  Extriangulated categories were introduced by Nakaoka and Palu as a
simultaneous generalization of exact categories and triangulated categories. In
this paper, we show that the idempotent completion of an extriangulated
category admits a natural extriangulated structure. As applications, we prove
that cotorsion pairs in an extriangulated category induce cotorsion pairs in
its idempotent completion under certain condition, and the idempotent
completion of a recollement of extriangulated categories is still a
recollement.
