*When is 144 closer to 8 than to 143?*

The usual definition of the *norm* of a real number is its modulus or absolute value . We measure the “distance” between two real numbers by means of the absolute value of their difference. This gives the Euclidean metric and, using it, we can define the usual topology on the real numbers .

The standard arrangement of the real numbers on a line automatically ensures that numbers with small Euclidean difference between them are geometrically close to each other. It may come as a surprise that there are other ways to define norms and distances, which provide other topologies, leading us to a radically different concept of closeness, and to completely new number systems, the p-adic numbers.