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Chapter 1: Q1P (page 1)

First simplify each of the following numbers to the $x + i y$ form or to the $r e^{i 胃}$ form. Then plot the number in the complex plane.
$\frac{1}{1 + i}$ .

Short Answer

Expert verified

The complex number becomes $\frac{\sqrt{2}}{2} e^{- i \frac{蟺}{4}}$ .

Step by step solution

01

Given Information

The complex number is $\frac{1}{1 + i}$ .

02

Definition of the Complex number

A complex number is a number that is made up of both real and imaginary numbers.

03

Find the value

Rationalize the complex number.

$\begin{array}{rcl} \frac{1}{1 + i} & = & \frac{1}{1 + i} 路 \frac{1 - i}{1 - i} \\ = & \frac{1 - i}{1^{2} - i^{2}} \\ = & \frac{1 - i}{2} \\ = & \frac{1}{2} - \frac{i}{2} \end{array}$

Find x and y

$x = \frac{1}{2} y = \frac{- 1}{2}$

Find the value of r .

$\begin{array}{rcl} r & = & \sqrt{x^{2} + y^{2}} \\ = & \sqrt{{(\frac{1}{2})}^{2} + {(\frac{- 1}{2})}^{2}} \\ = & \frac{1}{\sqrt{2}} \\ = & \frac{\sqrt{2}}{2} \end{array}$

Find the value of $胃$ $胃$ .

$\begin{array}{rcl} 胃 & = & \arctan (\frac{\frac{- 1}{2}}{\frac{1}{2}}) \\ = & \arctan (- 1) \\ = & \frac{- 蟺}{4} \end{array}$

$\frac{1}{1 + i} = \frac{\sqrt{2}}{2} e^{- i \frac{蟺}{4}}$

The complex number becomes $\frac{\sqrt{2}}{2} e^{- i \frac{蟺}{4}}$ .

The graph is mentioned below.

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Most popular questions from this chapter

In the following problems, find the limit of the given sequence as

$n 鈫� 鈭�$

Show that the binomial coefficients $(\begin{array}{l} - 1 \\ n \end{array}) = (- 1)^{n}$

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$x \sqrt{1 + x}$

Use Maclaurin series to evaluate each of the following. Although you could do them by computer, you can probably do them in your head faster than you can type them into the computer. So use these to practice quick and skillful use of basic series to make simple calculations.

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