Business Mathematics and Basic Statistics · Ch 5 — Coordinate Geometry (Two Dimensions)

The Cartesian Coordinate System

Business problems that involve two related quantities — for example, the price of a share plotted against time, or a firm’s total cost plotted against the number of units produced — are easiest to study with a picture. The Cartesian coordinate system, named after the French mathematician René Descartes, gives us exactly that picture: a way to locate any point in a flat plane using a pair of numbers.

Draw two number lines that cross each other at right angles. The horizontal line is the x-axis, the vertical line is the y-axis, and the point where they cross is the origin, written $O(0, 0)$ . Together the two axes divide the plane into four regions called quadrants, numbered I, II, III and IV in the anticlockwise direction starting from the upper-right region.

Every point $P$ in this plane corresponds to exactly one ordered pair of real numbers $(x, y)$ , and every ordered pair corresponds to exactly one point. The first number $x$ is called the abscissa — the point’s (signed) distance from the y-axis, measured parallel to the x-axis — and the second number $y$ is called the ordinate — the point’s (signed) distance from the x-axis, measured parallel to the y-axis. The pair is written $(x, y)$ and not $(y, x)$ ; because the two coordinates play different roles, reversing their order generally gives a completely different point — this is exactly why it is called an ordered pair.

Figure 1 — Cartesian plane showing points A(3,2), B(-4,3), C(-2,-5), D(4,-3) in the four quadrants, with each point's abscissa and ordinate marked by a perpendicular to the axes

The sign of $x$ and $y$ tells us at once which quadrant a point lies in, without needing to plot it first:

Note

Quadrant Sign Convention

Quadrant	Sign of $x$	Sign of $y$	Example
I	$+$	$+$	$(3,\ 2)$
II	$-$	$+$	$(-4,\ 3)$
III	$-$	$-$	$(-2,\ -5)$
IV	$+$	$-$	$(4,\ -3)$

A point can also lie exactly on an axis rather than inside a quadrant. If $y = 0$ , the point $(x, 0)$ lies on the x-axis; if $x = 0$ , the point $(0, y)$ lies on the y-axis. The origin itself, $(0, 0)$ , lies on both axes at once and belongs to none of the four quadrants.

This coordinate picture is what makes the rest of the chapter possible: once every point is just a pair of numbers, questions like “how far apart are two points?”, “what point lies exactly between them in a given ratio?”, and “where is the balance point of a triangle?” all turn into ordinary algebra performed on those numbers — which is exactly the three ideas the rest of this chapter builds, in order: the distance formula, the section formula, and the centroid formula.

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