Definition:Rational Number/Geometrical Definition

Definition

The definitions of rational numbers and irrational numbers as specified in Euclid's The Elements is different from the contemporary definitions:

In the words of Euclid:

With these hypotheses, it is proved that there exist straight lines infinite in multitude which are commensurable and incommensurable respectively, some in length only, and others in square also, with an assigned straight line. Let then the assigned straight line be called rational, and those straight lines which are commensurable with it, whether in length and in square or square only, rational, but those which are incommensurable with it irrational.

(The Elements: Book $\text{X}$: Definition $3$)

And let the square on the assigned straight line be called rational and those areas which are commensurable with it rational, but those which are incommensurable with it irrational, and the straight lines which produce them irrational, that is, in case the areas are squares, the sides themselves, but in case they are any other rectilineal figures, the straight lines on which are described squares equal to them.

(The Elements: Book $\text{X}$: Definition $4$)

Also see

• Results about rational numbers can be found here.
• Results about irrational numbers can be found here.