Alex's Adventures in Numberland - Alex Bellos [174]
And here217;s where we come to another strange result. We know that there are c points between 0 and 1, and yet we know that there are fractions on the totality of the number line. Since we have proved that c is bigger than , it must be the case that there are more points on a line between 0 and 1 than there are points that represent fractions on the entire number line.
Again, Cantor has led us to a very counter-intuitive world. Fractions, though they are infinite in number, are responsible for only a tiny, tiny part of the number line. They are much more lightly sprinkled along the line than the other type of number that makes up the number line, the numbers that cannot be expressed as fractions, which are our old friends the irrational numbers. It turns out that the irrational numbers are so densely packed that there are more of them in any finite interval on the number line than there are fractions on all of the number line.
We introduced c above as being the number of points on a number line between 0 and 1. How many points are there between 0 and 2, or between 0 and 100? Exactly c of them. In fact, between any two points on the number line there are exactly c points in between, no matter how far or close they are apart. What’s even more amazing is that the totality of points on the entire number line is also c, and this is shown by the following proof, illustrated opposite. The idea is to show that there is a one-to-one correspondence between the points that lie between 0 and 1, and the points that lie on the entirety of the number line. This is done by pairing off every point on the number line with a point between 0 and 1. First, draw a semicircle suspended above 0 and 1. This semicircle acts like a matchmaker in that it fixes the couplings of the points between 0 and 1 and the points on the number line. Take any point on the number line, marked a, and draw a straight line from a to the centre of the circle. The line hits the semicircle at a point that is a unique distance between 0 and 1, marked a', by drawing a line vertically down until it meets the number line. We can pair up every point marked a to a unique point a' in this way. As our chosen point a heads to plus infinity, the corresponding point between 0 and 1 closes in on 1, and as the chosen point heads to minus infinity, the corresponding point closes in on 0. If every point on the number line can be paired off with a unique point between 0 and 1, and vice versa, then the number of points on the number line must be equal to the number of points between 0 and 1.
The difference between and c is the difference between the number of points on the number line that are fractions and the total number of points, including fractions and irrationals. The leap between and c, however, is so immense that were we to pick a point at random from the number line, we have 0 percent probability of getting a fraction. There just aren’t enough of them, compared to the uncountably infinite number of irrationals.
Difficult as Cantor’s ideas were to accept at first, his invention of the aleph has been vindicated by history; not only is it now almost universally accepted into the numerical fold, but the zigzag and diagonal proofs are generally hailed as among the most dazzling in the whole of mathematics. David Hilbert said: ‘From the paradise created for us by Cantor, no one will drive us out.’
Unfortunately for Cantor, this paradise came at the expense of his mental health. After he recovered from his first breakdown, he began to focus on other subjects, such as theology and Elizabethan history, becoming convinced that the scientist Francis Bacon wrote the plays of