and phi, much was said about phi but all that I remember now is something about a golden-rectangle and a mention of the Fibonacci sequence. Most of what I heard about the number e has also receded into the memory's mists, but I have a faint recollection of a bank manager who was willing to pay 100 per cent interest per annum on an investment of £1.

He was also willing to have interest added every six months at 50 per cent compound interest, or even to have interest compounded every day at the appropriate rate. He was even willing to be pushed to the limits of his resources and his generosity and allow the £1 to gather interest "continuously".

I also remember that even with all this frantic adding of interest, the £1 does not exceed all known fortunes by the end of a year. Instead it becomes a modest £2.72. The decimal £2.72 is the number e 2.7182818 ... rounded to two decimal places. (By the way, the letter e is used in honour of the mathematician Leonhard Euler.)

As for our old school chum, pi, the latest is that its decimal expansion is now known to 206,158,430,000 digits!

We covered many other interesting topics in the Mathematical Excursions evening class. One was Floors and Ceilings (nothing to do with building); another was Magic Squares and knights hopping all around the chessboard.

Speaking of which, I bet you think that there are 64 squares on the chessboard. Well, there are 204. How about that?

A further topic was Quadratics and Projectiles, while in another a ball was dropping from far up in the sky and we were trying to figure out how fast it was going at any given instant - instantaneous speed. Calculus stuff.

But Dad was at his most dramatic when we discussed probability. This was when I realised that he was a closet gambler, if not a latent conman. During these talks, packs of cards, dice and other miscellaneous props such as toy cars and large mugs made their appearance. There was talk of winning lotteries, fellows getting on planes with bombs, birthday paradoxes, goats and cars behind screens, envelopes with sums of money in them, buying cornflake boxes to collect models of Robin Hood and his Merry Men and, of course, playing poker.

He revelled in confounding our intuitions with bizarre examples. At one stage he showed us three cards, one coloured red on both sides, another coloured blue on both sides and the other coloured red on one side and blue on the other. If you choose a card from one of these three and it's red on one side, what would you bet me that the other side is blue?" Don't be fooled ! It's not a fifty-fifty bet (to use bookmakers' language).

In Code, A Mathematical Journey by Sarah Flannery with David Flannery is published by Profile Books Ltd, price £14.99.