An irrational number is any number N that divides the set of rational numbers into two parts: those greater than N, and those less than N, and for which all rational numbers fall into one of those two categories.--hodgepodge
It's an intersting
trait but I don't think it's a
definition, simply because it doesn't rule out other
possibilities. For example, do
rational numbers not do the same thing? Any rational
number,
r divides the
set of irrational numbers into two parts : those greater than
r and those less than
r. This
characteristic comes from the
fact that no
irrational number can be a
rational number and
vice versa.
The best defintion is that listed above: assume p is an irrational number; then there exist no integers a and b such that a/b = p.