The Hirsch index is a number that synthesizes a researcher’s output. It is defined as the maximum number h such that the researcher has h papers with at least h citations each. Four characterizations of the Hirsch index are suggested. The most compact one relies on the interpretation of the index as providing the number of valuable papers in an output and postulates three axioms. One, only cited papers can be valuable. Two, the index is strongly monotonic:​ if output x has more papers than output y and each paper in x has more citations than the most cited paper in y, then x has more valuable papers than y. And three, the minimum amount of citations under which a paper becomes valuable is different for each paper.