Q1P For each of the following number... [FREE SOLUTION]

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 $x, y, r, 胃$ 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

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

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.

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

In the given number 1+i .

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

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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