Continued Fractions Without Tears

This "python descending a staircase" format has the advantage of historical priority, and it suggests important generalizations. However, there are approaches that are more conceptual, and I would like to outline one of them. (For the sake of comparison, we will untangle the infinite fraction in its traditional form and tie it together with our theory at the end of the paper.) If by the end of this paper you feel that this is all pretty trivial, then I have succeeded; if you think it's tough, then this approach isn't your cup of tea. Our goal will be to discover and prove the essential facts about continued fractions and to develop a computer algorithm for them. The spirit of this presentation is geometrical, but it is geometry in an unusual setting: that of one-dimensional space. Remember the traditional drill sergeant's complaint: "Can't you tell your left hand from your right hand?" Our approach will be, essentially, to keep careful track of which points lie right and which ones lie left.