WE WILL BEGIN with vocabulary.
First, a coördinate. A coördinate is a number. It labels a point on a line.
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, (x, y).
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 ordered pair.
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 on one of the axes.
Now the horizontal axis is always called the x-axis, and the vertical axis is always called the y-axis. The first coördinate, then, is called the x-coördinate; the second is called the y-coördinate. We always write (x, y).
Finally, the coördinate axes divide the plane into four quadrants:
The first, the second, the third, and the fourth. The quadrants are labeled 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 x-axis, what is the value of every y-coördinate? 0
On the x-axis, we don't move up or down. At every point, y = 0.
b) On the y-axis, what is the value of every x-coördinate? 0
On the y-axis, we don't move right or left. At every point, x = 0.
c) Where is the y-coördinate always 0? On the x-axis.
d) Where is the x-coördinate always 0? On the y-axis.
a) Where is the x-coördinate always 2?
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 x = 2, because on every point of that line, the x-coördinate is 2 and that x = 2 is the equation of that line.
b) Where is the y-coördinate always −3?
On the horizontal line 3 units below the origin.
That line is called the graph of y = −3. And y = −3 is called the equation of that line.
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 x-coördinate as B. Therefore its x-coördinate is 15. And C has the same y-coördinate as A. Therefore its y-coördinate is 3. The coördinates at C are (15, 3).
Actual versus potential infinities
The idea of an actually infinite straight line is that it has no extremities
Obviously, that can be only an idea, because it is impossible to draw one.
(Similarly, it is impossible to produce an actually infinite sequence of digits, as this symbol, "0.36363636. . . ," is meant to signify.)
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 warned 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. That has been the authoritative point of view since the 19th century, when it also became authoritative to say that the real numbers stretched continuously from "minus infinity to infinity." Therefore the x-axis -- that "straight line" -- also had to be actually infinite.
In classical plane geometry, however, a line is potentially infinite; we may extend it for as far as we please. Only what we have actually drawn, that is, actually produced, can we say exists for mathematics. That is the essence of the logical principle that we may not assume that just because something has been defined, it exists. For if we accept every idea without question, then we have entered the realm of fantasy mathematics.
See First Principles of Euclidean Geometry, Commentary on the Definitions.
A ray is the idea of a straight line with one extremity.
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Copyright © 2012 Lawrence Spector
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