## Thought Experiment: the Rational Plane

By convention, when we refer to the number line, the “numbers” in question are the real numbers $\mathbb{R}$.  This is very different from the number line that most of us learned in kindergarten, which consisted of the counting numbers together with 0 and their negatives.  There is a drastic qualitative difference between these two versions of the number line — one is continuous and one is discrete.  There are measurable and noticeable gaps between the integers.  It’s the difference between a ribbon of highway and the  mile markers along the way.

What about the intermediate case of the rational numbers, the numbers that can be written as fractions?  Are they more like the discrete number line or the continuous number line?

We saw in an earlier post that there are the same number of rational numberes as there are integers, but there are a LOT more real numbers.  So we would expect that the rational numbers will be rather sparse in the real number line.  In fact this is not quite true.  The truth is that the rational numbers are dense in the real  numbers, which means that every little interval in the real number line contains a rational number (actually infinitely many rational numbers).  Every real number can be approximated as well as you may wish by rational numbers.

In other words, if you put a black dot at every rational number on the number line, the line would look as though it were completely colored black.  There would be no gaps of any positive size without coloring.

This density means, among other things, that we could never decide whether a real-world quantity is a rational number or not.  Any measurement has an amount of uncertainty, and no matter how small that window is, there will be both rational and irrational numbers inside it.

Now we extend the picture into two dimensions.  The usual $xy$-plane consists of ordered pairs of real numbers.  For today, I’ll call this the real plane. How does it compare to the “rational plane”, which consists of all the points $(x,y)$ where $x$ and $y$ are rational numbers?

For starters, the rational plane is dense in the real plane.  Any little area in the plane contains rational points, and any point can be approximated as well as desired by a rational point.  Can we say more than that?

Consider the unit circle $x^2+y^2=1$.  In the real plane, of course, this should be a very familiar object.  What does the unit circle look like in the rational plane?  It turns out that the situation is much like the situation of the number line.  There are an infinite number of rational points on the circle, given by $\left(\frac{m^2-n^2}{m^2+n^2},\frac{2mn}{m^2+n^2}\right)$ and $\left(\frac{2mn}{m^2+n^2},\frac{m^2-n^2}{m^2+n^2}\right)$ for integers $m$ and $n$, not both zero.  (This description of the rational points on the unit circle is connected to the abundance of right triangles with whole number sides.)  Moreover these points are dense in the real circle, so that there are no gaps between them of any size.

The fact that the rational points on the unit circle are dense is by no means automatic.  For example, consider the “fattened circle” whose equation is $x^4+y^4=1$.  In the real plane, this is a smooth closed curve, but in the rational plane it contains only 4 points, $(\pm 1,0), (0,\pm 1)$.  In general, given an equation $f(x,y)=0$ which describes some sort of curve in the real plane, it is very hard to predict just by looking at $f$ how many rational points there will be and how they will be configured.   However, circles with rational center and rational radius behave as nicely as one could ever hope for in this context, which makes the next observation so surprising.

Now consider two circles.  For example, take the circles with radius 1 centered at $(0,0)$ and $(1,0)$.  Even in the rational plane, we can define the “inside” and “outside” of a circle.  The “inside” of a circle $(x-a)^2+(y-b)^2=r^2$ is the set of all the points for which $(x-a)^2+(y-b)^2, and the outside is the set of points where $(x-a)^2+(y-b)^2>r^2$.  With this (very natural) notion, each of our circles has points that are inside the other and points that are outside the other.  In the real plane, this has a very unsurprising consequence: the circles intersect at two points $\left(\frac{1}{2}, \pm\frac{\sqrt{3}}{2}\right)$.  But in the rational plane, these points do not exist.  So these curves, which have overlapping interiors, somehow “pass through” each other in the rational plane without actually intersecting!

This should feel unsettling.  This crossing without crossing is just not how lines work.  There is a theorem in topology called the Jordan Curve Theorem which states, roughly, that a curve in the plane which ends at the same point it begins without crossing itself, divides the plane into an “inside” and an “outside”, so that any path from the “inside” to the “outside” must cross the curve.  This theorem is easy to state and easy to believe, but very difficult to prove; this example gives some insight into why.  Any proof of the Jordan Curve Theorem must involve some specific property of the real number system, since we just saw that the theorem isn’t even true in the rational numbers.

Since the Jordan Curve Theorem is so intuitively reasonable, so deeply embedded in our sense of geometry, this provides a measure of motivation for considering the real numbers as making up the “geometric” number line, the one we use to model the real world.