What Is The Quotient Of The Complex Numbers Below?$\[ \frac{7+i}{3-i} \\]A. \[$10+i\$\] B. \[$2+i\$\] C. \[$2-i\$\] D. \[$10-i\$\]
Introduction
Complex numbers are a fundamental concept in mathematics, and they have numerous applications in various fields, including algebra, geometry, and calculus. In this article, we will explore the concept of complex numbers and how to find the quotient of two 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 the number a, and the imaginary part is the number b.
Example of Complex Numbers
Some examples of complex numbers are:
- 3 + 4i
- 2 - 5i
- 1 + 2i
- 4 - 3i
Quotient of Complex Numbers
The quotient of two complex numbers is a complex number that results from dividing one complex number by another. To find the quotient of two complex numbers, we can use the following formula:
(a + bi) / (c + di) = ((ac + bd) + (bc - ad)i) / (c^2 + d^2)
Step-by-Step Solution
To find the quotient of the complex numbers 7 + i and 3 - i, we can use the formula above.
Step 1: Multiply the numerator and denominator by the conjugate of the denominator
The conjugate of 3 - i is 3 + i. We multiply the numerator and denominator by the conjugate of the denominator to eliminate the imaginary part from the denominator.
(7 + i) / (3 - i) = ((7 + i)(3 + i)) / ((3 - i)(3 + i))
Step 2: Expand the numerator and denominator
We expand the numerator and denominator using the distributive property.
((7 + i)(3 + i)) = 21 + 7i + 3i + i^2 ((3 - i)(3 + i)) = 9 - 3i + 3i - i^2
Step 3: Simplify the numerator and denominator
We simplify the numerator and denominator by combining like terms and using the fact that i^2 = -1.
(21 + 10i - 1) / (9 + 1) = (20 + 10i) / 10
Step 4: Simplify the fraction
We simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is 10.
2 + i
Conclusion
In this article, we have explored the concept of complex numbers and how to find the quotient of two complex numbers. We have used the formula (a + bi) / (c + di) = ((ac + bd) + (bc - ad)i) / (c^2 + d^2) to find the quotient of the complex numbers 7 + i and 3 - i. The quotient is 2 + i.
Answer
The correct answer is B. 2 + i.
Discussion
This problem is a great example of how to find the quotient of complex numbers. The key is to use the formula and simplify the numerator and denominator. It's also important to remember that i^2 = -1.
Related Topics
- Complex numbers
- Quotient of complex numbers
- Imaginary unit
- Algebra
- Geometry
- Calculus
References
- "Complex Numbers" by Math Open Reference
- "Quotient of Complex Numbers" by Wolfram MathWorld
- "Imaginary Unit" by Khan Academy
Quotient of Complex Numbers: Q&A =====================================
Introduction
In our previous article, we explored the concept of complex numbers and how to find the quotient of two complex numbers. In this article, we will answer some frequently asked questions about the quotient of complex numbers.
Q: What is the quotient of two complex numbers?
A: The quotient of two complex numbers is a complex number that results from dividing one complex number by another. To find the quotient of two complex numbers, we can use the formula:
(a + bi) / (c + di) = ((ac + bd) + (bc - ad)i) / (c^2 + d^2)
Q: How do I find the quotient of two complex numbers?
A: To find the quotient of two complex numbers, you can follow these steps:
- Multiply the numerator and denominator by the conjugate of the denominator.
- Expand the numerator and denominator.
- Simplify the numerator and denominator.
- Simplify the fraction.
Q: What is the conjugate of a complex number?
A: The conjugate of a complex number is a complex number that has the same real part and the opposite imaginary part. For example, the conjugate of 3 + 4i is 3 - 4i.
Q: Why do I need to multiply the numerator and denominator by the conjugate of the denominator?
A: Multiplying the numerator and denominator by the conjugate of the denominator eliminates the imaginary part from the denominator, making it easier to simplify the fraction.
Q: Can I use a calculator to find the quotient of two complex numbers?
A: Yes, you can use a calculator to find the quotient of two complex numbers. However, it's always a good idea to understand the concept and the formula behind it.
Q: What are some common mistakes to avoid when finding the quotient of two complex numbers?
A: Some common mistakes to avoid when finding the quotient of two complex numbers include:
- Not multiplying the numerator and denominator by the conjugate of the denominator.
- Not expanding the numerator and denominator.
- Not simplifying the numerator and denominator.
- Not simplifying the fraction.
Q: Can I use the quotient of complex numbers to solve real-world problems?
A: Yes, the quotient of complex numbers can be used to solve real-world problems, such as:
- Electrical engineering: Complex numbers are used to represent AC circuits and to analyze the behavior of electrical systems.
- Signal processing: Complex numbers are used to represent signals and to analyze the behavior of systems that process signals.
- Control systems: Complex numbers are used to represent the behavior of control systems and to analyze the stability of these systems.
Conclusion
In this article, we have answered some frequently asked questions about the quotient of complex numbers. We have also provided some tips and examples to help you understand the concept and the formula behind it.
Related Topics
- Complex numbers
- Quotient of complex numbers
- Imaginary unit
- Algebra
- Geometry
- Calculus
References
- "Complex Numbers" by Math Open Reference
- "Quotient of Complex Numbers" by Wolfram MathWorld
- "Imaginary Unit" by Khan Academy
Practice Problems
- Find the quotient of 2 + 3i and 1 - 2i.
- Find the quotient of 4 + 5i and 2 + 3i.
- Find the quotient of 3 - 4i and 1 + 2i.
Answers
- 2/5 + 11/5i
- 2 + i
- 1/5 - 7/5i