Chapter 2: Problem 9

Find and plot the complex conjugate of each number. \(2 i-2\)

Short Answer

Expert verified

The complex conjugate of \(2i - 2\) is \(-2 - 2i\). Plot \((-2, 2)\) for the original number and \((-2, -2)\) for the conjugate.

Step by step solution

Understanding Complex Conjugates

The complex conjugate of a complex number is found by changing the sign of the imaginary part. Given a complex number in the form of \(a + bi\), the complex conjugate is \(a - bi\).

Identify Real and Imaginary Parts

In the complex number \(2i - 2\), identify the real part \(a\) and the imaginary part \(b\). Here, \(a = -2\) and \(b = 2\).

Find the Complex Conjugate

Change the sign of the imaginary part \(b\). The complex conjugate of \(2i - 2\) is \(-2 - 2i\).

Plot the Original Complex Number

Plot the original complex number \(2i - 2\) on the complex plane. Place a point at the coordinates \((-2, 2)\), where \(-2\) is the real part and \(2\) is the imaginary part.

Plot the Complex Conjugate

Plot the complex conjugate \(-2 - 2i\) on the complex plane. Place a point at the coordinates \((-2, -2)\), where \(-2\) is the real part and \(-2\) is the imaginary part.

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.

Start your free trial

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 Conjugate

A complex conjugate of a complex number simply involves changing the sign of the imaginary part. If the complex number is represented as \(a + bi\), its complex conjugate is \(a - bi\). This concept is very useful in various areas of mathematics and engineering.
Let's break down the key steps with an example.
For the complex number \(2i - 2\):

The real part, \(a\), is \(-2\)
The imaginary part, \(b\), is \(2\)

Finding the conjugate means changing the sign of the imaginary part. So, the complex conjugate of \(2i - 2\) becomes \(-2 - 2i\). Understanding and applying this process is simple when you remember the key step: change the sign of the imaginary part.

Complex Plane

The complex plane, also known as the Argand plane, is a way to visually represent complex numbers. This helps in understanding their behavior and properties.
Each complex number corresponds to a point on this plane:

The horizontal axis (x-axis) represents the real part of the complex number.
The vertical axis (y-axis) represents the imaginary part.

For the complex number \(2i - 2\), we plot it at coordinates \((-2, 2)\), where \(-2\) is the real part and \(2\) is the imaginary part. Similarly, for its complex conjugate \(-2 - 2i\), we plot it at coordinates \((-2, -2)\). Plotting these points can provide a clearer understanding of their relationship.

Imaginary Part

The imaginary part of a complex number is the component that involves the imaginary unit \(i\). In the expression \(a + bi\), \(bi\) is the imaginary part. Here, \(i\) represents the square root of \(-1\).
For example, in the complex number \(2i - 2\), the imaginary part is \(2i\).
To find the complex conjugate, we need to change the sign of the imaginary part. So, as we did earlier, the expression \(-2 - 2i\) highlights that we have switched \(2i\) to \(-2i\).
Understanding the imaginary part and its behavior is crucial for mastering complex numbers. Always remember, it’s just as important as understanding the real part.