Chapter 2: Q1MP (page 80)

Find one or more values of each of the following complex expressions and compare with a computer solution.
${(\frac{1 + i}{1 - i})}^{2718}$

Short Answer

Expert verified

The value of the complex number is, $z = - 1$ .

Step by step solution

Given Information.

The given expression is, ${(\frac{1 + i}{1 - i})}^{2718}$ .

Definition of complex series.

The numbers that are presented in the form of $a + i b$ where, a is real numbers and ' ib ' is an imaginary number called complex numbers.

Example: $3 + 2 i$ .

Convert into polar form.

Consider.

$z_{1} = 1 + i z_{2} = 1 - i$

Write both in polar form.

$z_{1} = 1 + i z_{1} = \sqrt{2} e^{(πi / 4)} z_{2} = 1 - i z_{2} = \sqrt{2} e^{(- πi / 4)}$

Substitute the values and solve.

Put both the value in the given question.

$z = {(\frac{1 + i}{1 - i})}^{2718} z = {(\frac{\sqrt{2} e^{(πi / 4)}}{\sqrt{2} e^{(- πi / 4)}})}^{2718} z = {[e^{(πi / 4)}]}^{2718} z = e^{(1359 πi)}$

$z = \cos (1359 π) + i \sin (1359 π) z = \cos (1359 π - 2 nπ) + i \sin (1359 π - 2 nπ) z = \cos (1359 π - 1358 π) + i \sin (1359 π - 1358 π) z = \cos (π) + i \sin (π)$

$z = - 1$

Hence the value is found to be ${(\frac{1 + i}{1 - i})}^{2718} = - 1$

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Find one or more values of each of the following complex expressions and compare with a computer solution.
${(\frac{1 + i}{1 - i})}^{2718}$

Short Answer

Step by step solution

Given Information.

Definition of complex series.

Convert into polar form.

Substitute the values and solve.

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Find one or more values of each of the following complex expressions and compare with a computer solution.(1+i1-i)2718

Short Answer

Step by step solution

Given Information.

Definition of complex series.

Convert into polar form.

Substitute the values and solve.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Rotational Dynamics

Dynamics

Electric Charge Field and Potential

Translational Dynamics

Scientific Method Physics

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Study anywhere. Anytime. Across all devices.

Find one or more values of each of the following complex expressions and compare with a computer solution.
${(\frac{1 + i}{1 - i})}^{2718}$