The Number Line

The number line gives rational numbers a geometric home: mark an origin for $0$ and a point one unit to its right for $1$; every other rational number $x>0$ sits $x$ units to the right of $0$, and every rational $-r$ (for $r>0$) sits $r$ units to the left. Order matches position: for $x,y\in Q$, $x<y$ exactly when $x$ lies to the left of $y$ -- and since $x<\dfrac{x+y}2<y$ always holds, there is always another rational number strictly between any two distinct rationals (the rationals are dense). …