The Paper Trail

 

I found this video in the Internet. Watch this video:

In my view, one reason why the aspect ratio (the width to height ratio) of the standard paper sizes remain same even after halving the longer side is due to the following:

The seemingly strange ratio of length to breadth is because the ratio remains the same when the longer side is folded by half. So, if the ratio is 1:√2, with the longer side being √2, when we fold the longer side by half we have a new ratio of 1:(√2/2). This is the same as  1:(1/√2). This is again the same as √2:1.

We know that the ratio does not change if we multiply both sides of the ratio with the same number. For example, if we have a ratio of 2:3, the ratio remains same when we multiply both sides of the ratio by the same number 2 (say). So, the new ratio 4:6 is the same as 2:3. So, if we multiply, 1:(1/√2) by √2 on both sides of the ratio, we get,  √2:(1/√2)x√2, or, √2:1. So, the ratio keeps oscillating between 1:√2 and √2:1, as we keep folding the paper on the longer side by half. So, when we start with 1 square metre of paper for A0, we need a ratio of length to breadth of √2:1, in such a way that the product of length into breadth is 1 square metre. So we have, 1189mm x 841mm = 9,99,949 square mm, or approximately 1 square metre. And the ratio of 1189:841 is 1.41..., and therefore approximately, √2. 

Because we started with 1 square metre for A0, A4  has seemingly strange dimensions.