Visualizing the square root of 2 A junior high school student I am tutoring asked me a question that stumped me - I was wondering if anyone could shed some light on it here.
We were talking about how the square root of 2 is an irrational number, and that means you can't write that value as the ratio of two integers. The decimal form of this value goes on forever, without repeating. And then we tried visualizing what it means when a decimal number "goes on forever."
I tried explaining it as imagining drawing a line, but you keep adding smaller and smaller pieces to its length. The pieces end up being so small, you effectively get a finite length. It was here that I could hear my own explanation breaking down. I realized I truly don't know how to visualize this concept.
The diagram I was using to describe this line was the diagonal of a square with a side length of 1. The length of this diagonal is the square root of 2, and clearly, it has a finite length (it fits inside the box, after all). Yet looking at that length in decimal form, apparently it isn't really finite. Sure those additional values that keep getting added to it as you go out from the right of the decimal point keep getting smaller and smaller, but they each have substance, adding to the overall length of the line.
What am I missing here? Or, is there a better way to explain this concept?

Here is another way of approximating the square root of two by rational numbers which doesn't depend on the decimal system.
Suppose $p^2-2q^2=\pm 1$ so that $\left(\cfrac pq\right)^2=2\pm\cfrac 1{q^2}$, then the larger we can make $q$ the closer $\cfrac pq$ is to $\sqrt 2$.
Consider now $(p+2q)^2-2(p+q)^2=p^2+4pq+4q^2-2p^2-4pq-2q^2=2q^2-p^2=\mp 1$ so that $\cfrac {p+2q}{p+q}$ is a better approximation.
From this we obtain the approximations $$\frac 11, \frac 32, \frac 75, \frac {17}{12}, \frac {41}{29},\frac {99}{70} \dots$$
The Wikipedia entry gives also that if $r$ is an approximation, $\frac r2+\frac 1r$ is a better one, which picks out $1, \frac 32, \frac {17}{12}, \frac {577}{408} \dots$ which converges very quickly, picking out a subsequence of the previous one..
This takes $$\frac pq \text{ to } \frac {p^2+2q^2}{2pq}$$
It also gives a geometric proof which is quite visual and may help - the problem with these kinds of proofs is that they often work by some form of descent, and therefore terminate, so don't give a sense of never-ending.
