Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 10: Problem 17
Perform the following operations and express your answer in the form \(a+b i\). $$\left(\frac{2}{3}-\frac{1}{2} i\right) \div\left(\frac{1}{3}+\frac{1}{2} i\right)$$
Short Answer
Expert verified
The simplest form is \(-\frac{1}{13} - \frac{18}{13} i\).
Step by step solution
01
Write the Division as a Multiplication
To simplify the expression, consider dividing two complex numbers as multiplying by the reciprocal. So we have:\[\left(\frac{2}{3}-\frac{1}{2} i\right) \div \left(\frac{1}{3}+\frac{1}{2} i\right) = \left(\frac{2}{3}-\frac{1}{2} i\right) \times \left(\frac{1}{3}+\frac{1}{2} i\right)^{-1}\]
02
Find the Reciprocal of the Denominator
To find the reciprocal of the denominator \(\frac{1}{3}+\frac{1}{2} i\), multiply it by its conjugate \(\frac{1}{3}-\frac{1}{2} i\) over itself:\[\left(\frac{1}{3}+\frac{1}{2} i\right)^{-1} = \frac{\frac{1}{3}-\frac{1}{2} i}{\left(\frac{1}{3}\right)^2 + \left(\frac{1}{2}\right)^2}\]The denominator here simplifies by using the formula \(a^2 + b^2\) to \(\frac{1}{9} + \frac{1}{4} = \frac{4}{36} + \frac{9}{36} = \frac{13}{36}\).Thus, the reciprocal becomes:\[\left(\frac{1}{3}+\frac{1}{2} i\right)^{-1} = \frac{\frac{1}{3}-\frac{1}{2} i}{\frac{13}{36}} = \frac{36}{13} \left(\frac{1}{3}-\frac{1}{2} i\right)\]
03
Multiply by the Reciprocal
Now multiply the two complex numbers:\[\left(\frac{2}{3}-\frac{1}{2} i\right) \times \frac{36}{13} \left(\frac{1}{3}-\frac{1}{2} i\right)\]Distribute the multiplication:\[= \frac{36}{13} \left(\left(\frac{2}{3}\right) \cdot \frac{1}{3} - \left(\frac{2}{3}\right) \cdot \frac{1}{2} i - \left(\frac{1}{2} i\right) \cdot \frac{1}{3} + \left(\frac{1}{2} i\right) \cdot \frac{1}{2} i\right)\]Simplify each part:\[= \frac{36}{13} \left(\frac{2}{9} - \frac{i}{3} - \frac{i}{6} - \frac{1}{4}\right)\]
04
Simplify the Expression
Combine like terms:- Real part: \(\frac{2}{9} - \frac{1}{4}\)- Imaginary part: \(-\frac{i}{3} - \frac{i}{6}\)Calculate the real part:\[\frac{2}{9} - \frac{1}{4} = \frac{8}{36} - \frac{9}{36} = -\frac{1}{36}\]Calculate the imaginary part:\[-\frac{i}{3} - \frac{i}{6} = -\frac{2i}{6} - \frac{i}{6} = -\frac{3i}{6} = -\frac{i}{2}\]Thus, the expression becomes:\[\frac{36}{13} \left(-\frac{1}{36} - \frac{i}{2}\right) = -\frac{1}{13} - \frac{18i}{13}\]
05
Present the Final Answer
The solution in the standard form \(a + bi\) is achieved by distributing the scalar \(\frac{36}{13}\) to each part of the simplified expression from Step 4, resulting in:\[-\frac{1}{13} - \frac{18i}{13}\]
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.
Complex Division
When dividing complex numbers, we follow a process that turns division into multiplication by using the reciprocal of the denominator. This approach simplifies our computations and helps us express complex numbers in the standard form. Consider a complex number division task:
- Begin by rewriting the division as a multiplication with the reciprocal of the denominator.
- Understand that the reciprocal of a complex number involves both real and imaginary parts.
- By performing conjugate multiplication, we prepare the denominator for simplification.
Conjugate Multiplication
Conjugate multiplication is a clever technique used in complex number division. It's all about transforming the denominator by using the conjugate, which helps eliminate the imaginary part. Here's how it works:
- The conjugate of a complex number \(a + bi\) is \(a - bi\).
- Multiplying a complex number by its conjugate results in a real number, making it easier to divide.
- This simplifies the division because the imaginary parts cancel each other out.
Real and Imaginary Parts
In any complex number, there are two components to consider: the real part and the imaginary part. Understanding these parts is crucial in working with complex numbers:
- The real part is the non-imaginary component, represented as \(a\) in \(a + bi\).
- The imaginary part involves the imaginary unit \(i\), represented as \(bi\).
- Each operation we perform, whether addition, subtraction, multiplication, or division, must account for these two parts separately.
Standard Form \(a+bi\)
Expressing complex numbers in the standard form \(a + bi\) is essential for clarity and simplicity. This form clearly communicates both the real and imaginary parts:
- After performing operations on complex numbers, like division, ensure the result is expressed in this format.
- This involves simplifying and combining like terms from each part: \(a\) and \(bi\).
- Presenting the solution in \(a + bi\) makes interpretation straightforward, allowing others to easily identify the real and imaginary components.
