Chapter 1.6, Problem 68PS

Solution

a.

To find: The sum using the graph.

The sum is $6+5i$

Given:

The expression is $\left(5+i\right)+\left(1+4i\right)$

Graph is as follows.

  

Calculation:

The sum of the two complex numbers $5+i$ and $1+4i$ is as given below.

  $\begin{array}{c}\left(5+i\right)+\left(1+4i\right)=\left(5+1\right)+\left(1+4\right)i\\ =6+5i\end{array}$

The graph given below shows two complex numbers $5+i$ and $1+4i$ can be added geometrically to find their sum $6+3i$ .

  

b.

To find: The sum using the graph.

The sum is $-5+i$ .

Given:

The expression is $\left(-7+3i\right)+\left(2-2i\right)$

Graph is as follows.

  

Calculation:

The sum of the two complex numbers $-7+3i$ and $2-2i$ is as given below.

  $\begin{array}{c}\left(-7+3i\right)+\left(2-2i\right)=\left(-7+2\right)+\left(3-2\right)i\\ =-5+i\end{array}$

The graph given below shows how two complex numbers $-7+3i$ and $2-2i$ can be added geometrically to find their sum $-5+i$ .

  

c.

To find: The sum using the graph.

The sum is $2-3i$ .

Given:

The expression is $\left(3-2i\right)+\left(-1-i\right)$

Graph is as follows.

  

Calculation:

The sum of the two complex numbers $3-2i$ and $-1-i$ is as given below.

  $\begin{array}{c}\left(3-2i\right)+\left(-1-i\right)=\left(3-1\right)+\left(-2-1\right)i\\ =2-3i\end{array}$

The graph given below shows two complex numbers $3-2i$ and $-1-i$ can be added geometrically to find their sum $2-3i$ .

  

d.

To find: The sum using the graph.

The sum is $-1-i$ .

Given:

The expression is $\left(4+2i\right)+\left(-5-3i\right)$

Graph is as follows.

  

Calculation:

The sum of the two complex numbers $4+2i$ and $-5-3i$ is as given below.

  $\begin{array}{c}\left(4+2i\right)+\left(-5-3i\right)=\left(4-5\right)+\left(2-3\right)i\\ =-1-i\end{array}$

The graph given below shows two complex numbers $4+2i$ and $-5-3i$ can be added geometrically to find their sum $-1-i$ .