Math Investigation eTwinning. Mission: Infinity...Affairs! Volume: Infinity | Page 13

Cantor’s Diagonal It now seems quite likely that we must be able to pair up any number set with the natural numbers, but this is not the case. One example are the real numbers , which include all the fractions but also numbers like Pi, e or √2. Of course there are infinitely many real numbers, but this infinity is of a “bigger kind” than before, and we say that the real numbers are uncountable. To prove that the real numbers can’t be paired up with the natural numbers, we use proof by contradiction. We start by assuming that the real numbers were countable. Then we deduce something we know is wrong: a contradiction. This means that our initial assumptions must have been wrong, i.e. that the real numbers are notcountable. All real numbers can be expressed as a decimal. The decimal expansion could go on forever, like 0.3333333…, or stop, in which case we can add 0s like in 0.12300000… Let us suppose that the real numbers were countable. Then we could write down all real numbers in a list. Below, every row is a new number, and the columns are the decimal expansions of the numbers: x1 . x11 x12 x13 x14 x15 … x2 . x21 x22 x23 x24 x25 … x3 . x31 x32 x33 x34 x35 … x4 . x41 x42 x43 x44 x45 … … We now construct a new number consisting of all the elements along the diagonal: 0 . x11 x22