Professor Stewart's Hoard of Mathematical Treasures (33 page)

For further information, take a look at:

And you may find it informative to look at what happened after that Channel 4 broadcast, at:

‘Ladies and gentlemen,’ the Great Whodunni announced, ‘My assistant Grumpelina will ask a member of the audience to place three cards in a row on the table, while I am blindfolded. I will then ask her to provide some limited information, after which I will name those cards.’

The cards were chosen and placed in a row. Grumpelina then recited a strange list of statements:

‘To the right of a King there’s a Queen or two.

‘To the left of a Queen there’s a Queen or two.

‘To the left of a Heart there’s a Spade or two.

‘To the right of a Spade there’s a Spade or two.’

Instantly, Whodunni named the three cards.

What were they? [Note that ‘two’ here means two cards, not a card with two spots.]

 

The first place most of us encounter mathematical infinity is when we study decimals. Not only do exotic numbers like π ‘go on for ever’ - so do more prosaic ones. Probably the first example We get to see is the fraction
. In decimals, this becomes 0.333333 . . . , and the only way to make the decimal exactly equal to
is to let it continue for ever.

The same problem arises for any fraction p/q where q is not just a lot of 2’s and 5’s multiplied together (which in particular includes all powers of 10). But unlike π, the decimal form of a fraction repeats the same pattern of digits over and over again, perhaps after some initial digits that don’t fit that pattern. For instance,
= 2.3714285714285714285 . . . , repeating the 714285 indefinitely. These are called recurring decimals, and the part that repeats is usually marked with a dot, or dots at each end if it involves several digits:

All this sounds reasonable, but the number 0.999999 . . . , or 0.9, often causes trouble. On the one hand, it is obviously equal to 3 times 0.3, which is 3 ×
, which is 1. On the other hand, 1 in decimals is 1.000000 . . . , which doesn’t look the same.

It seems to be widely believed that 0.9 is slightly less than 1. The reason for thinking that is presumably that whenever you stop, say at 0.9999999999, the resulting number differs from 1. The difference isn’t very big - here it’s 0.0000000001 - but it’s not zero. But, of course, the point is that you shouldn’t stop. So that argument doesn’t hold water. Nevertheless, many people get a sneaky feeling that 0.9 still ought to be less than 1. How much less? Well, by a number that is smaller than anything looking like 0.000 . . . 01, no matter how many 0’s there are.

A friend of mine, who worked in mathematics education, used to ask people how big 0.3 is, and then how big 0.9 is. Everyone was happy that the first decimal is exactly
, but on
being told to multiply by 3, they became nervous. One said: ‘That’s sneaky! At first I thought that point three recurring is exactly one-third, but now I see it must be slightly less than one-third!’

We get confused about this point because it’s a subtle feature of infinite series, and though we all do decimals, we don’t do infinite series at school. To see the connection, observe that
This series converges, that is, it has a well-defined sum, and the rules of algebra apply. So we can use a standard trick. If the sum is s, then
so 9s = 9, and s = 1.

There are lots of other calculations like this. They all tell us that 0.9 = 1.

So what about that number that is smaller than anything looking like 0.000 . . . 01, no matter how many 0’s there are? Is it an ‘infinitesimal’ - whatever that may mean?

Not in the real number system, no. There the only such number is 0. Why? Any (small) non-zero number has a decimal representation with a lot of 0’s, but eventually some digit must be non-zero - otherwise the number is 0.000 . . . , which is 0. As soon as we reach that position, we see that the number is greater than or equal to 0.000 . . . 01 with the appropriate number of 0’s. So it doesn’t satisfy the definition. In short: the difference between 1 and 0.9 is 0, so they are equal.

This is an annoying feature of the decimal representation: some numbers can be written in two apparently different ways. But the same goes for fractions:
and
are equal, for instance. No worries. You get used to it.