/*! 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 45 Add or subtract as indicated. Wr... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 45

Add or subtract as indicated. Write each sum or difference in standard form. $$(3+2 i)+(4-3 i)$$

Short Answer

Expert verified
The sum is \(7 - i\).

Step by step solution

01

Identify the Expression Components

We are given the expression \((3 + 2i) + (4 - 3i)\). This expression consists of two complex numbers, where \(3 + 2i\) is the first complex number and \(4 - 3i\) is the second complex number.
02

Separate Real and Imaginary Parts

Separate the real parts and the imaginary parts from both complex numbers. For \(3 + 2i\), the real part is 3, and the imaginary part is 2i. For \(4 - 3i\), the real part is 4, and the imaginary part is -3i.
03

Add the Real Parts Together

Add the real parts of the two complex numbers: \(3 + 4 = 7\).
04

Add the Imaginary Parts Together

Add the imaginary parts of the complex numbers: \(2i + (-3i) = -i\).
05

Combine into Standard Form

Combine the results from steps 3 and 4 to write the sum in standard form. The standard form of a complex number is \(a + bi\), where \(a\) and \(b\) are real numbers. Thus, the sum is \(7 - i\).

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.

Addition of Complex Numbers
Complex numbers are composed of a real part and an imaginary part, typically written in the form \( a + bi \). To add complex numbers, you simply add their real components together and their imaginary components together. Let's consider the example given:
  • The complex numbers are given as \( (3 + 2i) \) and \( (4 - 3i) \).
  • To add them, you first look at the real parts: 3 and 4.
  • Adding the real parts gives you 7.
Next, focus on the imaginary parts:
  • The imaginary components are \( 2i \) and \( -3i \).
  • Adding these gives us \( 2i + (-3i) = -i \).
Thus, by combining these results, the final sum of the complex numbers is \( 7 - i \). It's like adding two separate columns: one for the real and one for the imaginary parts.
Imaginary Unit
The imaginary unit, denoted by \( i \), is a fundamental component of complex numbers. Understanding it is crucial to working with complex numbers effectively. By definition:
  • \( i \) is the square root of \(-1\), meaning \( i^2 = -1 \).
When dealing with imaginary numbers, the operations may seem unusual at first but are quite logical:
  • \( 0i \) represents no imaginary part, simplifying to zero.
  • Multiplying by \( i \) rotates a point on the complex plane 90 degrees, which gives it a unique property in geometry and engineering.
In our problem, \( i \) appears in expressions \( 2i \) and \(-3i\). Here, it's treated as a regular number in arithmetic operations within the context of these expressions.
Standard Form
Writing complex numbers in standard form ensures consistency and clarity. In standard form, a complex number is expressed as \( a + bi \), where:
  • \( a \) is the real part.
  • \( b \) is the coefficient of the imaginary part, \( i \).
This format is beneficial as it separates the real and imaginary components clearly, making it easier to perform further mathematical operations like addition, subtraction, and multiplication.
In our solution, after calculating the sum of the given complex numbers, we find that the sum \( 7 - i \) is already in standard form. Here:
  • 7 is the real part, represented by \( a \).
  • -1 is the imaginary coefficient, represented by \( b \) (the number before \( i \)).
Expressing complex solutions in standard form simplifies interpretation and application in various mathematical contexts.

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

Use the rational zeros theorem to completely factor \(P(x)\). $$P(x)=21 x^{4}+13 x^{3}-103 x^{2}-65 x-10$$

Solve each problem. Selected values of the stopping distance \(y\) in feet of a car traveling \(x\) mph are given in the table. $$\begin{array}{c|c}\begin{array}{c}\text { Speed } \\\\\text { (in mph) }\end{array} & \begin{array}{c} \text { Stopping Distance } \\\\\text { (in feet) }\end{array} \\\\\hline 20 & 46 \\\30 & 87 \\\40 & 140 \\\50 & 240 \\ 60 & 282 \\\70 & 371\end{array}$$ (a) Plot the data. (b) The quadratic function $$f(x)=0.056057 x^{2}+1.06657 x$$ is one model of the data. Find and interpret \(f(45)\) (c) Use a graph of the function in the same window as the data to determine how well \(f\) models the stopping distance.

For each polynomial function, (a) list all possible rational zeros, (b) use a graph to eliminate some of the possible zeros listed in part (a), (c) find all rational zeros, and (d) factor \(P(x)\). $$P(x)=x^{3}+6 x^{2}-x-30$$

Solve each formula for the indicated variable. Leave \(\pm\) in answers when appropriate. Assume that no denominators are \(0 .\) $$S=2 \pi r h+2 \pi r^{2} \quad \text { for } r$$

Solve each formula for the indicated variable. Leave \(\pm\) in answers when appropriate. Assume that no denominators are \(0 .\) $$s=\frac{1}{2} g t^{2} \quad \text { for } t$$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

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.