Integral Line

Why

We are constantly thinking of the integers as the endpoints of equal length segments of a line.

Discussion

We commonly associate elements of the integers with the endpoints of equal-length segments of a real line. Take segment $S_0$ of $L$ with endpoints $p$ and $q$. Associate the point $p$ with $0$. Associate the point $q$ with $1$. Take a segment $S_1$ of equal length, non-overlapping with $S_0$, who shares the endpoint $q$. Associate the second endpoint of this segment $2$. Continue with the rest. We call the line so formed the integral line of unit $S_0$.

Integral Distance

Let $f: \Z \to \Z $ be defined by $f(a, b) = a - b$ if $a > b$ and $f(a, b) = b - a$ if $b > a$. Notice that $f$ is symmetric: $f(a,b) = f(b, a)$. The (geometric) interpretation of $f$ is the distance between the points associated with the two integers $a, b \in \Z $ in some integral line. We call $f$ the integral distance. Notice that $f(a, b) > 0$ for all $a, b \in \Z $.

Notation

We denote the distance between $a, b \in \Z $ by $\abs{a - b}$.