Rothe was born in 1773 in
Dresden, and in 1793 became a docent at the
University of Leipzig. He became an extraordinary professor at Leipzig in 1796, and in 1804 he moved to Erlangen as a full professor, taking over the chair formerly held by
Karl Christian von Langsdorf. He died in 1842, and his position at Erlangen was in turn taken by Johann Wilhelm Pfaff, the brother of the more famous mathematician
Johann Friedrich Pfaff.
In the study of
permutations, Rothe was the first to define the inverse of a permutation, in 1800. He developed a technique for visualizing permutations now known as a
Rothe diagram, a square table that has a dot in each cell (i,j) for which the permutation maps position i to position j and a cross in each cell (i,j) for which there is a dot later in row i and another dot later in column j. Using Rothe diagrams, he showed that the number of
inversions in a permutation is the same as in its inverse, for the inverse permutation has as its diagram the
transpose of the original diagram, and the inversions of both permutations are marked by the crosses. Rothe used this fact to show that the
determinant of a
matrix is the same as the determinant of the transpose: if one expands a determinant as a
polynomial, each term corresponds to a permutation, and the sign of the term is determined by the
parity of its number of inversions. Since each term of the determinant of the transpose corresponds to a term of the original matrix with the inverse permutation and the same number of inversions, it has the same sign, and so the two determinants are also the same.
In his 1800 work on permutations, Rothe also was the first to consider permutations that are
involutions; that is, they are their own inverse, or equivalently they have symmetric Rothe diagrams. He found the
Systematisches Lehrbuch der Arithmetik, Leipzig, 1811
^Bekemeier, Bernd (1987), Martin Ohm, 1792-1872: Universitäts- und Schulmathematik in der neuhumanistischen Bildungsreform, Studien zur Wissenschafts-, Sozial- und Bildungsgeschichte der Mathematik (in German), vol. 4, Vandenhoeck & Ruprecht, p. 83,
^Jahnke, Hans Niels (1990), Mathematik und Bildung in der Humboldtschen Reform, Studien zur Wissenschafts-, Sozial- und Bildungsgeschichte der Mathematik (in German), vol. 8, Vandenhoeck & Ruprecht, p. 175,
^Rowe, David E. (1997), "In search of Steiner's Ghosts : Imaginary elements in the nineteenth-century geometry", in Flament, Dominique (ed.), Le Nombre : une Hydre à n visages, Entre nombres complexes et vecteurs, Fondation Maison des sciences de l'homme, pp. 193–208.