TITLE: Archimedean Property and Real Numbers
QUESTION [25 upvotes]: I have few confusions:
a) What exactly is Archimedean Property. What does infinitesimal and infinite numbers do not exist in Archimedian ordered fields mean? Are not 0 and infinity such numbers?
b) What are the surreal numbers? Do they have anything to do with extended real numbers? I mean real numbers and positive and negative infinity. Rudin introduces extended real numbers with these two additional numbers. Does it mean in the field of reals, infinite means undefined and in extended, infinite means defined?
Does this mean extended real numbers are not Archimedian ?
Thank You.

REPLY [6 votes]: Similar to the other answers.  The Archimedean property for an ordered field $F$ states: if $x,y>0$, then there is $n \in \mathbb N$ so that
$$
x+x+\dots+x \ge y
$$
where we have added $n$ terms all equal to $x$.  
Consequences:  There are no infinite elements $u \in F$, that is, there is no $u$ so that $1+1+\dots+1 \lt u$ (with $n$ terms) for all $n$.  
There are no infinitesimal elements $v \in F$, that is, there is no $v$ so that $v>0$ and $v+v+\dots+v < 1$ (with $n$ terms) for all $n$.
There is no real number called $\infty$, so we say the real numbers satisfy the Archimedean property.  The "extended real numbers" do not form a field, but may be useful for certain computations in analysis.  Instead of saying $\infty$ is defined or undefined maybe it is better to say whether $\infty$ is an element of the set you are talking about.   
The surreal numbers No does form an ordered Field, but has infinite and infinitesimal elements, so the Archimedian property fails in No.