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March 19, 2018

Wheel of Numbers - Open Slate

[et_pb_section fb_built="1" _builder_version="3.0.47"][et_pb_row _builder_version="3.0.47" background_size="initial" background_position="top_left" background_repeat="repeat"][et_pb_column type="4_4" _builder_version="3.0.47" parallax="off" parallax_method="on"][et_pb_text _builder_version="3.0.106" border_radii="on|5px|5px|5px|5px" module_alignment="center" custom_padding="|20px||20px" box_shadow_style="preset1"]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. [/et_pb_text][et_pb_image src="https://www.cheenta.com/wp-content/uploads/2018/03/Screen-Shot-2018-03-19-at-9.55.18-AM.png" align="center" _builder_version="3.0.106" box_shadow_style="preset1"][/et_pb_image][et_pb_text _builder_version="3.0.106" border_radii="on|5px|5px|5px|5px" module_alignment="center" custom_padding="|20px||20px" box_shadow_style="preset1"]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. [/et_pb_text][et_pb_image src="https://www.cheenta.com/wp-content/uploads/2018/03/Screen-Shot-2018-03-19-at-10.00.32-AM.png" align="center" _builder_version="3.0.106" box_shadow_style="preset1"][/et_pb_image][et_pb_text _builder_version="3.0.106" border_radii="on|5px|5px|5px|5px" module_alignment="center" custom_padding="|20px||20px" box_shadow_style="preset1"]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. [/et_pb_text][et_pb_image src="https://www.cheenta.com/wp-content/uploads/2018/03/Screen-Shot-2018-03-19-at-10.18.34-AM.png" align="center" _builder_version="3.0.106" box_shadow_style="preset1"][/et_pb_image][et_pb_text _builder_version="3.0.106" border_radii="on|5px|5px|5px|5px" module_alignment="center" custom_padding="|20px||20px" box_shadow_style="preset1"]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. [/et_pb_text][et_pb_video src="https://youtu.be/UB55r8ZL5LY" _builder_version="3.0.106"][/et_pb_video][/et_pb_column][/et_pb_row][/et_pb_section]

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