Definition: Def. 10.03: Rational and Irrational Magnitudes

These things being assumed, it is proved that there exist an infinite multitude of straight lines commensurable and incommensurable with an assigned straight line - those (commensurable and incommensurable) in length only, and those also (commensurable or incommensurable) in square)1. Therefore, let the assigned straight line be called rational. And (let) the (straight lines) commensurable with it, either in length and square, or in square only, (also be called) rational. But let the (straight lines) incommensurable with it be called irrational.2

Modern Formulation

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References

Adapted from (subject to copyright, with kind permission)

1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"

Adapted from CC BY-SA 3.0 Sources:

1. Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016

Footnotes

1. To be more exact, straight lines can either be commensurable in square only, incommensurable in length only, or commensurable/incommensurable in both length and square, with an assigned straight line (translator's note).

2. Let the length of the assigned straight line be unity. Then rational straight lines have lengths expressible as $$k$$ or $$\sqrt{k}$$, depending on whether the lengths are commensurable in length, or in square only, respectively, with unity. All other straight lines are irrational (translator's note).