Chapter 2: Problem 1
First simplify each of the following numbers to the \(x+i y\) form or to the \(r e^{i \theta}\) form. Then plot the number in the complex plane. $$\frac{1}{1+i}$$
Short Answer
Expert verified
\( \frac{1}{2} - \frac{i}{2} \) or \( \frac{\sqrt{2}}{2} e^{-i \frac{\pi}{4}} \)
Step by step solution
01
- Multiply numerator and denominator
To simplify the given expression \ \(\frac{1}{1+i}\), multiply both the numerator and the denominator by the complex conjugate of the denominator, which is \(1-i\). The expression becomes: \ \[ \frac{1\cdot(1-i)}{(1+i)\cdot(1-i)} \]
02
- Simplify the product in the numerator
Calculate the product in the numerator: \ \[ 1 \cdot (1 - i) = 1 - i \] So, we have: \ \[ \frac{1-i}{(1+i)(1-i)} \]
03
- Calculate the product in the denominator
Expand the denominator using the difference of squares formula: \ \[ (1+i)(1-i) = 1^2 - i^2 = 1 - (-1) = 1 + 1 = 2 \] So, the expression becomes: \ \[ \frac{1-i}{2} \]
04
- Simplify the expression
Divide both terms in the numerator by 2: \ \[ \frac{1-i}{2} = \frac{1}{2} - \frac{i}{2} \] So, the simplified form is: \ \[ \frac{1}{2} - \frac{i}{2} \]
05
- Express in polar form
To convert \( \frac{1}{2} - \frac{i}{2} \) to polar form, find the magnitude \( r \): \ \[ r = \sqrt{ \left( \frac{1}{2} \right)^2 + \left( - \frac{1}{2} \right)^2 } = \sqrt{ \frac{1}{4} + \frac{1}{4} } = \sqrt{ \frac{2}{4} } = \frac{\sqrt{2}}{2} \] Then, find the argument \( \theta \): \ \[ \theta = \tan^{-1} \left( \frac{ - \frac{1}{2} }{ \frac{1}{2} } \right) = \tan^{-1} ( -1 ) = - \frac{\pi}{4} \] So, the polar form is: \ \[ \frac{\sqrt{2}}{2} e^{-i \frac{\pi}{4}} \]
06
- Plot in the complex plane
Plot \( \frac{1}{2}-\frac{i}{2} \) on the complex plane, which corresponds to the point \( \left( \frac{1}{2}, - \frac{1}{2} \right) \).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with Vaia!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Complex Number Simplification
To simplify a complex number like \(\frac{1}{1+i}\), the key is to eliminate the imaginary number in the denominator. We use the *complex conjugate* of the denominator to achieve this. For a number like \(1 + i\), its complex conjugate is \(1 - i\). Multiplying the numerator and the denominator by the complex conjugate transforms the expression into a form where the denominator is a real number.
For our example, \(\frac{1 \cdot (1 - i)}{(1 + i) \cdot (1 - i)} = \frac{1 - i}{1 - (-1)} = \frac{1 - i}{2}\).
Next, we write the simplified result in the form \(\frac{1}{2} - \frac{i}{2}\). This form is much easier to handle in calculations and can be converted to polar form if necessary.
For our example, \(\frac{1 \cdot (1 - i)}{(1 + i) \cdot (1 - i)} = \frac{1 - i}{1 - (-1)} = \frac{1 - i}{2}\).
Next, we write the simplified result in the form \(\frac{1}{2} - \frac{i}{2}\). This form is much easier to handle in calculations and can be converted to polar form if necessary.
Polar Form
Converting a complex number to its polar form \(r e^{i \theta}\) involves finding the magnitude \(r\) and the angle \(\theta\).
First, we find the magnitude \(r\) using the formula: \(\[\begin{equation} r = \sqrt{x^2 + y^2} \end{equation}\]\). For \( \frac{1}{2} - \frac{i}{2} \), \(\[\begin{equation} r = \sqrt{\bigg(\frac{1}{2}\bigg)^2 + \bigg(-\frac{1}{2}\bigg)^2} = \sqrt{\frac{1}{4} + \frac{1}{4}} = \sqrt{\frac{2}{4}} = \frac{\sqrt{2}}{2} \end{equation}\]\).
Then, we find the angle \(\theta\) as: \( \theta = \tan^{-1} \bigg( \frac{-\frac{1}{2}}{\frac{1}{2}} \bigg) = \tan^{-1}(-1) = - \frac{\pi}{4} \).
Combining these, the polar form is \( \frac{\sqrt{2}}{2} e^{-i \frac{\pi}{4}} \). This form is useful because it simplifies multiplication and division of complex numbers.
First, we find the magnitude \(r\) using the formula: \(\[\begin{equation} r = \sqrt{x^2 + y^2} \end{equation}\]\). For \( \frac{1}{2} - \frac{i}{2} \), \(\[\begin{equation} r = \sqrt{\bigg(\frac{1}{2}\bigg)^2 + \bigg(-\frac{1}{2}\bigg)^2} = \sqrt{\frac{1}{4} + \frac{1}{4}} = \sqrt{\frac{2}{4}} = \frac{\sqrt{2}}{2} \end{equation}\]\).
Then, we find the angle \(\theta\) as: \( \theta = \tan^{-1} \bigg( \frac{-\frac{1}{2}}{\frac{1}{2}} \bigg) = \tan^{-1}(-1) = - \frac{\pi}{4} \).
Combining these, the polar form is \( \frac{\sqrt{2}}{2} e^{-i \frac{\pi}{4}} \). This form is useful because it simplifies multiplication and division of complex numbers.
Complex Conjugate
A complex conjugate is a key concept in simplifying complex numbers and performing operations like division. For a complex number \(a + bi\), the complex conjugate is \(a - bi\). Using complex conjugates helps us turn a complex denominator into a real number.
For example, in \(\frac{1}{1+i}\), the conjugate of \(1+i\) is \(1-i\). Multiplying by the conjugate gives a real denominator: \( (1+i)(1-i)=1^2-i^2=1-(-1)=2 \).
This process is indispensable in many aspects of complex number operations, making calculations feasible and straightforward.
For example, in \(\frac{1}{1+i}\), the conjugate of \(1+i\) is \(1-i\). Multiplying by the conjugate gives a real denominator: \( (1+i)(1-i)=1^2-i^2=1-(-1)=2 \).
This process is indispensable in many aspects of complex number operations, making calculations feasible and straightforward.
Complex Plane Plotting
Plotting complex numbers on the complex plane helps visualize them as points or vectors. The horizontal axis represents the real part, and the vertical axis represents the imaginary part.
For \( \frac{1}{2} - \frac{i}{2} \), the real part is \ \frac{1}{2} \ and the imaginary part is \ -\frac{1}{2} \. So, it corresponds to the point \( \big( \frac{1}{2}, -\frac{1}{2} \big) \) on the complex plane.
Plotting this point helps understand the magnitude and direction of the complex number. The point represents both the distance from the origin and the angle made with the positive real axis, reinforcing the connections between algebraic and geometric properties of complex numbers.
For \( \frac{1}{2} - \frac{i}{2} \), the real part is \ \frac{1}{2} \ and the imaginary part is \ -\frac{1}{2} \. So, it corresponds to the point \( \big( \frac{1}{2}, -\frac{1}{2} \big) \) on the complex plane.
Plotting this point helps understand the magnitude and direction of the complex number. The point represents both the distance from the origin and the angle made with the positive real axis, reinforcing the connections between algebraic and geometric properties of complex numbers.