The rectangular coordinates of a point 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 coordinate -- Right or left -- is always entered first.  The vertical coordinate -- Up or down -- is always entered second.  For that reason, (2, 3) is called an ordered pair.

The coordinates 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 coordinate.  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 coordinate, then, is called the x-coordinate; the second is called the y-coordinate.  We always write (x, y).

Finally, the coordinate 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 coordinates of each point.

To see the answer, pass your mouse over the colored area.
To cover the answer again, click "Refresh" ("Reload").
Do the problem yourself first!

a)

b)

c)

(2, 3)

(3, 2)

(−1, 3)

d)

e)

f)

(3, −1)

(−2, −1)

(0, 3)

g)

h)

i)

(3, 0)

(−2, 0)

(0, −2)

Problem 2.   Coordinate 0.   

a)   On the x-axis, what is the value of every y-coordinate?  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-coordinate?  0

On the y-axis, we don't move right or left.  At every point, x = 0.

The extremities of a straight line AB have coordinates (4, 3) and (15, 8), and that line is the hypotenuse of a right triangle ABC. Name the coordinates of the right angle at C.

C has the same x-coordinate as B. Therefore its x-coordinate is 15. And C has the same y-coordinate as A. Therefore its y-coordinate is 3. Its coordinates are (15, 3).

Actual versus potential infinities

The idea of an actually infinite straight line, is that it has no extremities
-- no end points.

We are to imagine that such a line actually exists.  Now.

A potentially infinite straight line, on the other hand, has two extremities.

It is potentially infinite in the sense that we may extend in either direction for as far as we please.  It is a line that 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 is a segment of an actually infinite one.  That has become the standard modern point of view.  In classical plane geometry, however, lines are potentially infinite.  Only what we have actually drawn can be said to exist.  That is consistent with the logical principle that we may not assume that what has been defined -- every idea -- exists.  If we do not assume that, then we have entered the realm of fantasy mathematics.

This quarrel about actual versus potential infinities arises only when a line is abstracted from the boundary of a figure. For we can speak of the "side" of a square -- as if there were a side apart from the square! The words are distinct, but they do not refer to any distinction in actuality. Analytic geometry, and modernism in general, gets distracted by these verbal and then logical distinctions. Plane geometry, on the other hand, is the study of figures. Our idea of a figure -- a whole form: a square, a pentagon -- does come first, and then we logically describe its physical construction. Plane geometry is thus concerned with what we actually perceive. But in the ultimate theorems, straight lines appear only as the boundaries of figures.

Finally, apart from the question of the practical value of actual infinities, if we imagine something to be actually infinite, then it is never complete, never whole. Is it meaningful to say that something that is never whole exists?

A ray is the idea of a straight line with one extremity.