I was reading about the formal construction of the rational numbers as pairs of integers on wikipedia and was interested in the diagram (see below) illustrating the equivalence classes geometrically as the gradients of lines going through the origin. It got me thinking maybe there's some interesting geometry that can be done on this. For example computing the sum of 2 rational numbers would be equivalent to "adding 2 of the lines to get a 3rd line", and maybe there's some geometric intuition behind how the 2 lines combine to produce the 3rd line. No doubt this has been studied before, does anyone know where I can learn more about this? I don't know what words I should look up exactly, couldn't find anything.
