Rulers in a bounded model

We are going to figure out what distance measurement can look like if we scrunch a whole line into a finite interval.  To start with, we will review a familiar ratio that we use to locate points on a line.

Graph CA/CB, ratio on a line

• On graph paper, on the x-axis, mark points A and B.  Then for other points on that line, mark variable points C and label them with the value of CA/CB (remember this is a SIGNED ratio).

• Then plot the values that you have marked (i.e., for each x-value C use the ratio as the y-value). 
• Then sketch in what you think the graph looks like.  There are a few things that are particularly significant: (a) Where is the ratio 0 and where does it approach infinity? (b) Where is the ratio positive and where is it negative? (c) Is there a missing value for the ratio?  (d) What happens when the point C is very far from A and B in either direction?

Graph a "scrunched distance"

This time, imagine that you begin with a number line with numbers going 0, 1, 2, 3, … (and all the numbers in between) in the positive direction and  …, -3, -2, -1, 0 in the other direction.  Now suppose that you compress the whole line into a segment. (Technically, this means a function from the number line to the segment, but don't think of the details of that for now.)

• Start again with an A and B on an x-axis on graph paper.  Mark a point 0 somewhere on the segment and then start marking 1, 2, 3, etc as you move towards B.  The labeled points will have to get closer and closer together, of course.  Now do the same towards A but with negative numbers.
• Sketch the graph that takes as x-values the points in the segment and the y-value the label from the number line.
• What does this graph look like?  Where is the graph above the line and where below?  What are the values near the end points A and B?
• Suppose you want to create a scrunched number line with a precise rule.  You can do this by starting with a graph like the one you have just drawn.  What must be the properties of this graph to give a good scrunched number line?

Graph logarithm of |CA/CB|

Return to the first graph.  Consider the part of the graph between A and B.

• Graph the logarithm of |CA/CB| for C between A and B.  Note: For C between A and B, the ratio CA/CB < 0, so |CA/CB| = -CA/CB = AC/CB.
• How does this compare with your scrunched graph? Can you make a number line, a ruler from this?