Coordinate Geometry

In the previous chapters we saw that the relation between 2 variables in a pattern can be represented as an algebraic expression or a graph.

The invention of Coordinate Geometry by the French mathematician Rene Descartes enabled combining both these ideas. Coordinate Geometry can be considered as the integration of Euclid's Geometry with algebra.

Story goes that René Descartes famously loved to lounge in bed in the morning and think. It was on one such morning—as the story goes—while dreamily watching the path of a fly flitting around on the ceiling, that he came up with the xy plane of Cartesian coordinates.

Coordinate Geometry & Our World

Coordinate systems allowed more readily for measurements of natural events in the universe and paved the way for such greats as Newton and Leibniz who further developed these ideas in Calculus.

Coordinate Geometry extended the idea of a number line into a number plane. To the existing number line, another was added, intersecting the existing line at right angles, passing through the 0. By convention, in the second line the numbers going upwards were considered positive and the opposite side as negative. Hence any point on the plane could be described by a pair of numbers (x,y) where x represents the distance of the point from the original number line (now called x axis) and y represents the distance of the point from the new number line (now called the y axis). In memory of its inventor, the plane is called Cartesian Plane and the x & y readings of any point are called Cartesian Coordinates.

Hence the relation between 2 variables, say x & y, can be plotted as points where the relation holds good. The line joining these points gives a visual image or a graph of the relation.

The relation between the radius and the area of a circle A = π and between the radius and circumference of a circle C = 2πR can be plotted in the coordinate plane as an infinite series of points where R can be a point on the X axis and C & A can be points on the Y axis. They would seem to form lines or curves which at every point on the line or curve, obey the above relations.

Coordinate Geometry allows algebraic expressions to be plotted as graphs and their properties studied visually and in terms of geometrical ideas.

It also allows 2 equations to be plotted on the same graph and the points of intersection studied as solutions to the 2 equations.

Coordinate Geometry & Algebra opened the field of analysis of relations between expressions & equations.

Calculus

The development of coordinate geometry also led to the invention of calculus by Newton & Leibnitz. Calculus was invented to understand quantities which changed over time or with respect to other quantities. For example, when a car is driven, its velocity keeps on changing from moment to moment.

In calculus a function (or relation) involving x & y can be plotted as a graph. Similarly, the rate of change of y, at every point with respect to x, can also be plotted as a graph. Hence a lot of relations governing parameters in real life can be plotted as graphs and their behavior analysed. Hence it is also called Analytic Geometry.