Position of the irrational numbers on the number line explanation.

Question

To start off, I have a nice little discovery about the position of an irrational number on the number line. Lets draw a straight line and divide it into ten equal parts (with red ink in this case to mark the equal parts) .

Now in each part, divide them into another ten equal parts.

Here the interesting thing is that if we keep doing the same process perpetually, the irrational numbers would never lie into the division mark (the red marks in the pictures). How is that happening ?

Answer 1

The reason is simple. Any tick marks you're making are numbers of the form

$$\frac{n}{10^m}$$

for some nonnegative integers $n,m$.

This inherently make those tick marks rational numbers, at each and every stage.