Coordinates and Lines in the Plane

Abstract

In Chapter I, we studied primarily individual points in the plane, or at most finite sets of points: for example, the vertices of triangles, parallelograms, and regular polygons. The sole exception was a straight line, which we encountered in defining the product of a vector by a number. In this second chapter, we take up the description of lines in coordinate form. We will first show that the simplest possible relation between coordinates x and y yields a line. We then define the graph of a function and functions themselves. We will also obtain the equation of a line from the geometric properties of lines. We will do the same for the circle, the ellipse, the hyperbola, and the parabola, defining these curves first by their geometric properties. After this, we take up the question of defining a line parametrically. We define the winding number of a closed curve with respect to the origin. In the last section of this chapter, we present a geometric interpretation of the behavior of polynomials with complex coefficients. Using the winding number, we prove the fundamental theorem of algebra: a polynomial of degree n admits n roots.