René Descartes (1596 – 1650) developed the idea of graphical representation, whereby, he represented ‘real numbers’ by the length of a line and accordingly defined a co-ordinate system. Thus a horizontal line can represent ‘real numbers’ from zero onwards. See Fig.5 below :Next let us look at ‘fractions’. By a ‘fraction’ we mean ‘the ratio of two natural numbers’ e.g. 1:4, 2:7, 1:125, 23:2365, ……. and so on. Fractions are also called ‘rational numbers’.[8]

By contrast , there are irrational numbers, like P, √2, √3 etc.[9] When we express these decimals they are infinitely long.In other words the decimal expansion of ‘irrational numbers’ neither terminates nor is periodic; it extends interminably e.g. P = 3.14159…. computers have checked it to a million decimal places without it ending or repeating.

‘Rational numbers’ always terminate or give a periodically recurring decimal. Conversely, a recurring or exact decimal will always be a fraction.

Now we make a very surprising statement : There is the same number of fractions as natural numbers ! Cantor made a grid as follows:-