Chapter 2, Problem 2.46P

(a)

To determine

The real and imaginary parts of the complex number, the modulus and phase of the complex number, the complex conjugate, and sketch $z{\text{and z}}^{*}$ in the complex plane for the complex number $z=1+i$.

Answer to Problem 2.46P

The real part of complex number is $\underset{\_}{1}$ and imaginary parts of the complex number is $\underset{\_}{1}$, the modulus of the complex number is $\underset{\_}{\sqrt{2}}$, phase angle of the complex number is $\underset{\_}{\frac{\pi }{4}}$, the complex conjugate of the given complex number is $\underset{\_}{z^{*}=1-i}$, and $z{\text{and z}}^{*}$ are sketched in the complex plane in Figure 1.

Explanation of Solution

Write the general form of complex number.   

    $z=x+iy$        (I)

Here, $z$ is the complex number, $x$ is the real part of complex number, and $y$ is the imaginary part of complex number.

The given complex number is $z=1+i$.

Compare the above equation with equation (I).

    $\text{Re}\left(z\right)=1\text{and Im}\left(z\right)=1$        (II)

Write the expression for modulus of the complex number.

    $\left|z\right|=\sqrt{x^2+y^2}$

Here, $\left|z\right|$ is the modulus of the complex number.

Use equation (II) in the above equation.

    $\begin{array}{c}\left|z\right|=\sqrt{{\left(1\right)}^2+{\left(1\right)}^2}\\ =\sqrt{2}\end{array}$        (III)

Write the expression for phase angle.

    $\theta ={\tan }^{-1}\left(\frac{y}{x}\right)$

Here, $\theta$ is the phase angle of the complex number.

Use equation (II) in the above equation.

    $\begin{array}{c}\theta ={\tan }^{-1}\left(\frac{1}{1}\right)\\ ={\tan }^{-1}\left(1\right)\\ =\frac{\pi }{4}\end{array}$

Write the complex conjugate of the complex number $\left(z=1+i\right)$.

    $z^{*}=1-i$

Here, $z^{*}$ is the complex conjugate.

Write the other form of complex number $\left(z=1+i\right)$.

    $z=\sqrt{2}{e}^{i\pi /4}$

Here, $z$ is the complex number.

Figure 1 represents the sketch of $z{\text{and z}}^{*}$.

Conclusion:

Therefore, the real part of complex number is $\underset{\_}{1}$ and imaginary parts of the complex number is $\underset{\_}{1}$, the modulus of the complex number is $\underset{\_}{\sqrt{2}}$, phase angle of the complex number is $\underset{\_}{\frac{\pi }{4}}$, the complex conjugate of the given complex number is $\underset{\_}{z^{*}=1-i}$, and $z{\text{and z}}^{*}$ are sketched in the complex plane in Figure 1.

(b)

To determine

The real and imaginary parts of the complex number, the modulus and phase of the complex number, the complex conjugate, and sketch $z{\text{and z}}^{*}$ in the complex plane for the complex number $z=1-i\sqrt{3}$.

Answer to Problem 2.46P

The real part of complex number is $\underset{\_}{1}$ and imaginary parts of the complex number is $\underset{\_}{-\sqrt{3}}$, the modulus of the complex number is $\underset{\_}{2}$, phase angle of the complex number is $\underset{\_}{-\frac{\pi }{3}}$, the complex conjugate of the given complex number is $\underset{\_}{z^{*}=1+i\sqrt{3}}$, and $z{\text{and z}}^{*}$ are sketched in the complex plane in Figure 2.

Explanation of Solution

Write the general form of complex number.   

    $z=x+iy$        (IV)

Here, $z$ is the complex number, $x$ is the real part of complex number, and $y$ is the imaginary part of complex number.

The given complex number is $z=1+i$.

Compare the above equation with equation (IV).

    $\text{Re}\left(z\right)=1\text{and Im}\left(z\right)=-\sqrt{3}$        (V)

Write the expression for modulus of the complex number.

    $\left|z\right|=\sqrt{x^2+y^2}$

Here, $\left|z\right|$ is the modulus of the complex number.

Use equation (V) in the above equation.

