Vector space/Complex numbers as real vector space/Example

The complex numbers ${}$ form a field, and therefore they form also a vector space over the field ${}\mathbb {C}$ itself. However, the set of complex numbers equals ${}\mathbb {R} ^{2}$ as an additive group. The multiplication of a complex number ${}a+b{\mathrm {i} }$ with a real number ${}s=(s,0)$ is componentwise, so this multiplication coincides with the scalar multiplication on ${}\mathbb {R} ^{2}$. Hence, the set of complex numbers is also a real vector space.