This week, the test is over, and it’s back to work, on the real numbers.
In keeping with the general theme of the course, real numbers are not defined as Dedekind cuts or Cauchy sequences, but as something much more familiar: infinite decimals.
I started off in the first lecture explaining the Pythagorean proof that the square root of 2 is irrational. So the rational numbers are not yet big enough to contain the numbers we need to do mathematics with (e.g. to measure the diagonal of a unit square). But is the set of infinite decimals big enough? I explained how we can find an infinite decimal expansion of the square root of 2. It turned out that very few of the students had learned the method of finding square roots at school, and none at primary school.
So I was able to tell them a little story. When I was at primary school, at a certain point the teacher took a dislike to me, and one day when I was off sick he taught the class square roots; when I came back he set a test on them. I had to invent an algorithm for finding square roots during the test. I felt very pleased with myself, and this should count as one of my earliest mathematical discoveries, even if not original.
I think my algorithm was very simple and unsophisticated, something like the one I showed the class in the lecture:
- 12 = 1 < 2, 22 = 4 > 2, so √2 = 1.…
- 1.42 = 1.96 < 2, 1.52 = 2.25 > 2, so √2 = 1.4…
- 1.412 = 1.9881 < 2, 1.422 = 2.0164 > 2, so √2 = 1.41…
and so on; this process can in principle be continued to find any digit. (Actually I hadn’t brought along the relevant numbers, so I had to ask a student with a calculator in the front row to call out the last two numbers.)
A little thought shows that this argument actually establishes that any positive real number has a real square root.
Of course we have to ask how infinite decimals represent numbers (which involves some talk of limits). I had introduced this by asking the class to be ready to explain why 0.4999… = 0.5000…. We had skirted around this question before, and I was aware that some people felt (or were prepared to argue) that the first number is not the same as the second, even if it is only infinitesimally smaller.
This turned out to be a good discussion; we went over Achilles and the tortoise (which is one approach to the problem), and then considered it more formally as a question about limits. At the end, even the doubters seemed to be convinced. My comment on this was that a mathematician’s response to the question would be, “I can’t answer the question until you tell me what you mean by 0.4999….”
In the final lecture I talked about the Principle of the Supremum: a non-empty set of real numbers which is bounded above has a least upper bound, or supremum. This is quite abstract, but I hope that the students will get something from this; at least when they meet it again they might recognise it. I didn’t even attempt to give a general proof, but merely a “proof by example”. So how do you show that a set, such as the set of positive real numbers x satisfying x2 < 2, has a supremum? You generate the supremum one decimal place at a time, exactly as for square roots. 1.4 is not an upper bound, but 1.5 is, so the supremum begins 1.4…; and so on.
I pointed out that this is the fundamental property of the real numbers, showing that we have filled in all the gaps in the rational numbers, and that it is the foundation for calculus and analysis. I was challenged to show how this could be the case. So I took the infinite series for e (the base of natural logarithms), and used the Principle of the Supremum to show that this series converges (to the supremum of the set of partial sums; all you have to do is show that the partial sums are bounded above, for example by 3, and then the supremum turns out to be the sum of the series).