Chapter 6.8, Problem 56WE

To determine

Prove that in the complex number system , multiplication is distributive over addition.

Explanation of Solution

Calculation:

To show : Multiplication is distributive over addition in the complex number system , that is ,

  $\begin{array}{l}\forall z_1,z_2,z_3\in ℂ:\\ z_1\left(z_2+z_3\right)=z_1{z}_2+z_1{z}_3\\ \left(z_2+z_3\right)z_1=z_2{z}_1+z_3{z}_1\end{array}$

Proof:

Let us define :

  $\begin{array}{l}{z}_1:=(x_1,y_1)\\ z_2:=(x_2,y_2)\\ z_3:=(x_3,y_3)\\ where{x}_1,x_2,x_3,y_1,y_2,y_3\in ℝ\end{array}$

So,

  $\begin{array}{l}{z}_1\left(z_2+z_3\right)=(x_1,y_1)((x_2,y_2)+(x_3,y_3))\mathrm{.....................................}[\text{complex numbers defined in this manner]}\\ =(x_1,y_1)((x_2+x_3,y_2+y_3))\mathrm{.......................................................}[\text{addition of complex numbers]}\\ =\left(x_1\left(x_2+x_3\right)-y_1\left(y_2+y_3\right),x_1(y_2+y_3)+y_1(x_2+x_3)\right)\mathrm{..............}[\text{multiplication of complex numbers]}\\ =\left(x_1{x}_2+x_1{x}_3-y_1{y}_2-y_1{y}_3,x_1{y}_2+x_1{y}_3+y_1{x}_2+y_1{x}_3\right)\mathrm{.............}[\text{real multiplication distributes over addition]}\\ =\left(x_1{x}_2-y_1{y}_2,x_1{y}_2+y_1{x}_2\right)+\left(x_1{x}_3-y_1{y}_3,x_1{y}_3+y_1{x}_3\right)\mathrm{.............}[\text{complex addition definition]}\\ =\left(x_1,y_1\right)\left(x_2,y_2\right)+\left(x_1,y_1\right)\left(x_3,y_3\right)\mathrm{............................................}[\text{complex multiplication definition]}\\ {\text{=z}}_1{z}_2+z_1{z}_3\end{array}$ So,

  $\begin{array}{l}\forall z_1,z_2,z_3\in ℂ:\\ z_1\left(z_2+z_3\right)=z_1{z}_2+z_1{z}_3\end{array}$

Also, complex multiplication is commutative , hence :

  $\begin{array}{l}\forall z_1,z_2,z_3\in ℂ:\\ \left(z_2+z_3\right)z_1=z_1\left(z_2+z_3\right)=z_1{z}_2+z_1{z}_3=z_2{z}_1+z_3{z}_1\\ \Rightarrow \left(z_2+z_3\right)z_1=z_2{z}_1+z_3{z}_1\end{array}$

Hence , in complex number system , multiplication is distributive over addition.

Hence proved.