Chapter 6.8, Problem 53WE

To determine

To show that the addition of complex numbers is associative.

Explanation of Solution

Associative property of addition states that we can add the group of addendeds in any order that would not affect the final result. For example if we take three variables $a.b$ and $c$ they would follow associative property if $\left(a+b\right)+c=a+\left(b+c\right)$

Let us assume three complex numbers as $x=a+bi,y=c+di$ and $z=e+fi$

Now we calculate $\left(x+y\right)+z$

  $\begin{array}{l}\Rightarrow \left(x+bi+c+di\right)+e+fi\\ \Rightarrow \left(x+c+e\right)+\left(b+d+f\right)i\end{array}$

Now we calculate $x+\left(y+z\right)$

  $\begin{array}{l}\Rightarrow a+bi+\left(c+di+e+fi\right)\\ \Rightarrow \left(x+c+e\right)+\left(b+d+f\right)i\end{array}$

Hence the result is same in both cases. Therefore we can confirm that the addition of complex number is associative.