Vector space/Complex numbers as real vector space/Example

The complex numbers ${}\mathbb {C}$ form a field, and therefore they form also a vector space over the field 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 complex numbers are also a real vector space.