What has zero dimension and has infinite length?

Answer 1

A straight line with no end points.

We say that a point has zero dimension, because we cannot move at any direction within the point, we just stay there. But a point can live in any dimension, depends on what coordinate system we introduce to represent that point.

For example, if we want to make a picture of a real number, say 5, we introduce a variable, say x, to represent the value of the number, then we say we impose an equation x = 5. Then, we plot a point, say A, on a number line (that we call it x-axis), exactly at the location that corresponds to the x-value, 5, and so we call 5, the x-coordinate value of the point A. We can generalize this for any real number a, by writing x = a. In this way, we say that we can plot infinitely many points on the x-axis, and if we connect those points, a line with no endpoints is formed, the line with equation x = a, which we call a linear equation in one variable. Now we can move to only one direction within the line, to the left, or to the right.

Thus, we started with points which are zero dimensional, we imposed an equation, and we ended in a one dimensional line, which is exactly the shaded of the x-axis. Thus, a point is a zero dimensional object, which lives in another object, which is one dimensional ( in our case, a horizontal line).

In order to be able to also move in another direction, up or down, we introduce another axis, perpendicular to the x-axis, which intersects the x-axis at zero location. We call the intersection point the origin. If we introduce another variable y, and call the vertical axis as y-axis, we say we constructed a new coordinate system, xy coordinate system, which determines an infinite flat plane, the xy-plane. So that any point on this plane will have two coordinates, x- and y-coordinates.

For example, the origin will have the same value, zero, for both x- and y coordinates, since it is the only points which lies at both axis, exactly at their zero location. To represent the coordinates of the origin, we write (0, 0) as a pair of numbers in order, where the first number is the value of x-coordinate, and the second number is the value of y-coordinate. In general, we represent any point on the xy-plane, as an ordered pair of numbers, of the (x, y) form. And to plot any point on the plane, we start moving far away from the origin in two directions: we move along x-axis, x-units to the right (if x is positive), or x-units to the left (if x is negative), and along y-axis, y-units up (if y is positive), or y-units down (if y is negative).

Now things become more interesting.

For example, if I want to plot the point (1, 1), these coordinates shows me the way of moving from the origin to get its unique location on the plane. I move along the x-axis 1 unit to the right, I stop there, and draw a dashed line parallel to y-axis. Then I move along y-axis 1 unit up, stop there, and draw a dashed line parallel to x-axis. Thus, the location of the point (1, 1), is exactly at the intersection of the two dashed line, so I plot it there. After that I observe that I can obtain the point (1, 1) also by moving first along the x-axis 1 unit to the right, then by continue of moving 1 unit vertically up. I also observe that if I continue to move in the same pattern, but now starting form the location of the point (1, 1), I can obtain another point with new coordinates (2, 2). And since two points determine a line, I connect them with a straight line, which contains all points that will have the same characteristic, y = x. Thus, I drew a unique line which still permits me to move only in one direction within the line. In this way I can draw infinitely many lines if I plot just one point, or at least two points. But now the line which is an one dimensional object, lives in a two dimensional object, a plane.

It is very important to note here that all points on the plane with coordinates for example (3, y) or (x, 5), where x-coordinate is a constant and y-coordinate can takes on many values, and where y- coordinate is a constant and x-coordinate can takes on many values, determine vertical and horizontal lines.

For example, all the points of the form (3, y) will determine a perpendicular line to the x-axis, which cuts the x-axis at x = 3, and is parallel to the y-axis, and all the points of the form (x, 5) will determine a horizontal line perpendicular to the y-axis, which cuts the y-axis at y = 5, and is parallel to the x-axis.

This helps to understand that if we introduce a new coordinate system, for example the xyz-coordinate system, all points of the form (constant, y, z), (x, constant, z) and (x, y, constant) will determine planes perpendicular to the missing variable axis, and parallel to the planes formed by the other two variables.

Therfore, planes which are two dimensional object, can live in a three dimensional space.

Answer 2

A line.