Since the real numbers together with addition and multiplication form a field (\(\mathbb R,+,\cdot)\) and the complex numbers form a non-empty set, we can define the maps
\[ \cases{ \mathbb C\times \mathbb C\mapsto \mathbb C:x,y\mapsto x + y:=(a+c,b+d) & \text{(the vector addition)}\\ \mathbb R\times \mathbb C\mapsto \mathbb C:\alpha,x\mapsto \alpha \cdot x:=(\alpha \cdot a,\alpha \cdot b) & \text{(the scalar multiplication)} }\]
for any real number \(\alpha\in\mathbb R\) and any complex numbers \(x,y\in\mathbb C\), identified by some ordered pairs of real numbers \(x:=(a,b)\), \(y:=(c,d)\), \(a,b,c,d\in\mathbb R\). Therefore, we can consider the complex numbers \(\mathbb C\) as a vector space over the field \(\mathbb R\).
Proofs: 1
