Chapter 2: Q1P (page 51)

For each of the following numbers, first, visualize where it is in the complete 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 numbers 1+i .

Short Answer

Expert verified

The right values for the complex numbers are

$x = 1, y = 1 r = \sqrt{2} θ = \frac{π}{4} 1 + i = \sqrt{2} (\cos \frac{π}{4} + i \sin \frac{π}{4}) = \sqrt{2} e^{i \frac{π}{4}}$

Step by step solution

Given information and the complex numbers:

The given complex number is 1+i.

The general form of complex numbers is x+iy.

Complex numbers and the complex conjugate:

The general form of complex numbers is x +iy.

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

In the given number 1+i .

$x = 1, y = 1 r = \sqrt{1^{2} + 1^{2}} = \sqrt{2} θ = \cos^{- 1} \frac{1}{\sqrt{2}} = \frac{π}{4} 1 + i = \sqrt{2} (\cos \frac{π}{4} + i \sin \frac{π}{4}) = \sqrt{2} e^{i \frac{x}{4}}$

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

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Short Answer

Step by step solution

Given information and the complex numbers:

Complex numbers and the complex conjugate:

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For each of the following numbers, first, visualize where it is in the complete 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 numbers 1+i .

Short Answer

Step by step solution

Given information and the complex numbers:

Complex numbers and the complex conjugate:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Physical Quantities And Units

Quantum Physics

Electricity and Magnetism

Classical Mechanics

Engineering Physics

Further Mechanics and Thermal Physics

Study anywhere. Anytime. Across all devices.