  Nonnegative matrix factorization (NMF) is the problem of decomposing a given
nonnegative $n \times m$ matrix $M$ into a product of a nonnegative $n \times
d$ matrix $W$ and a nonnegative $d \times m$ matrix $H$. A longstanding open
question, posed by Cohen and Rothblum in 1993, is whether a rational matrix $M$
always has an NMF of minimal inner dimension $d$ whose factors $W$ and $H$ are
also rational. We answer this question negatively, by exhibiting a matrix for
which $W$ and $H$ require irrational entries.
