I think the prerequisite of this question relies on assuming that we haven't give irrational numbers a rigorous definition .
Rational numbers and the points of a horizontal straight line becomes a real correspondence when we select upon the straight line a definite origin or zero-point and a definite unit of length for the measurement of segments. Of the greatest importance, however, is the fact that in the straight line there are infinitely many points which correspond to no rational number. If we name these points non-rational points, and say every of these points corresponds to one and only one non-rational number. Similarly , we define the point corresponding to a rational number a rational point.
If a non-rational point lies to the right(or left) of a rational point, we say the non-rational number corresponding to the non-rational point greater( or less) than the rational number corresponding to the rational point. If we denote the union of all these non-rational numbers and all rational numbers $K$, can we show that between two different numbers $a_1$ and $a_2$ ($a_1<a_2$) in $K$ such that there is a rational number $a\in K$ satisfying $a_1<a<a_2$?
I've read a proof on this result within real numbers , the proof relies on the Archimedean property for real numbers , so I think my question may lead to set up the Archimedean property for $K$, so how to set up Archimedean property for $K$?