/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 25 Performing Operations with Compl... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 25

Performing Operations with Complex Numbers. Perform the operation and write the result in standard form. $$(9-i)-(8-i)$$

Short Answer

Expert verified
The result of the operation \((9-i)-(8-i)\) is \(1\).

Step by step solution

01

Identify the components

Identify the real and imaginary parts of each complex number. For the complex number \(9-i\), the real part is \(9\) and the imaginary part is \(-1\). For the complex number \(8-i\), the real part is \(8\) and the imaginary part is \(-1\).
02

Subtraction of the Real Parts

Subtract the real part of the second complex number from the real part of the first complex number. This leads to \(9 - 8 = 1\).
03

Subtraction of the Imaginary Parts

Subtract the imaginary part of the second complex number (referred to as \(b_2\)) from the imaginary part of the first complex number (referred to as \(b_1\)). This action follows the subtraction of the real parts - \(-1 - (-1) = 0\). Here, \(i\) is the imaginary unit.
04

Writing it in standard form

The result after subtracting the complex numbers is a new complex number which is \(1 + 0i\). Note that since the imaginary part is \(0\), it can also be written in short form as \(1\) since \(0i = 0\). This is the final answer.

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 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Standard Form
Complex numbers have a unique way of being expressed called the "standard form." This form is represented as \(a + bi\), where \(a\) is the real part, and \(b\) is the imaginary part.
The letter \(i\) stands for the imaginary unit. It is important because it forms the basis of complex numbers, being equal to the square root of \(-1\).
  • Example: In the expression \(9 - i\), \(9\) is the real part, and \(-1\) is the imaginary part.
  • Every complex number can be written in this standard form.
This format allows for easy addition, subtraction, and other operations.
Real Part
The real part of a complex number is simply the component without the imaginary unit \(i\).
It behaves just like a regular number. For instance, if we look at the complex number \(9 - i\), the real part is \(9\).
  • When performing operations, treat the real part as any other number.
  • The real parts are added or subtracted independently of the imaginary parts.
This separation helps keep operations organized and clear.
Imaginary Part
In a complex number, the imaginary part is the component accompanied by the imaginary unit \(i\).
In the example \(9 - i\), the imaginary part is \(-1\).
Here, \(-1\) is multiplied by \(i\), giving us the complete term \(-i\).
  • Imaginary parts must be handled separately when performing operations.
  • Think of them as something that complements the real part.
By understanding imaginary parts, we can easily manage complex calculations.
Imaginary Unit
The imaginary unit \(i\) is a special mathematical symbol that represents \(\sqrt{-1}\).
It allows mathematicians to work with square roots of negative numbers.
This innovation opens the door to complex number calculations and solutions.
  • Using \(i\), we create expressions like \(3 + 2i\).
  • It's essential for understanding imaginary parts in complex numbers.
Grasping \(i\) helps demystify complex numbers and their operations.
Subtraction of Complex Numbers
Subtracting complex numbers involves dealing with both the real and imaginary components.
To subtract \((9-i)\) and \((8-i)\):
  • Subtract Real Parts: \(9 - 8 = 1\)
  • Subtract Imaginary Parts: \(-1 - (-1) = 0\)
The result is \(1 + 0i\), which simplifies to \(1\).
Remember that the imaginary part can be zero, making it disappear from the expression.
Mastering this subtraction process is a key skill for handling complex mathematical problems.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

pH Levels In Exercises \(51-56\) , use the acidity model given by \(p \mathbf{H}=-\log \left[\mathbf{H}^{+}\right],\) where acidity \((\mathbf{p} \mathbf{H})\) is a measure of the hydrogen ion concentration \(\left[\mathbf{H}^{+}\right]\) (measured in moles of hydrogen per liter) of a solution. Compute \(\left[\mathrm{H}^{+}\right]\) for a solution in which \(\mathrm{pH}=5.8\)

Finding the Domain of an Expression In Exercises \(61-66\) , find the domain of the expression. Use a graphing utility to verify your result. $$\sqrt{x^{2}-9 x+20}$$

True or False? In Exercises \(61-64\) , determine whether the statement is true or false. Justify your answer. The domain of a logistic growth function cannot be the set of real numbers.

Geometry You want to make an open box from a rectangular piece of material, 15 centimeters by 9 centimeters, by cutting equal squares from the corners and turning up the sides. (a) Let \(x\) represent the side length of each of the squares removed. Draw a diagram showing the squares removed from the original piece of material and the resulting dimensions of the open box. (b) Use the diagram to write the volume \(V\) of the box as a function of \(x .\) Determine the domain of the function. (c) Sketch the graph of the function and approximate the dimensions of the box that will yield a maximum volume. (d) Find values of \(x\) such that \(V=56 .\) Which of these values is a physical impossibility in the construction of the box? Explain.

True or False?, determine whether the statement is true or false. Justify your answer. The graph of the function $$ f(x)=x^{4}-6 x^{3}-3 x-8 $$ falls to the left and to the right.
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.