Trying to understand irrational numbers in a rational way

Most of us like playing Jenga. For a moment, imagine that you are in a room with infinitely many identical Jenga blocks.

Jenga Blocks

At some point, you started wondering a Jenga block’s length-to-width ratio. Let’s forget about depth. But you have no access to any measurement tool. However, you easily figure out how to find it and you start placing Jenga blocks vertically and horizontally.

A solution finding length-to-width ratio.

Once they align, you count how many horizontal and vertical blocks are used. (number of vertical blocks / number of horizontal blocks) is length-to-width ratio.

However you notice no alignment happens for these Jenga blocks. Actually you don’t know it is going to align or not. They get closer but they never align. What is special about the length-to-width ratio of these blocks?

The ratio is irrational number.

To represent an irrational number, you need infinitely many blocks. We are not able to represent an irrational number in finite quantity domain. The other fact is that the length-to-width ratio is real in your hand but there is no way to represent them in finite world. They are like symbols \sqrt{2}, \pi etc. We can only know they exist.

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