Dedekind's method for irrational number I am now reading the definition of irrational number, which we can describe by the following terms: suppose that we have divided  all rational numbers into two classes, a lower class and an upper class, such that every number of the lower class is less then all numbers  in the upper class. From the book,  it is remarked that we may have three   different cases:

1$\text{}$. The lower class can have a greatest number and the upper class no smallest number. 

It is clear   that if we take number 5 as a division number, in lower class we will have 4,3,2,1  so  largest is  4  and in upper  just  7,7,7,7,7  no smallest number right yes?

2$\text{}$. The  upper class  can have  a smallest  number and lower class  no greater number. 

It is clear that  in case of 5, upper  class  can  be  6,7,8,9,10, so smallest is  6, while  lower  4,4,4,4,4,4,4  no  greater number and third one.

3$\text{}$.  The lower  class  can have no greatest  number and  the  upper  class no smallest number.

I did not understand the method  which is  used in the book    for the third case, namely: if we arrange  positive numbers and their squares so that  each  square number is  underneath  it's corresponding  number, then since   the square of a fraction in it's lowest terms is a fraction whose numerator and denominator   are perfect squares, we see that  there are not rational    numbers whose  squares are 2,3,5,6,7,8,10,11 and so on. Then the author used an approximation method  to show that there are rational numbers which are   as near to these numbers as we please and  finally  he divided so that all negative  numbers, 0, and positive numbers  whose square is less then 2, and all numbers whose square are more than 2. For  rational numbers which are near  to numbers which I mentioned  are looking like this:
 2, 1.5, 1.42, 1.415, 1.4143
  1, 1.4, 1.41, 1.414, 1.4142

I did not understand why author used these numbers, so please  help me, sorry  if text  is too much big, I couldn't express my question otherwise.
 A: You may have missed the fact that this requires you to split all rational numbers (that is numbers which can be written as $\frac{a}{b}$ with $a$ an integer and $b$ a positive integer where they have no common factor above $1$) into two subsets. 
If you divide them into two sets where the lower set is everything negative or whose square is less than or equal to 25, you will find numbers like $-1, 5, \frac{3}{2}, \frac{500}{101}$ in the lower subset and numbers like $6, 999, \frac{5000}{999}$ in the upper subset. 
If you divide them into two sets where the lower set is everything negative or whose square is strictly less than 25, you will find numbers like $-1, \frac{3}{2}, \frac{500}{101}$ in the lower subset and numbers like $6, 5, 999, \frac{5000}{999}$ in the upper subset. 
If you divide them into two sets where the lower set is everything negative or whose square is less than 2, you will find numbers like $-1, 1, \frac{141421}{100000}$ in the lower subset and numbers like $6, 2, \frac{141421357}{100000000}$ in the upper subset. 
A: These number are approximations of $\sqrt 2$, which is an irrational number. Hence it is an example of case 3.
A: The way we do it would best be viewed by drawing a line and imagining their are only rationals marked on it.
Then mark a point at root(2) - you know this isn't rational - and then you have seperated the rationals into (where intervals are taken in the rationals):
$(-\infty,0]$ U { $ p > 0 | p \in \mathbb Q , p^2 < 2 $} as a 'lower' set and {$p > 0 | p \in \mathbb Q , p^2 > 2$ }