    $\begin{array}{c}\left|z\right|=\sqrt{{\left(1\right)}^2+{\left(-\sqrt{3}\right)}^2}\\ =\sqrt{1+3}\\ =2\end{array}$        (VI)

Write the expression for phase angle.

    $\theta ={\tan }^{-1}\left(\frac{y}{x}\right)$

Here, $\theta$ is the phase angle of the complex number.

Use equation (V) in the above equation.

    $\begin{array}{c}\theta ={\tan }^{-1}\left(\frac{-\sqrt{3}}{1}\right)\\ ={\tan }^{-1}\left(-\sqrt{3}\right)\\ =-{\tan }^{-1}\left(\sqrt{3}\right)\\ =-\frac{\pi }{3}\end{array}$

Write the complex conjugate of the complex number $\left(z=1-i\sqrt{3}\right)$.

    $z^{*}=1+i\sqrt{3}$

Here, $z^{*}$ is the complex conjugate.

Write the other form of complex number $\left(z=1+i\right)$.

    $z=\sqrt{2}{e}^{-i\pi /3}$

Here, $z$ is the complex number.

Figure 2 represents the sketch of $z{\text{and z}}^{*}$.

Conclusion:

Therefore, the real part of complex number is $\underset{\_}{1}$ and imaginary parts of the complex number is $\underset{\_}{-\sqrt{3}}$, the modulus of the complex number is $\underset{\_}{2}$, phase angle of the complex number is $\underset{\_}{-\frac{\pi }{3}}$, the complex conjugate of the given complex number is $\underset{\_}{z^{*}=1+i\sqrt{3}}$, and $z{\text{and z}}^{*}$ are sketched in the complex plane in Figure 2.

(c)

To determine

The real and imaginary parts of the complex number, the modulus and phase of the complex number, the complex conjugate, and sketch $z{\text{and z}}^{*}$ in the complex plane for the complex number $z=\sqrt{2}{e}^{-i\pi /4}$.

Answer to Problem 2.46P

The real part of complex number is $\underset{\_}{1}$ and imaginary parts of the complex number is $\underset{\_}{-1}$, the modulus of the complex number is $\underset{\_}{\sqrt{2}}$, phase angle of the complex number is $\underset{\_}{-\frac{\pi }{4}}$, the complex conjugate of the given complex number is $\underset{\_}{z^{*}=1+i}$, and $z{\text{and z}}^{*}$ are sketched in the complex plane in Figure 3.

Explanation of Solution

Write the general form of complex number.   

    $z=x+iy$        (VII)

Here, $z$ is the complex number, $x$ is the real part of complex number, and $y$ is the imaginary part of complex number.

The given complex number is $z=\sqrt{2}{e}^{-i\pi /4}$.

The above complex number can be written as

    $\begin{array}{c}z=\sqrt{2}\left[\cos \left(-\pi /4\right)+i\sin \left(-\pi /4\right)\right]\\ =\sqrt{2}[\cos (\pi /4)+i(-\sin (\pi /4))]\\ =\sqrt{2}\left[\cos \frac{\pi }{4}-i\sin \frac{\pi }{4}\right]\\ =\sqrt{2}\left[\frac{1}{\sqrt{2}}-i\left(\frac{1}{\sqrt{2}}\right)\right]\end{array}$

Rearrange the above equation.

    $z=1-i$

Compare the above equation with equation (IV).

    $\text{Re}\left(z\right)=1\text{and Im}\left(z\right)=-1$        (VIII)

Write the expression for modulus of the complex number.

    $\left|z\right|=\sqrt{x^2+y^2}$

Here, $\left|z\right|$ is the modulus of the complex number.

Use equation (VIII) in the above equation.

    $\begin{array}{c}\left|z\right|=\sqrt{{\left(1\right)}^2+{\left(-1\right)}^2}\\ =\sqrt{1+1}\\ =\sqrt{2}\end{array}$        (IX)

Write the expression for phase angle.

    $\theta ={\tan }^{-1}\left(\frac{y}{x}\right)$

Here, $\theta$ is the phase angle of the complex number.

Use equation (VIII) in the above equation.

