Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 10: Problem 22
Add or subtract. Write the sum or difference in the form \(a+b.\) $$ (8-3 i)+(-8+3 i) $$
Short Answer
Expert verified
0
Step by step solution
01
Identify and Write the Terms
Look at the expression \( (8 - 3i) + (-8 + 3i) \) and identify the terms. You have two complex numbers: \(8 - 3i\) and \(-8 + 3i\). These numbers are written in the form of \(a + bi\).
02
Group Real and Imaginary Parts
Separate the real parts and the imaginary parts from both complex numbers. The real parts are \(8\) and \(-8\). The imaginary parts are \(-3i\) and \(+3i\).
03
Add the Real Parts
Perform the addition on the real parts: \(8 + (-8) = 0\).
04
Add the Imaginary Parts
Add the imaginary parts: \(-3i + 3i = 0i\).
05
Combine the Results
Combine the results from Steps 3 and 4 to form a complex number in the form of \(a + bi\). You get \(0 + 0i\).
06
Simplify the Expression
Simplify \(0 + 0i\) to just \(0\), since the imaginary part is zero and doesn't need to be written.
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
Adding complex numbers follows a straightforward process. A complex number is generally represented as \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. To add complex numbers, simply combine their real and imaginary parts separately. Consider the problem from the exercise: \((8-3i)+(-8+3i)\). First, identify the terms for addition. Each complex number has its own real number and imaginary coefficient. The primary goal is to combine like terms, which means:
- Add the real numbers together.
- Add the imaginary numbers together.
Imaginary Numbers
Imaginary numbers are the cornerstone of complex numbers. An imaginary number is defined as a number that gives a negative result when squared. This concept may initially seem a bit abstract, but consider the basic imaginary unit \(i\), which is defined as \(i = \sqrt{-1}\). Thus, \(i^2 = -1\). Imaginary numbers appear in mathematical problems where negative square roots are involved.
- They are typically denoted as \(bi\).
- The \(b\) represents the imaginary coefficient.
- Imaginary numbers, combined with real numbers, make up complex numbers.
Real and Imaginary Parts
Complex numbers consist of both real and imaginary parts. In the expression \(a + bi\), the \(a\) component is the real part, and \(bi\) is the imaginary part. Understanding the distinction between these parts is critical in operations involving complex numbers. In the exercise:
- The numbers \(8\) and \(-8\) are the real parts.
- The numbers \(-3i\) and \(+3i\) are the imaginary parts.
Simplification of Expressions
Simplifying expressions is crucial to presenting complex numbers in their simplest form. After performing operations on the separate parts, you combine them into a final expression. In our exercise, both the real and imaginary parts sum to zero:
- Real part: \(8 + (-8) = 0\)
- Imaginary part: \(-3i + 3i = 0i\)
