Definition:Rational Line Segment

Definition
A rational line segment is a line segment the square of whose length is a rational number of units of area.

In other words, a rational line segment is a line segment whose length belongs to the set $\left\{{x \in \R_{>0} : x^2 \in \Q}\right\}$.



Also known as
This is also known as a rational straight line.

Euclid's Definition of Rational
Note that this usage of rational differs from the contemporary definition of rational number.

Let $AB$ be a straight line whose length $\rho$ is a rational number of units.

Then a straight line whose length $\rho \sqrt k$, where $k$ is an integer, is also rational straight line.

Thus, to, a straight line of length $\sqrt 2$ is defined as a rational straight line, despite the fact that its length is an irrational number of units.

In, the lengths of rational line segments are called rationally expressible instead, in order to distinguish from the standard usage of rational.

Also see

 * Definition:Rationally Expressible Number
 * Definition:Irrational Line Segment