    $\begin{array}{c}\theta ={\tan }^{-1}\left(\frac{-1}{1}\right)\\ ={\tan }^{-1}\left(-1\right)\\ =-{\tan }^{-1}\left(1\right)\\ =-\frac{\pi }{4}\end{array}$

Write the complex conjugate of the complex number $\left(z=\sqrt{2}{e}^{-i\pi /4}\right)$.

    $z^{*}=1+i$

Here, $z^{*}$ is the complex conjugate.

Figure 3 represents the sketch of $z{\text{and z}}^{*}$.

Conclusion:

Therefore, the real part of complex number is $\underset{\_}{1}$ and imaginary parts of the complex number is $\underset{\_}{-1}$, the modulus of the complex number is $\underset{\_}{\sqrt{2}}$, phase angle of the complex number is $\underset{\_}{-\frac{\pi }{4}}$, the complex conjugate of the given complex number is $\underset{\_}{z^{*}=1+i}$, and $z{\text{and z}}^{*}$ are sketched in the complex plane in Figure 3.

(d)

To determine

The real and imaginary parts of the complex number, the modulus and phase of the complex number, the complex conjugate, and sketch $z{\text{and z}}^{*}$ in the complex plane for the complex number $z=5e^{i\omega t}$.

Answer to Problem 2.46P

The real part of complex number is $\underset{\_}{5\cos \omega t}$ and imaginary parts of the complex number is $\underset{\_}{5\sin \omega t}$, the modulus of the complex number is $\underset{\_}{5}$, phase angle of the complex number is $\underset{\_}{\omega t}$, the complex conjugate of the given complex number is $\underset{\_}{z^{*}=5e^{-i\omega t}}$, and $z{\text{and z}}^{*}$ are sketched in the complex plane in Figure 4.

Explanation of Solution

Write the general form of complex number.   

    $z=x+iy$        (X)

Here, $z$ is the complex number, $x$ is the real part of complex number, and $y$ is the imaginary part of complex number.

The given complex number is $z=5e^{i\omega t}$.

The above complex number can be written as

    $\begin{array}{c}z=5\left[\cos \omega t+i\sin \omega t\right]\\ =5\cos \omega t+5i\sin \omega t\end{array}$

Compare the above equation with equation (X).

    $\text{Re}\left(z\right)=5\cos \omega t\text{and Im}\left(z\right)=5\sin \omega t$        (XI)

Write the expression for modulus of the complex number.

    $\left|z\right|=\sqrt{x^2+y^2}$

Here, $\left|z\right|$ is the modulus of the complex number.

Use equation (XI) in the above equation.

    $\begin{array}{c}\left|z\right|=\sqrt{{\left(5\cos \omega t\right)}^2+{\left(5\sin \omega t\right)}^2}\\ =\sqrt{25{\cos }^2\omega t+25{\sin }^2\omega t}\\ =\sqrt{25\left({\cos }^2\omega t+{\sin }^2\omega t\right)}\\ =5\end{array}$        (XII)

Write the expression for phase angle.

    $\theta ={\tan }^{-1}\left(\frac{y}{x}\right)$

Here, $\theta$ is the phase angle of the complex number.

Use equation (XI) in the above equation.

    $\begin{array}{c}\theta ={\tan }^{-1}\left(\frac{\sin \omega t}{\cos \omega t}\right)\\ ={\tan }^{-1}\left(\tan \omega t\right)\\ =\omega t\end{array}$

Write the complex conjugate of the complex number $\left(z=5e^{i\omega t}\right)$.

    $z^{*}=5e^{-i\omega t}$

Here, $z^{*}$ is the complex conjugate.

Figure 4 represents the sketch of $z{\text{and z}}^{*}$.

Conclusion:

Therefore, the real part of complex number is $\underset{\_}{5\cos \omega t}$ and imaginary parts of the complex number is $\underset{\_}{5\sin \omega t}$, the modulus of the complex number is $\underset{\_}{5}$, phase angle of the complex number is $\underset{\_}{\omega t}$, the complex conjugate of the given complex number is $\underset{\_}{z^{*}=5e^{-i\omega t}}$, and $z{\text{and z}}^{*}$ are sketched in the complex plane in Figure 4.