Divide. Choose The Correct Answer From The Choices Below.$\frac{(5-i)}{(-3+2i)}$A. $\frac{-13-7i}{5}$ B. $\frac{-13-7i}{13}$ C. $\frac{-17-7i}{5}$ D. $\frac{-17-7i}{13}$
Introduction
Complex numbers are a fundamental concept in mathematics, and they play a crucial role in various fields, including algebra, geometry, and calculus. When dealing with complex numbers, division is an essential operation that requires careful attention to detail. In this article, we will explore the process of dividing complex numbers and provide a step-by-step guide on how to choose the correct answer from the given choices.
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. Complex numbers can be represented graphically on a complex plane, with the real part on the x-axis and the imaginary part on the y-axis.
The Division of Complex Numbers
To divide two complex numbers, we can use the following formula:
(z1 / z2) = (z1 * conjugate(z2)) / (z2 * conjugate(z2))
where z1 and z2 are complex numbers, and conjugate(z2) is the conjugate of z2.
Step-by-Step Guide to Dividing Complex Numbers
Let's consider the given problem:
To divide this complex number, we will follow the steps below:
Step 1: Find the Conjugate of the Denominator
The conjugate of the denominator (-3 + 2i) is (-3 - 2i).
Step 2: Multiply the Numerator and the Conjugate of the Denominator
We will multiply the numerator (5 - i) by the conjugate of the denominator (-3 - 2i):
(5 - i) * (-3 - 2i) = -15 - 10i + 3i + 2i^2
Using the fact that i^2 = -1, we can simplify the expression:
-15 - 10i + 3i - 2 = -17 - 7i
Step 3: Multiply the Denominator and the Conjugate of the Denominator
We will multiply the denominator (-3 + 2i) by the conjugate of the denominator (-3 - 2i):
(-3 + 2i) * (-3 - 2i) = 9 + 6i - 6i - 4i^2
Using the fact that i^2 = -1, we can simplify the expression:
9 - 4(-1) = 13
Step 4: Divide the Result from Step 2 by the Result from Step 3
We will divide the result from step 2 (-17 - 7i) by the result from step 3 (13):
Conclusion
In this article, we have explored the process of dividing complex numbers and provided a step-by-step guide on how to choose the correct answer from the given choices. By following the steps outlined above, we can divide complex numbers and simplify expressions involving complex numbers.
Answer
Based on the steps outlined above, the correct answer is:
This answer matches option D in the given choices.
Discussion
Complex numbers are a fundamental concept in mathematics, and they play a crucial role in various fields, including algebra, geometry, and calculus. When dealing with complex numbers, division is an essential operation that requires careful attention to detail. In this article, we have provided a step-by-step guide on how to divide complex numbers and choose the correct answer from the given choices.
Additional Resources
For further reading on complex numbers, we recommend the following resources:
Introduction
In our previous article, we explored the process of dividing complex numbers and provided a step-by-step guide on how to choose the correct answer from the given choices. In this article, we will answer some frequently asked questions about dividing complex numbers and provide additional insights and examples.
Q&A
Q: What is the difference between dividing complex numbers and multiplying complex numbers?
A: When dividing complex numbers, we need to multiply the numerator and the denominator by the conjugate of the denominator, whereas when multiplying complex numbers, we simply multiply the two numbers together.
Q: Why do we need to multiply the numerator and the denominator by the conjugate of the denominator?
A: We need to multiply the numerator and the denominator by the conjugate of the denominator to eliminate the imaginary part from the denominator. This is because the conjugate of a complex number is the number with the opposite sign of the imaginary part.
Q: How do I know which option is the correct answer?
A: To determine which option is the correct answer, you need to follow the steps outlined in our previous article and simplify the expression. You can also use a calculator or a computer program to check your answer.
Q: Can I use a calculator to divide complex numbers?
A: Yes, you can use a calculator to divide complex numbers. However, make sure to enter the complex numbers in the correct format, with the real part first and the imaginary part second.
Q: What if the denominator is zero?
A: If the denominator is zero, the expression is undefined, and you cannot divide complex numbers. In this case, you need to simplify the expression or use a different approach.
Q: Can I divide complex numbers with different magnitudes?
A: Yes, you can divide complex numbers with different magnitudes. However, make sure to follow the steps outlined in our previous article and simplify the expression.
Q: How do I simplify complex numbers?
A: To simplify complex numbers, you need to combine the real and imaginary parts. You can do this by adding or subtracting the real and imaginary parts separately.
Q: Can I use complex numbers in real-world applications?
A: Yes, complex numbers are used in many real-world applications, including engineering, physics, and computer science. They are used to model and analyze complex systems, such as electrical circuits and mechanical systems.
Examples
Example 1:
Divide the complex number (3 + 4i) by (2 - 3i).
Solution:
(3 + 4i) / (2 - 3i) = (3 + 4i) * (2 + 3i) / (2 - 3i) * (2 + 3i) = (6 + 9i + 8i + 12i^2) / (4 + 6i - 6i - 9i^2) = (-6 + 17i) / 13
Example 2:
Divide the complex number (5 - 2i) by (3 + 4i).
Solution:
(5 - 2i) / (3 + 4i) = (5 - 2i) * (3 - 4i) / (3 + 4i) * (3 - 4i) = (15 - 20i - 6i + 8i^2) / (9 - 12i + 12i - 16i^2) = (-13 - 26i) / 25
Conclusion
In this article, we have answered some frequently asked questions about dividing complex numbers and provided additional insights and examples. We hope this article has been helpful in clarifying the process of dividing complex numbers and choosing the correct answer from the given choices.
Additional Resources
For further reading on complex numbers, we recommend the following resources:
We hope this article has provided a helpful guide on how to divide complex numbers and choose the correct answer from the given choices. If you have any questions or need further clarification, please don't hesitate to ask.