Stiefel–Whitney class of the underlying real bundle of a complex line ($w(L_{\mathbb{R}})=1+(c_1(L)\mathrm{mod}2)$)

A complex line bundle has a canonically oriented underlying real vector bundle of rank two. Its first Stiefel–Whitney class vanishes, and its second Stiefel–Whitney class is the mod-two reduction of its First Chern class.