Chapter 6.5, Problem 20E

To determine

To calculate:

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

Answer to Problem 20E

The complex number is $6+4i$ that has the greatest distance from the origin that is $7.21$ .

Explanation of Solution

Given information:

Complex numbers:

  $\begin{array}{l}{z}_1=-5+i\\ z_2=-7i\\ z_3=6+4i\end{array}$

Formula used:

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

Calculation:

Consider the $z_1=-5+i$

  $\begin{array}{l}\left|a+bi\right|=\sqrt{a^2+b^2}\\ \left|-5+i\right|=\sqrt{{\left(-5\right)}^2+{\left(1\right)}^2}\\ =\sqrt{25+1}\\ =\sqrt{26}\\ =5.09\end{array}$

The absolute value of $-5+i$ is $\mathrm{5.09...}\left(i\right)$ .

Now, consider $z_2=-7i$

  $\begin{array}{l}\left|a+bi\right|=\sqrt{a^2+b^2}\\ \left|-7i\right|=\sqrt{{(-7)}^2}\\ =\sqrt{49}\\ =7\end{array}$

The absolute value of $-7i$ is $\mathrm{7...}\left(ii\right)$ .

Now, consider $z_3=6+4i$

  $\begin{array}{l}\left|a+bi\right|=\sqrt{a^2+b^2}\\ \left|6+4i\right|=\sqrt{{\left(6\right)}^2+{(4)}^2}\\ =\sqrt{36+16}\\ =\sqrt{52}\\ =7.21\\ \end{array}$

The absolute value of $6+4i$ is $\mathrm{7.21...}\left(iii\right)$ .

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 $5.09,7$ and $7.21$ .

As getting the absolute value it is easy to determine that the greatest distance from the origin is $7.21$ .