René Descartes (1596–1650) had developed the idea of graphical representation, whereby, he had represented ‘real numbers’ by the length of a line and defined a co-ordinate system accordingly. Thus a horizontal line can represent ‘real numbers’ from zero onwards.

Cantor pushed his mathematical reflections on set theory and the ‘infinitely many’ further. He showed that there exist sets with an ‘infiniteness’ which is ‘beyond countability’. For this purpose, Cantor used a line segment as a measure for ‘real numbers’. Whereas ‘natural numbers’ belong to the sequence 0, 1, 2, 3, 4, and so on.. the ‘real numbers’ are those which are integers with a decimal placed between them somewhere e.g. 1.291 or 0.3125 or 365.89 and so on and so forth. These are also known as decimal numbers. In real life, we use ‘real numbers’ for measurements, weights, scientific calculations, cash transactions etc.

Cantor showed that if we take, let us say, just the line segment from ‘zero’ to 1, we have a set of numbers like:

a₁         0.1 3 5 8 6 4 3 2 …….
a₂         0.3 6 4 9 2 5 4 1 …….
a₃         0.2 8 7 2 1 6 3 6 …….
a₄         0.8 3 1 5 7 6 4 2 …….     Ω
a₅         0.7 1 4 3 1 8 5 7 …….
a₆         0.6 2 4 7 8 3 1 9 …….
a₇         0.4 3 7 6 3 9 4 5 …….
a₈         0.3 6 7 2 8 9 8 9 …….

And so on,