p-adic Numbers
See also: P-adic NumberIn addition to the absolute value metric mentioned above, there are other metrics which turn Q into a topological field:
Let p be a prime number and for any non-zero integer a, let |a|p = p−n, where pn is the highest power of p dividing a.
In addition set |0|p = 0. For any rational number a/b, we set |a/b|p = |a|p / |b|p.
Then dp(x,y) = |x − y|p defines a metric on Q.
The metric space (Q,dp) is not complete, and its completion is the p-adic number field Qp. Ostrowski's theorem states that any non-trivial absolute value on the rational numbers Q is equivalent to either the usual real absolute value or a p-adic absolute value.
Read more about this topic: Rational Number
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