This is a note from Cheenta Open Seminar – Wheel of Numbers. 

Rational Numbers are interesting in their own right. They are numbers that can be expressed as a ratio of two integers.

$$ \frac {p}{q} $$

We will construct rational numbers by folding paper.

Getting a Square out of a Rectangle.

Tear off a piece of paper from your notebook. Most probably it will be a rectangle. Can you fold it in a way, such that you get a square out of it?

Suppose ABCD is the rectangle in question.

One way to do this is, take A and put it along BC to get the diagonal crease. Suppose the new position of A is \( A_1 \) and the other end is at Y.
Exercise: Rigorously prove that \( ABA_1Y \)  is a square.

Marking the rationals

Mark Y as (0,0),, \(A_1\) as (1, 0), A as (0, 1) and B as (1,1).

Our goal is to fold the paper in a clever way to mark rational numbers on the segment \( Y A_1 \). Start by marking the 1/2 point. This is easy. Just fold the paper in half (put A on B and Y on \( A_1\) ).

Suppose the mid point is M as in the picture below. Note that we have marked the creases by dotted lines. In our construction, creases play a very important role.

Warning: The paper foldings are very precise. They are not eye estimation. We actually specify where each end of the fold goes. For example, in the previous step, A goes to B and Y goes to \(A _1 \).

Next, can you find the 1/3 mark on \( Y A_1\)? Certainly, it should be somewhere in between Y and M. The trick is to make the AM crease.

With this set-up in mind, you may watch the video lecture recording.