Chapter 2: Q3P (page 51)

For each of the following numbers, first visualize where it is in the complex plane. With a little practice you can quickly find $x, y, r, θ$ in your head for these simple problems. Then plot the number and label it in five ways as in Figure 3.3. Also, plot the complex conjugate of the number $1 - I \sqrt{3}$ .

Short Answer

Expert verified

The right values for the complex numbers are

$x = 1, y = - \sqrt{3} r = 2 θ = \frac{5 π}{3} 1 - i \sqrt{3} = 2 (\cos \frac{5 π}{3} + i \sin \frac{5 π}{3}) = 2 e^{i \frac{5 π}{3}}$

Step by step solution

Given information and the general form of complex numbers:

The given complex number is $1 - i \sqrt{3}$ .

The general form of complex numbers is x+iy .

Complex numbers and the given conjugate:

The general form of complex numbers is x+iy .

$x + i y = r (\cos θ + i \sin θ) = r e^{i θ} r = \sqrt{x^{2} + y^{2}} θ = \sin^{- 1} \frac{y}{r}$

In the given number $1 - i \sqrt{3}$ .

$x = 1, y = - \sqrt{3} r = \sqrt{1^{2} + {(- \sqrt{3})}^{2}} = 2 θ = \sin^{- 1} \frac{- \sqrt{3}}{2} = \frac{5 π}{3} 1 - i \sqrt{3} = 2 (\cos \frac{5 π}{3} + i \sin \frac{5 π}{3}) = 2 e^{i \frac{5 π}{3}}$

Plot the given complex number and complex conjugate of the number on the graph.

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Short Answer

Step by step solution

Given information and the general form of complex numbers:

Complex numbers and the given conjugate:

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For each of the following numbers, first visualize where it is in the complex plane. With a little practice you can quickly find x,y,r,θin your head for these simple problems. Then plot the number and label it in five ways as in Figure 3.3. Also, plot the complex conjugate of the number 1-I3.

Short Answer

Step by step solution

Given information and the general form of complex numbers:

Complex numbers and the given conjugate:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Electric Charge Field and Potential

Medical Physics

Magnetism and Electromagnetic Induction

Famous Physicists

Astrophysics

Dynamics

Study anywhere. Anytime. Across all devices.