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Problems

  1. Prove that in PZ geometry, every PZ line has an equation of the form ax+by=c, where a, b, c are all rational numbers and a and b are not both zero.

  2. Prove that every equation of the form ax+by=c, where a, b, and c are all rational numbers and a and b are not both zero, is in fact a PZ line.

  3. Prove that if two distinct PZ lines m and n are not parallel in the usual Euclidean sense, then they have a PZ point in common.

  4. In terms of the possible parallel axioms for geometry, which geometry does PZ geometry most resemble: projective, Euclidean, or hyperbolic geometry?

Progress

PZ means Pixel + Zoom geometry. The PZ points are $(x,y)$ of the coordinate plane such that $x$ and $y$ are rational numbers and the $PZ$ lines are the lines of Euclidean geometry which pass through (at least) two $PZ$ points.

I have tried using geometer's sketchpad to come up with PZ lines, but I'm not sure how to start the proof. Should I just use random variables to represent ratios for a and b? How do I show that a and b cannot be zero?

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    Thanks for the clarification. For future reference, it's better to edit your original post or put more information about a problem in comments than it is to post the new information as a solution.2012-03-07

1 Answers 1

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Hint for the first question: if $(x_1, y_1)$ and $(x_2,y_2)$ are on the same line, then either:

a) $x_1=x_2$ (what is the equation for the line in this case?)

or

b)this line has the form $y-y_1=\frac{y_1-y_2}{x_1-x_2}(x-x_1)$ And all the numbers in sight are rational. I'll leave it to you to manipulate this equation into the appropriate form.

For the second question: What does it mean, in terms of $a$, $b$, and $c$ for the lines to be parallel? When the lines are not parallel, can you find a common solution to the two equations? Keep track of your steps to show that the common solution has rational coordinates.

The last question is pretty much answered by the second.