Chapter 6.5, Problem 19E

To determine

To calculate:

The complex number which is the greatest distance from the origin.

Answer to Problem 19E

The complex number is $z_2=4-i$ that has the greatest distance from the origin that is $4.12$ .

Explanation of Solution

Given information:

Complex numbers:

  $\begin{array}{l}{z}_1=3+2i\\ z_2=4-i\\ z_3=i\end{array}$

Formula used:

Absolute Value: $\left|a+bi\right|=\sqrt{a^2+b^2}$

Calculation:

Consider the $z_1=3+2i$

  $\begin{array}{l}\left|a+bi\right|=\sqrt{a^2+b^2}\\ \left|3+2i\right|=\sqrt{3^2+2^2}\\ =\sqrt{9+4}\\ =\sqrt{13}\\ =3.60\end{array}$

The absolute value of $3+2i$ is $\mathrm{3.60...}\left(i\right)$ .

Now, consider $z_2=4-i$

  $\begin{array}{l}\left|a+bi\right|=\sqrt{a^2+b^2}\\ \left|4-i\right|=\sqrt{4^2+{(-1)}^2}\\ =\sqrt{16+1}\\ =\sqrt{17}\\ =4.12\end{array}$

The absolute value of $4-i$ is $\mathrm{4.12...}\left(ii\right)$ .

Now, consider $z_3=i$

  $\begin{array}{l}\left|a+bi\right|=\sqrt{a^2+b^2}\\ \left|i\right|=\sqrt{{(1)}^2}\\ =\sqrt{1}\\ =1\\ \end{array}$

The absolute value of $i$ is $\mathrm{1...}\left(iii\right)$ .

With the help of $\left(i\right)\left(ii\right)$ and $\left(iii\right)$ the absolute values of the values of the complex numbers are $3.60,4.12$ and $1$ .

As getting the absolute value it is easy to determine that the greatest distance from the origin is $4.12$ .