Find The Quotient Of These Complex Numbers: { (6-i) \div (4+3i) =$}$A. { \frac{27}{7}-\frac{22}{7} I$}$B. { \frac{27}{25}-\frac{22}{25} I$}$C. ${ 3-\frac{22}{7} I\$} D. { \frac{21}{25}-\frac{22}{25} I$}$
Introduction
Complex numbers are a fundamental concept in mathematics, and they have numerous applications in various fields, including engineering, physics, and computer science. In this article, we will focus on dividing complex numbers, which is a crucial operation in complex analysis. We will use the given problem to illustrate the steps involved in dividing complex numbers.
What are Complex Numbers?
Complex numbers are numbers that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit, which satisfies the equation i^2 = -1. The real part of a complex number is a, and the imaginary part is bi.
The Division of Complex Numbers
To divide two complex numbers, we can use the following formula:
(a + bi) ÷ (c + di) = (ac + bd) / (c^2 + d^2) + (bc - ad) / (c^2 + d^2)i
where a, b, c, and d are real numbers.
Step-by-Step Solution
Now, let's apply the formula to the given problem:
(6 - i) ÷ (4 + 3i)
First, we need to identify the real and imaginary parts of the two complex numbers:
a = 6, b = -1, c = 4, d = 3
Next, we plug these values into the formula:
(ac + bd) / (c^2 + d^2) = (6*4 + (-1)*3) / (4^2 + 3^2) = (24 - 3) / (16 + 9) = 21 / 25
(bc - ad) / (c^2 + d^2) = ((-1)4 - 63) / (4^2 + 3^2) = (-4 - 18) / (16 + 9) = -22 / 25
Therefore, the quotient of the two complex numbers is:
21/25 - 22/25i
Conclusion
In this article, we have discussed the division of complex numbers, which is a fundamental operation in complex analysis. We have used the given problem to illustrate the steps involved in dividing complex numbers. By following the formula and plugging in the values, we have obtained the quotient of the two complex numbers.
Answer
The correct answer is:
B. 21/25 - 22/25i
Why is this the correct answer?
The correct answer is B. 21/25 - 22/25i because it satisfies the formula for dividing complex numbers. The real part of the quotient is 21/25, and the imaginary part is -22/25i, which is obtained by plugging in the values into the formula.
What are the other options?
The other options are:
A. 27/7 - 22/7i C. 3 - 22/7i D. 21/25 - 22/25i
These options are incorrect because they do not satisfy the formula for dividing complex numbers.
Why are these options incorrect?
Option A is incorrect because the real part is 27/7, which is not equal to 21/25. Option C is incorrect because the real part is 3, which is not equal to 21/25. Option D is incorrect because the imaginary part is -22/25i, which is correct, but the real part is 21/25, which is not equal to 21/25.
What is the significance of this problem?
This problem is significant because it illustrates the importance of following the formula for dividing complex numbers. By plugging in the values into the formula, we can obtain the correct quotient of the two complex numbers.
What are the applications of complex numbers?
Complex numbers have numerous applications in various fields, including:
- Engineering: Complex numbers are used to analyze and design electrical circuits, control systems, and signal processing systems.
- Physics: Complex numbers are used to describe the behavior of physical systems, such as the motion of particles and the propagation of waves.
- Computer Science: Complex numbers are used in algorithms for solving linear systems, finding eigenvalues, and performing numerical computations.
Conclusion
Introduction
In our previous article, we discussed the division of complex numbers, which is a fundamental operation in complex analysis. In this article, we will provide a Q&A guide to help you understand the concept of complex number division better.
Q: What is the formula for dividing complex numbers?
A: The formula for dividing complex numbers is:
(a + bi) ÷ (c + di) = (ac + bd) / (c^2 + d^2) + (bc - ad) / (c^2 + d^2)i
where a, b, c, and d are real numbers.
Q: How do I apply the formula to divide complex numbers?
A: To apply the formula, you need to identify the real and imaginary parts of the two complex numbers, plug in the values into the formula, and simplify the expression.
Q: What are the steps involved in dividing complex numbers?
A: The steps involved in dividing complex numbers are:
- Identify the real and imaginary parts of the two complex numbers.
- Plug in the values into the formula.
- Simplify the expression.
Q: What is the significance of the real and imaginary parts in complex number division?
A: The real and imaginary parts play a crucial role in complex number division. The real part represents the real number, while the imaginary part represents the imaginary number.
Q: How do I handle complex numbers with zero imaginary part?
A: If the complex number has a zero imaginary part, you can simply ignore the imaginary part and divide the real numbers.
Q: Can I divide complex numbers with zero real part?
A: Yes, you can divide complex numbers with zero real part. In this case, the result will be a purely imaginary number.
Q: What are the applications of complex number division?
A: Complex number division has numerous applications in various fields, including:
- Engineering: Complex number division is used to analyze and design electrical circuits, control systems, and signal processing systems.
- Physics: Complex number division is used to describe the behavior of physical systems, such as the motion of particles and the propagation of waves.
- Computer Science: Complex number division is used in algorithms for solving linear systems, finding eigenvalues, and performing numerical computations.
Q: What are some common mistakes to avoid when dividing complex numbers?
A: Some common mistakes to avoid when dividing complex numbers include:
- Not identifying the real and imaginary parts correctly.
- Not plugging in the values into the formula correctly.
- Not simplifying the expression correctly.
Q: How can I practice complex number division?
A: You can practice complex number division by using online resources, such as calculators and worksheets, or by working on problems from textbooks and online resources.
Conclusion
In conclusion, complex number division is a fundamental operation in complex analysis. By following the formula and plugging in the values, you can obtain the correct quotient of the two complex numbers. This Q&A guide provides a comprehensive overview of complex number division, including the formula, steps, and applications.
Frequently Asked Questions
- Q: What is the difference between complex number addition and division? A: Complex number addition involves adding the real and imaginary parts separately, while complex number division involves dividing the real and imaginary parts separately.
- Q: Can I divide complex numbers with complex numbers? A: Yes, you can divide complex numbers with complex numbers. In this case, you need to use the formula for dividing complex numbers.
- Q: What are some real-world applications of complex number division? A: Complex number division has numerous real-world applications, including engineering, physics, and computer science.
Additional Resources
- Online Calculators: There are many online calculators available that can help you practice complex number division.
- Textbooks: There are many textbooks available that provide a comprehensive overview of complex number division.
- Online Resources: There are many online resources available that provide practice problems and worksheets for complex number division.