Chapter 2: Q6P (page 52)

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.
${(\frac{1 + i}{1 - i})}^{2}$ .

Short Answer

Expert verified

The complex number can be written as $e^{i π}$ .

The graph is mentioned below:

Step by step solution

Given Information

The complex number is ${(\frac{1 + i}{1 - i})}^{2}$ .

Definition of the Complex number.

A complex number is a number that has real and imaginary numbers and is written in the form as:

$z = a + {}_{ι}b$ where a and b are real, and z is the complex number.

Find the values

$θ$ Factorize the complex number as:

$z = {(\frac{1 + i}{1 - i})}^{2} = (\frac{{(1 + i)}^{2}}{{(1 - i)}^{2}}) = \frac{1 + 2 i - 1}{1 - 2 i - 1} = - 1$

Find x and y:

$x = - 1 y = 0$

Find the value of r .

$r = \sqrt{x^{2} + y^{2}} r = \sqrt{1 + 0} = 1$

Find the value of $θ$ .

localid="1658487417780" $θ = \cos^{- 1} (\frac{x}{r}) θ = \cos^{- 1} (- \frac{1}{1}) = θ {(\frac{1 + i}{1 - i})}^{2} = e^{i π}$

The complex number will be written as localid="1658487424374" $e^{i π}$ .

The graph is mentioned below as:

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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.
${(\frac{1 + i}{1 - i})}^{2}$ .

Short Answer

Step by step solution

Given Information

Definition of the Complex number.

Find the values

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First simplify each of the following numbers to the x+iyform or to the reiθform. Then plot the number in the complex plane.(1+i1-i)2.

Short Answer

Step by step solution

Given Information

Definition of the Complex number.

Find the values

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Wave Optics

Electromagnetism

Measurements

Electric Charge Field and Potential

Force

Particle Model of Matter

Study anywhere. Anytime. Across all devices.

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.
${(\frac{1 + i}{1 - i})}^{2}$ .