What is an Ordered Pair and Cartesian Coordinates?

One of my most frequently requested explanations has to do with graphing concepts. I have many things to write about for this topic. However, in this first post, I am going to lay the groundwork for those by starting with the basics. I figure that I should construct my site by following my own advice: start with the basics, and build up from there! So here, I will discuss the Cartesian coordinate system and what is an ordered pair.

The Cartesian Coordinate System

So, what is the most basic concept of graphing? I would say it is probably the axis system and coordinate system. The standard blank graph that you have undoubtedly seen countess times is composed of a y-axis (the vertical line) and an x-axis (the horizontal line). These lines intersect at the origin, or the zero point of both lines, which denotes the boundary between positive and negative values of the axes. (That should answer your first question – what is a Cartesian graph?) Now, consider this: if the zero is somewhere in the middle of these lines, there is no actual starting point at all! They extend to infinity in all directions – up, down, left, right – and it is for this reason that the convention is to include arrowheads on the ends of both axes. It is also common to have ticks on the axes to indicate the scale of the axes. Like a number line, the x-axis extends with positive values to the right and negative values to the left. Similarly, the y-axis has positive values above the origin, and negative values below it. This numbering concept is specific to this Cartesian (or rectangular) coordinate system. The intersection of these lines creates four areas, known as quadrants, denoted by Roman numerals I to IV starting in the upper right portion of the graph (where x and y values are both positive) and then increasing in a counter-clockwise direction. Take a look at the following diagram, which should hopefully explain all of what I just said in a far more concise package!

This is a standard Cartesian coordinate system, with the ordered pair at point A shown.
This is a standard Cartesian coordinate system, with the ordered pair at point A shown.

Note that:

  • Quadrant I: x and y > 0
  • Quadrant II: x < 0, y > 0
  • Quadrant III: x < 0, y < 0
  • Quadrant IV: x > 0, y < 0

In this two-dimensional graph, any point that you can place corresponds to both an x and y value. These numbers just tell you how far away you are, vertically and horizontally, from the origin. If you only have this location information, you can figure out where the point goes. If you only have the point, you can look at the axes to figure out its location. Sometimes, graphs will be shown on a grid, or graph paper, whereas other times, the axes have labels and you have to extend some lines to determine the precise numbers.

What is an Ordered Pair?

Once you know the precise location of your point, you have its coordinate – and a coordinate is represented by an ordered pair. The convention of writing an ordered pair is to put the x-value and y-value in a set of parentheses, separated by a comma. So, for example, a point with x-value of 3 and y-value of 5 is denoted as (3,5). The first number is always the x, followed by the y. (When you get more advanced and enter into three-dimensional math, the z-coordinate follows the y.) Furthermore, if the point has an actual label – say, A – it can go in front of the ordered pair: A (3,5). Also, going back a bit now, the origin has the ordered pair (0,0). Every point has a different ordered pair, or a different coordinate – it is what defines the point.

If you have graph paper, it is easy to practice drawing points and getting used to these naming conventions. All you have to do is count the appropriate number in each direction and you know where your point goes. As I said, a solid understanding of the basics is a great place to start! Hopefully, this post has helped you to understand what is an ordered pair, and also given you a primer on the Cartesian graphing system. I will have more to say about related concepts in subsequent posts, so check back soon for those!

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