Counting All Pairs

The collection of points displayed is a portion of the set of all ordered pairs of integers (lattice points). Your goal is to establish a counting pattern (a path) that sets up a one-to-one correspondence between the counting numbers (natural numbers) and infinite sets of ordered pairs of integers.

Start a path by clicking on points; the list you are making appears in the space at the left.

With the red axes in their initial location, the points displayed are first quadrant points (both coordinates are positive integers). The arrow buttons at the bottom and on the right shift the portion of the plane that appears in the workspace, so that moving the axes to the center of the workspace displays points in all four quadrants.

Activities

1. Describe or create a path that goes through all the first quadrant lattice points starting at the point (1, 1) and consisting entirely of horizontal and vertical segments. Could your path first go through all of the lattice points i unit above the x-axis (the lattice points on the line y = 1)? That is, could your list begin {(1, 1), (2, 1), (3, 1), . . .} and get all of the lattice points on the line y = 1 before getting any points on the line y = 2?

2. Describe another path that consists mostly of diagonal segments.

3. Draw the path that begins {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (3, 4), (1, 3), (2, 4), . . .}. Will the continuation of such a path allow you to include all of the lattice points of the first quadrant?

4. Adjust the workspace so the origin appears at the center. Indicate how you would create a list that contains every lattice point in the plane by describing a path that begins at (0, 0) and goes through all the points. How can your path be used to set up a one-to-one correspondence between the set of lattice points of the whole plane and the set of lattice points in the first quadrant? Could your path go through all the points in the first quadrant before picking up points in the other quadrants?

5. Can you describe a path that goes through every lattice point in the plane and that consists entirely of horizontal and vertical segments? How about a path that contains no horizontal or vertical segments? Can you describe a path that consists mostly of triangles (concentric or not)?

6. Adjust the workspace so that the y-axis is on the left edge of the workspace and the x-axis goes through the center, (first and fourth quadrant points are visible). Using only first quadrant points, if we don't count duplications, and pairing the point (a, b ) with the positive rational number a/b, a path through all first quadrant points sets up a one-to-one correspondence between the counting numbers and the positive rationals. Now make a path to go through all points in the first and fourth quadrants and explain how to get a one-to-one correspondence between the counting numbers and all rationals, positive and negative.