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On the real number line, each point P is uniquely associated with a real number p. Such a real number p is the coordinate of the associated point P. If p is positive, then P is p units to the right of the origin 0. If p is negative, then P is |p| (absolute value of p) units to the left of the origin 0. In other words, the position of a point P on the number line is uniquely determined by its coordinate p.
The word "coordinate" is commonly used. For instance, someone might say: "My coordinates are: 700, Mississippi Street, St. Paul, MN 55106." This statement refers to this individual's address, her exact location in a given area. It is this meaning of the word that we use mathematically to refer to exact position of a given point on the number line, the plane, or space. Â How can we determine uniquely the position of a point P on a plane? A moment of reflection might prompt you to ask: in reference to what? On the number line, the point of reference is 0 and there are only two directions one might go (left or right). On the plane however, there are infinitely many directions one might go from a given point of reference (east, north, west, south, northeast, etc.) There are a few ways we can determine the position of a point on the plane with a specific reference (you might even invent your own way). In what follows, we will look at a specific way of doing that: the Cartesian Coordinate System.
Cartesian Coordinate SystemThe basic idea behind the Cartesian Coordinate System is to use two real number lines that are perpendicular and intersecting (meeting) at their origins just as depicted below. We use the point where the two lines meet as the reference point. Notice that, since this point is on both number lines, its coordinate on the horizontal line is 0 and on the vertical number line, its coordinate is also 0. This point qualifies naturally for the point of reference and we will call it the origin.
Figure 1
Now, given a point P on the plane, how can we characterize in a unique way its location with respect to the origin (point of reference)? Consider the point P that is depicted on the figure above. How does this point relate to the numbers 5 on the horizontal number line and 3 on the vertical number line? Starting at the origin, one might go 5 units (steps) east (to the right) and then 3 steps north (upward) to get to the point P. Of course, one might go 3 steps upward and then 5 steps to the right to land on P.
Exercise 1 (i) Can you think of another way to get to P from the origin? (ii) Describe in words how you would get to the points Q, R, and S.
As you can see, no matter the way to get to the point P, it has to involve the numbers 5 and 3. These two numbers determine in a unique way the position of the point P on the plane in reference to the origin. We say that the coordinates of P are (5,3). 5 is the horizontal coordinate and 3 is the vertical coordinate.
Exercise 2 Refer to Figure 1. What are the coordinates of the points Q, R, and S? What are the coordinates of the origin?
Each of those points P, Q, R, and S is therefore associated with an ordered pair (P with (5,3), Q with (-4,2), R with (-7,-4), and S with (8,-7)). More specifically, if you are given an ordered pair (a,b), you can locate a unique point M whose horizontal coordinate is a and whose vertical coordinate is b.
Exercise 3 On the figure below, locate the points A, B, C, and D whose coordinates are given by (-2,5), (6,-4), (-7,-1), and (5,5) respectively.
We may also go the other way. That is, given a point on the plane, we can find its coordinates. Consider the point M in the figure above. As you can see, to get to M, we need to go 6 units right and 6 units downward. So that the coordinates of M are (6,-6).
Example 4 Use Figure 2 to determine the coordinates of the points N, T, and V.
Solution N(5,7), T(-3,3), and V(-6,-4).
The figure above is referred to as the Cartesian Coordinate System. In summary, in this coordinate system, each point on the plane determines a unique ordered pair referred to as the coordinates of that point, and conversely, each ordered pair determines the coordinates of a unique point on the plane. Therefore, we have this nice one to one correspondence between points on the plane and ordered pairs.
The basic idea behind this coordinate system is attributed to the French philosopher Rene Descartes (1596-1650). The horizontal number line seen in this coordinate system is commonly referred to as the horizontal axis, and the vertical as the vertical axis. It is also common to speak generically of a point with coordinates (x,y). Because of that, the horizontal axis is also commonly called the x-axis, and the vertical axis called the y-axis.
Definition 1
Note 1 Because the number line does not end on either side, what we see in the above figure is just a small window of the Cartesian Coordinate System. Just like the number line, the plane continues without bound. On the exploration page, the Cartesian coordinate system is set so that the x values (x-range) are from -5 to 5 and the y values (y-range) are from -5 to 5. We only see a part (or window) of the Cartesian coordinate system. Any point whose abscissa is less than -5 or greater than 5 or whose ordinate is less than -5 or greater than 5 is not be visible in this window. We refer to such a window as the viewing window. The viewing window can be adjusted by entering the x-range in the form "xmin,xmax" and the y-range in the same form "ymin,ymax" (do not include the quotation marks) and then pushing the button "set viewing window". Of course xmin, xmax, ymin, and ymax are values you choose. For example, if we want to set the viewing window so that x runs between -10 and 10 and y runs between -5 and 8, we enter -10,10 in the text-field indicated by x-range and -5,8 in the text-field indicated by y-range, and then we push the button "set viewing window".
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Exploration 1