We review the extended rational numbers, which extend the rational numbers to all expressions of the form a/b, where a and b are integers---even b=0. Then we give some examples of how these strange beasts might prove useful in mathematics. But first we give one example of where they are unlikely to be useful---in economics, where there is a big difference between a very big positive number and a very big negative number!When we graph rational polynumbers, we see situations where the graph seems to want to ``go up to infinity'' and then immediately ``come up from infinity''. This suggests to us that the mathematics wants us to connect these seemingly divergent arms of a graph, by introducing a point at infinity. We give some initial suggestions that this might allow us to compactly visualize the entire graph of a rational polynumber on a finite square with opposite sides identified. Topologically this is a torus, so perhaps studying infinity will naturally lead to calculus being more naturally visualized, at least for some problems, on a donut!The extended rational numbers also naturally arise in projective geometry, particularly the one dimensional projective line, which may be viewed as an affine line with one point (infinity) attached at both ends, or, as Mobius and Plucker realized, as the space of lines through the origin in a two dimensional affine space. This naturally connects with our earlier picture of extended rationals as arising from grouping integer points together into teams, along lines.Finally we show that the extended rationals can be used to parametrize a circle, or more general conic, as long as it passes through the origin. In fact we have seen this parametrization before (with axes interchanged!) but now infinity can be also used to close up and unify the parametrization. I end with two very interesting challenges: how to geometrically describe addition and multiplication on points on the circle.
Does mathematics make logical sense? No, it does not. Foundational issues have been finessed by modern mathematicians, and this series aims to turn things around. And it will have interesting things to say also about mathematics education---especially at the primary and high school level. The plan is to start right from the beginning, and to define all the really important concepts of basic mathematics without any waffling or appeals to authority. Roughly we discuss first arithmetic, then geometry, then algebra, then analysis, then set theory. This course is aimed for a general audience, interested in mathematics, or willing to learn.