Is God a Mathematician_ - Mario Livio [45]
The Mathematics of a New York City Map
Take a look at the partial map of Manhattan in figure 24. If you are standing at the corner of Thirty-fourth Street and Eighth Avenue and you have to meet someone at the corner of Fifty-ninth Street and Fifth Avenue, you will have no trouble finding your way, right? This was the essence of Descartes’ idea for a new geometry. He outlined it in a 106-page appendix entitled La Géométrie (Geometry) to his Discourse on the Method. Hard to believe, but this remarkably simple concept revolutionized mathematics. Descartes started with the almost trivial fact that, just as the map of Manhattan shows, a pair of numbers on the plane can determine the position of a point unambiguously (e.g., point A in figure 25a). He then used this fact to develop a powerful theory of curves—analytical geometry. In Descartes’ honor, the pair of intersecting straight lines that give us the reference system is known as a Cartesian coordinate system. Traditionally, the horizontal line is labeled the “x axis,” the vertical line the “y axis,” and the point of intersection is known as the “origin.” The point marked “A” in figure 25a, for instance, has an x coordinate of 3 and a y coordinate of 5, which is symbolically denoted by the ordered pair of numbers (3,5). (Note that the origin is designated (0,0).) Suppose now that we want to somehow characterize all the points in the plane that are at a distance of precisely 5 units from the origin. This is, of course, precisely the geometrical definition of a circle around the origin, with a radius of five units (figure 25b). If you take the point (3,4) on this circle, you find that its coordinates satisfy 32 + 42 = 52. In fact, it is easy to show (using the Pythagorean theorem) that the coordinates (x, y) of any point on this circle satisfy x2 + y2 = 52. Furthermore, the points on the circle are the only points in the plane for whose coordinates this equation (x2 + y2 52) holds true. This means that the algebraic equation x2 + y2 = 52 precisely and uniquely characterizes this circle. In other words, Descartes discovered a way to represent a geometrical curve by an algebraic equation or numerically and vice versa. This may not sound exciting for a simple circle, but every graph you have ever seen, be it of the weekly ups and downs of the stock market, the temperature at the North Pole over the past century, or the rate of expansion of the universe, is based on this ingenious idea of Descartes’. Suddenly, geometry and algebra were no longer two separate branches of mathematics, but rather two representations of the same truths. The equation describing a curve contains implicitly every imaginable property of the curve, including, for instance, all the theorems of Euclidean geometry. And this was not all. Descartes pointed out that different curves could be drawn on the same coordinate system, and that their points of intersection could be found simply by finding the solutions that are common to their respective algebraic equations. In this way, Descartes managed to exploit the strengths of algebra to correct for what he regarded as the disturbing shortcomings of classical geometry. For instance, Euclid defined a point as an entity that has no parts and no magnitude. This rather obscure definition became forever obsolete once Descartes defined a point in the plane simply as the ordered pair of numbers (x,y). But even these new insights were just the tip of the iceberg. If two quantities x and y can be related in such a way that for every value of x there corresponds a unique value of y, then they constitute what is known as a function,