(related to Proposition: Distributivity Law for Complex Numbers)
By the definition of complex numbers, the complex numbers \(x,y,z \in \mathbb C\) are represented by pairs \(x:=(a,b)\), \(y:=(c,d)\) and \(z:=(e,f)\) of some real numbers \(a,b,c,d,e,f\in\mathbb R\). Because multiplying complex numbers is commutative, it is sufficient to show the left-distributivity law \[x\cdot(y+z)=(x\cdot y)+(x\cdot z).\]
The left-distributivity law can be proven using the following mathematical definitions and concepts: * definition of adding complex numbers, * definition of multiplying complex numbers, * distributivity law for real numbers, * associativity law for adding real numbers, and * commutativity law for adding real numbers. The proof follows:
\[\begin{array}{ccll} x\cdot(y+z)&=&(a,b)\cdot((c,d)+ (e,f))&\text{by definition of complex numbers}\\ &=&(a,b)\cdot(c+e,d+f)&\text{by definition of adding complex numbers}\\ &=&(a(c+e)-b(d+f),a(d+f)+b(c+e))&\text{by definition of multiplying complex numbers}\\ &=&(ac+ae-bd-bf,ad+af+bc+be)&\text{by distributivity law of real numbers}\\ &=&(ac-bd+ae-bf,ad+bc+af+be)&\text{by commutativity of adding (and subtracting) real numbers}\\ &=&((ac-bd)+(ae-bf),(ad+bc)+(af+be))&\text{by associativity of adding complex numbers}\\ &=&(ac-bd,ad+bc)+(ae-bf,af+be)&\text{by definition of adding complex numbers}\\ &=&((a,b)\cdot (c,d))+ ((a,b)\cdot(e,f))&\text{by definition of multiplying complex numbers}\\ &=&(x\cdot y)+ (x\cdot z)&\text{by definition of complex numbers}\\ \end{array} \]
