31 ## RECTANGULAR COÖRDINATESActual versus potential infinities WE WILL BEGIN with vocabulary. First, a coördinate. A coördinate is a The coördinate 0 is called the origin of coördinates. Distances to the right of 0 are labeled with positive coördinates: 1, 2, 3, etc. Distances to the left are labeled with negative numbers: −1, −2, −3, etc. Each coördinate is the "address" of a distance and direction from 0. A coördinate axis is a line with coördinates. Now, to label a point in a plane (a flat surface), we will need more than one coördinate axis, and we place a second at right angles to the first. Distances above the origin will have positive coördinates; distances below, negative coördinates. Those axes are called rectangular coördinate axes, because they are at right angles to one another. The coördinates on them are called rectangular coördinates. They are also called Cartesian coördinates, after the 17th century philosopher and mathematician René Descartes; for he was one of the first to realize the possibility of solving problems of geometry by means of algebra with the coördinates. Hence we have the name coördinate geometry or, as it is often called, analytic geometry. Rectangular coördinates are an ordered pair, ( The pair (2, 3) -- over 2 and up 3 -- labels a different point than (3, 2): over 3 and up 2. The horizontal coördinate -- Right or left -- is always entered first. The vertical coördinate -- Up or down -- is always entered second. For that reason, (3, 2) is called an The coördinates of the origin are (0, 0). We don't move right or left and we don't move up or down. We will see that 0 is an extremely important coördinate. It means that the point is Now the horizontal axis is always called the Finally, the coördinate axes divide the plane into four quadrants: The first, the second, the third, and the fourth. We label the quadrants counter-clockwise. Problem 1. Name the coördinates of each point. To see the answer, pass your mouse over the colored area.
Problem 2. Coördinate 0. a) On the
b) On the
c) Where is the d) Where is the Problem 3. a) Where is the On the vertical line 2 units to the right of the origin. In fact, we say that that vertical line is the graph of the equation
That is the equation, because at every point of that line, the b) Where is the On the horizontal line 3 units below the origin. That line is called the graph of Problem 4. In which quadrant does each point lie? Or is it on an axis; if so, which axis?
The extremities of a straight line AB have coördinates (4, 3) and (15, 8), and that line is the hypotenuse of a right triangle ACB, whose sides are parallel to the axes. Name the coördinates of the right angle at C.
C has the same Actual versus potential infinities An actually infinite straight line is a line that it has no extremities -- no endpoints. Obviously, such a line is a mental object only. It is not possible to draw one. (Similarly, it is not possible to produce an actually infinite sequence of digits, as this symbol, "0 Since the A potentially infinite straight line, on the other hand, has two extremities. It is potentially infinite in the sense that we may extend it in either direction for as far as we please. It is a line we could actually draw. The student should be advised that when writers use the expression "straight line" these days, they invariably mean an actually infinite line. Hence they refer to any finite line, with its two extremities, as a line segment. They imagine that every finite straight line -- even the side of a square you might draw -- is a segment of an actually infinite line. In classical plane geometry, however, a line is potentially infinite. We may extend it for as far as we please, which in practice is all we ever require. That is consistent with the logical principle that we may not assume that, just because something has been defined, it exists. We define the objects of Euclidean geometry: the angles, circles, squares. The word "exists," however, applies only when we have drawn them; only when we have brought them into this world. See First Principles of Euclidean Geometry, Commentary on the Definitions. A ray is also a mental object. It is the idea of a straight line with one extremity. Next Lesson: The Pythagorean distance formula Please make a donation to keep TheMathPage online. Copyright © 2014 Lawrence Spector Questions or comments? E-mail: themathpage@nyc.rr.com |