Definition:Rational Line Segment

From ProofWiki
Jump to navigation Jump to search


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 $\left\{{x \in \R_{>0} : x^2 \in \Q}\right\}$.

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 $3$)

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 Euclid, 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 $\mathsf{Pr} \infty \mathsf{fWiki}$, the lengths of rational line segments are called rationally expressible instead, in order to distinguish from the standard usage of rational.

Also see