  How many permutations of the natural numbers are needed so that every
conditionally convergent series of real numbers can be rearranged to no longer
converge to the same sum? We define the \emph{rearrangement number}, a new
cardinal characteristic of the continuum, as the answer to this question. We
compare the rearrangement number with several natural variants, for example one
obtained by requiring the rearranged series to still converge but to a new,
finite limit. We also compare the rearrangement number with several
well-studied cardinal characteristics of the continuum. We present some new
forcing constructions designed to add permutations that rearrange series from
the ground model in particular ways, thereby obtaining consistency results
going beyond those that follow from comparisons with familiar cardinal
characteristics. Finally, we deal briefly with some variants concerning
rearrangements by a special sort of permutation and with rearranging some
divergent series to become (conditionally) convergent.
