We discuss further manifestations of the Stern-Brocot tree, which we discussed in the previous lecture. Here we introduce the matrix form of the tree, which sheds new light on how to move smoothly down the tree. To do this we introduce some basic definitions on 2x2 matrices and their determinants, including the operations of addition and multiplication, and the three special matrices I,L and K.Then we exhibit a somewhat novel planar, geometrical representation of the Stern-Brocot tree, identifying fractions with visible integral points in the positive quadrant. This gives an explanation of the curious role of the pseudo-fraction 1/0 which we used in the last lecture to generate the tree. Finally we introduce a new geometrical structure that results from combining the matrix and geometrical forms of the Stern-Brocot tree; the theory of wedges. A wedge is a particular triangle associated to a fraction, or to a matrix in the matrix form of the S-B tree, and these provide a curious tesselation of the visible positive quadrant, rather challenging to visualize!
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.