We tend to think primarily about visual algebraic models in terms of marks on a sheet of paper. Which makes difficult any attempt to draw the model of a Real line as distinct from a Rational or Irrational line.
However, if we consider a computer display, then we can introduce the possibility of animation. On a computer screen, the model of the Real number line would appear much as a drawn model on paper. In order to display a model of the Rational number line, the line might blink (red, say), and an Irrational line might blink blue. If that capability were implemented, then a line that blinked alternately red & blue would be an additional model of the Real line.
Now - what about a line minus a point? On paper usually shown by a small circle on the line. However a computer screen model might consist of a blinking green pixel. So - what would we make of a line that that blinked red & blue, with a single pixel blinking green?
From a rather naive perspective, it seems that arrows visually differentiate between segments, rays and lines. I'm more interested in how one might draw a rational (or irrational) number 'line'. (as distinct from the Real counterparts.)