Introduction to Coordinate Geometry

The Cartesian plane is one of the greatest inventions in mathematics.  Had Rene Descartes not invented the rectangular coordinate system, calculus would not have progressed immensely as we are using it in our time.

The Number Line

The coordinate system was derived from the correspondence of the real numbers and the points on number line.  Each point on the number line represents a unique number coordinate and any real number can be located — at least in principle — on the number line.

The number line represents the ‘entirety’ of the real numbers.  By convention, the number line is a horizontal line where the negative numbers are placed on the left of 0, and  the positive numbers on the right.

Creating another  number line, this time a vertical line passing through 0, whose negative coordinates are below the horizontal line and positive coordinates above, creates the rectangular coordinate system.  Notice that now,  any number on the coordinate plane can be represented by two numbers: its distance from the x-axis, and its distance from the y-axis. The coordinates (2,3) is the point 2 units away from the y-axis and 3 units away from the x-axis. 

The coordinate system enabled mathematicians to ‘marry’ geometry and algebra.  After the invention of the rectangular coordinate system, algebraic representations became available to points, lines, curves, and other geometric constructions and drawing.

After Descartes invention of the coordinate system, other systems followed. The coordinate system in three dimensions and the polar coordinate system are among the earliest derivatives.

One of the applications of the coordinate system, and perhaps the most practical, is the mapping of the earth into latitude and longitudes.  The mapping was believed to be first proposed by Amerigo Vespucci, the man who discovered the New World. Ironically, Vespucci, lived more than 100 before Descartes was born.

Another practical application of coordinate systems is the invention of pixel in computers and television screens. Each pixel in the screen has its own coordinate, where in most system, the corner at the upper-left is (0,0) and both axes are positive. Hence, the coordinates (100, 20) means that you travel 100 pixels to the right, and then 20 pixels down.

From the discussion above, it seems that there is truth in Leopold Kronecker’s famous quote that “God made the integers; all else is the work of man.” Mathematics is a system, the most consistent among the sciences (although not entirely consistent), but the consequences of everything is based on the axioms, and postulates.  It’s like chess. The game is played based on predetermined rules.

What if a different coordinate system was invented? What if the polar system was invented first and the Cartesian coordinate system was not invented? Surely, mathematics would have been different.