Chapter 8.3, Problem 19E

To determine

To calculate: To sketch the complex numbers $z_1,z_2,z_1+z_2,z_1{z}_2$ on the same complex plane

Answer to Problem 19E

Complex numbers are sketched on complex plane

Explanation of Solution

Given information: Complex numbers are $z_1=2-i;z_2=2+i$

Formula Used:

Complex number is a number that can be expressed in the form of $a+bi$ , where $a$ and $b$ are real numbers

  $z=a+bi$

Calculation:

Complex numbers are given as

  $\begin{array}{l}{z}_1=2-i\\ z_2=2+i\end{array}$

Value of below complex numbers are:

  $\begin{array}{l}{z}_1+z_2=\left(2-i\right)+\left(2+i\right)\\ z_1+z_2=4\end{array}$

  $\begin{array}{l}{z}_1{z}_2=\left(2-i\right)\left(2+i\right)\\ z_1{z}_2=4+2i-2i-i^2\\ z_1{z}_2=4-i^2\\ z_1{z}_2=4-{\left(\sqrt{-1}\right)}^2\\ z_1{z}_2=4-\left(-1\right)\\ z_1{z}_2=5\end{array}$

On comparing the above complex numbers with standard form $z=a+bi$ ,

For complex number $z_1=2-i$

  $\begin{array}{l}a=2\\ b=-1\end{array}$

For complex number $z_2=2+i$

  $\begin{array}{l}a=-2\\ b=2\sqrt{3}\end{array}$

For complex number $z_1+z_2=4$

  $\begin{array}{l}a=4\\ b=0\end{array}$

For complex number $z_1{z}_2=5$

  $\begin{array}{l}a=5\\ b=0\end{array}$

In order to graph the complex number, plot $\left(2,-1\right)$ , $\left(2,1\right)$ , $\left(4,0\right)$ and $\left(5,0\right)$ on the complex plane

  

Conclusion:

Hence, complex numbers are sketched