A way to well-order real line How is well-ordering in real line possible?
I know that the axiom of choice provides possible well-ordering, but intuitively, this does not seem to make sense.
How can you compare 1.111111.... and 1.111111... when we can never know the full digits of each number?

Using the axiom of choice there exists a well-ordering of the real numbers. The axiom of choice does not say anything about whether this well-ordering has any relation to the actually linear ordering on $\mathbb{R}$. 
Also you don't need to know decimal representation. A real number is some object. In this well-ordering given by the axiom of choice, it could well happen that $\pi <_W \sqrt{2}$. 
In fact the well-ordering is constructed intuitively as follows:
Pick an element $a_0 \in \mathbb{R}$. Define $A_0 = \{a_0\}$. Pick an element $a_1 \in \mathbb{R} - A_0$. Define $A_1 = A_0 \cup \{a_1\}$. Defined $a_0 <_W a_1$. Continue this process. To formally do this in set theory requires the axiom of choice and some transfinite recursion. 
