Chapter A, Problem 1P

To determine

The conjugate of $z$, $\overline{z}$ and modulus of $z$, $\left|z\right|$ for the given complex number.

Answer to Problem 1P

Solution:

The value of $\overline{z}=2-5i$ and $\left|z\right|=\sqrt{29}$ for the given complex number.

Explanation of Solution

Given:

The given complex number is,

$z=2+5i$

Approach:

The definition of complex conjugate states that,

If $z=a+ib$, then the complex number $\overline{z}$ defined by $\overline{z}=a-ib$ is called the conjugate of $z$.

The definition of modulus of $z$ is as follows,

If $z=a+ib$, then real number $\sqrt{a^2+b^2}$ is often called modulus of $z$ or the absolute value of $z$ and is denoted $\left|z\right|$.

Calculation:

Consider the given complex number,

$z=2+5i$

Compare the above complex number with the standard complex number $z=a+ib$.

Here, $a=2$ and $b=5$.

By the definition of complex conjugate of $z$ is $\overline{z}$ and $\overline{z}=a-ib$

Hence, the complex of conjugate of $z$ is $\overline{z}=2-5i$.

By the definition of modulus of $z$,

$\left|z\right|=\sqrt{a^2+b^2}$

Substitute $2$ for $a$ and $5$ for $b$ in above expression to find $\left|z\right|$.

$\begin{array}{rll}\left|z\right|& =& \sqrt{{\left(2\right)}^2+{\left(5\right)}^2}\\ & =& \sqrt{4+26}\\ & =& \sqrt{29}\end{array}$

Therefore, the value of $\overline{z}=2-5i$ and $\left|z\right|=\sqrt{29}$ for the given complex number.

Conclusion:

Hence, the value of $\overline{z}=2-5i$ and $\left|z\right|=\sqrt{29}$ for the given complex number.