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Rational Numbers

  A rational number, in mathematics, is one which can be represented by a fraction, the ratio of two integers (the one which does the dividing, the denominator, cannot be 0). The rational numbers were introduced to solve the problem that there were equations of the form a/b = ? with no solutions. It is usual to write rational numbers ‘in their lowest terms’, meaning that the two numbers in the fraction cannot have any integers which divide both of them. One of the problems which most distressed the Greeks was that there are still equations with no rational number solutions (the example they discovered that there was no rational number whose square is 2, which means that there is no solution of the equation x2 - 2 = 0). This problem, which the Greeks did not attempt to solve, was overcome by the introduction of the algebraic numbers (this was not possible until systematic notation for polynomials was introduced).

It may seem to be difficult to think of numbers which are not rational, but in fact, very few numbers really are. One of the first controversial results of the set theory of Georg Cantor (1845 - 1918) was that the number of rational numbers was not the same as the number of real numbers. (This was controversial because mathematicians felt that any idea that there were different sizes of infinity was impossible to consider.) Cantor showed that the rational numbers were countable (that is, could be counted or written in a list) whereas the real numbers were not. The proof that the rational numbers were countable relies on a famous ‘diagonalisational argument’, in which he gave a way in which they could be listed. The rationals are written in an infinite square by writing all those with denominator 1 in the top row, as 0/1, 1/1, 2/1 and so on, then all those with denominator 2 on the next row, and so on. The way to list them is to start at the top, left-hand corner, to go one to the right (to 1/1), then diagonally down and to the left (to 0/2), then down (to 0/3) and then diagonally up and to the right (to 1/2) and again (to 3/1); then to 4/1 and diagonally down and right again and so on. Every number will at some point be included in such a list. SMcL



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