Chapter 1: Q1P (page 1)

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

Short Answer

Expert verified

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

Step by step solution

Given Information

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

Definition of the Complex number

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

Find the value

Rationalize the complex number.

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

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 $θ$ $θ$ .

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

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

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

The graph is mentioned below.

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

Short Answer

Step by step solution

Given Information

Definition of the Complex number

Find the value

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First simplify each of the following numbers to the x+iyform or to thereiθ form. Then plot the number in the complex plane.11+i.

Short Answer

Step by step solution

Given Information

Definition of the Complex number

Find the value

One App. One Place for Learning.

Most popular questions from this chapter

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