Is the distance between two real numbers p and q equal to the absolute value of p and q?

Answer 1

No.

First, the answer depends on how distance is defined. There are many ways of defining metrics.

With the common definition "Euclidean distance", in 2 or more dimensions, the distance between P and Q is calculated using the generalised form of Pythagoras's theorem.

Thus, if P = (p1, p2, ..., pn) and Q = (q1, q2, ... , qn) are the coordinates of points P and Q in n-dimensional space, then the distance PQ is qiven by

|PQ|2 = (p1 - q1)2 + (p2 - q2)2 + ... + (pn - qn)2

and so |PQ| is the principal square root of the above.

In 1-dimensional space (along a line), this simplifies to

|PQ|2 = (p1 - q1)2

so that

|PQ| = |p1 - q1| and this will always be less than or equal to abs(p and q).