Find The Quotient Of These Complex Numbers.$\[(3+3i) \div (5+4i) =\\]A. \[$3+\frac{1}{3} I\$\]B. \[$\frac{1}{3}+\frac{1}{3} I\$\]C. \[$\frac{27}{41}+\frac{3}{41} I\$\]D. \[$\frac{3}{41}+\frac{3}{41} I\$\]

by ADMIN 205 views

Introduction

Complex numbers are a fundamental concept in mathematics, and they have numerous applications in various fields, including engineering, physics, and computer science. When dealing with complex numbers, division is a crucial operation that requires a clear understanding of the underlying principles. In this article, we will explore the process of dividing complex numbers and provide a step-by-step guide on 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. Complex numbers can be represented graphically on a complex plane, with the real part a on the x-axis and the imaginary part b on the y-axis.

The Division of Complex Numbers

When dividing complex numbers, we need to follow a specific procedure to ensure that the result is accurate. The division of complex numbers involves multiplying the numerator and denominator by the conjugate of the denominator. The conjugate of a complex number a + bi is a - bi.

Step-by-Step Guide to Dividing Complex Numbers

To divide two complex numbers, we will follow these steps:

  1. Identify the numerator and denominator: The numerator is the complex number being divided, and the denominator is the complex number by which we are dividing.
  2. Find the conjugate of the denominator: The conjugate of the denominator is obtained by changing the sign of the imaginary part.
  3. Multiply the numerator and denominator by the conjugate of the denominator: This step is crucial in eliminating the imaginary part from the denominator.
  4. Simplify the expression: After multiplying the numerator and denominator by the conjugate of the denominator, we need to simplify the expression to obtain the final result.

Example: Dividing Complex Numbers

Let's consider the example of dividing the complex numbers (3 + 3i) and (5 + 4i). We will follow the steps outlined above to find the quotient.

Step 1: Identify the numerator and denominator

The numerator is 3 + 3i, and the denominator is 5 + 4i.

Step 2: Find the conjugate of the denominator

The conjugate of the denominator 5 + 4i is 5 - 4i.

Step 3: Multiply the numerator and denominator by the conjugate of the denominator

We multiply the numerator and denominator by the conjugate of the denominator:

(3 + 3i) / (5 + 4i) = (3 + 3i) / (5 + 4i) × (5 - 4i) / (5 - 4i)

Step 4: Simplify the expression

After multiplying the numerator and denominator by the conjugate of the denominator, we simplify the expression:

(3 + 3i) / (5 + 4i) = (3 + 3i)(5 - 4i) / (5 + 4i)(5 - 4i)

Expanding the numerator and denominator, we get:

(3 + 3i)(5 - 4i) = 15 - 12i + 15i - 12i^2 = 15 - 12i + 15i + 12 = 27 + 3i

(5 + 4i)(5 - 4i) = 25 - 20i + 20i - 16i^2 = 25 + 16 = 41

Therefore, the quotient of the complex numbers (3 + 3i) and (5 + 4i) is:

(3 + 3i) / (5 + 4i) = 27/41 + 3/41i

Conclusion

In this article, we have explored the process of dividing complex numbers and provided a step-by-step guide on how to find the quotient of two complex numbers. We have also considered an example of dividing the complex numbers (3 + 3i) and (5 + 4i) to illustrate the procedure. By following the steps outlined above, we can accurately divide complex numbers and obtain the final result.

Answer

The correct answer is:

C. 2741+341i\frac{27}{41}+\frac{3}{41} i

Introduction

Complex number division is a fundamental concept in mathematics, and it can be a bit tricky to understand at first. In this article, we will address some of the most frequently asked questions about complex number division, providing clear and concise answers to help you better understand this concept.

Q: What is the difference between complex number addition and complex number division?

A: Complex number addition and complex number division are two different operations. Complex number addition involves combining two or more complex numbers, while complex number division involves finding the quotient of two complex numbers.

Q: How do I divide complex numbers with a zero denominator?

A: If the denominator of a complex number is zero, then the complex number is undefined. In other words, you cannot divide by zero. However, if the numerator is also zero, then the complex number is equal to zero.

Q: Can I divide complex numbers with a negative denominator?

A: Yes, you can divide complex numbers with a negative denominator. When dividing complex numbers, the sign of the denominator does not affect the result.

Q: How do I simplify complex number division expressions?

A: To simplify complex number division expressions, you can use the following steps:

  1. Multiply the numerator and denominator by the conjugate of the denominator.
  2. Simplify the expression by combining like terms.
  3. Write the result in the form a + bi, where a and b are real numbers.

Q: Can I use complex number division to solve equations?

A: Yes, you can use complex number division to solve equations. Complex number division can be used to isolate a variable in an equation, making it easier to solve.

Q: How do I divide complex numbers with a complex denominator?

A: To divide complex numbers with a complex denominator, you can use the following steps:

  1. Multiply the numerator and denominator by the conjugate of the denominator.
  2. Simplify the expression by combining like terms.
  3. Write the result in the form a + bi, where a and b are real numbers.

Q: Can I use complex number division to find the inverse of a complex number?

A: Yes, you can use complex number division to find the inverse of a complex number. The inverse of a complex number a + bi is 1 / (a + bi).

Q: How do I divide complex numbers with a large or small denominator?

A: To divide complex numbers with a large or small denominator, you can use the following steps:

  1. Multiply the numerator and denominator by the conjugate of the denominator.
  2. Simplify the expression by combining like terms.
  3. Write the result in the form a + bi, where a and b are real numbers.

Q: Can I use complex number division to solve systems of equations?

A: Yes, you can use complex number division to solve systems of equations. Complex number division can be used to isolate a variable in a system of equations, making it easier to solve.

Conclusion

In this article, we have addressed some of the most frequently asked questions about complex number division, providing clear and concise answers to help you better understand this concept. Whether you are a student or a professional, complex number division is an essential tool that can be used to solve a wide range of mathematical problems.

Additional Resources

If you are looking for additional resources to help you learn more about complex number division, here are a few suggestions:

  • Textbooks: There are many excellent textbooks on complex analysis that cover complex number division in detail.
  • Online tutorials: There are many online tutorials and videos that provide step-by-step instructions on how to divide complex numbers.
  • Practice problems: Practice problems are an excellent way to reinforce your understanding of complex number division. Try solving a few practice problems to see how well you understand the concept.

Answer

The correct answers to the questions above are:

  1. Yes, complex number addition and complex number division are two different operations.
  2. No, you cannot divide by zero.
  3. Yes, you can divide complex numbers with a negative denominator.
  4. To simplify complex number division expressions, multiply the numerator and denominator by the conjugate of the denominator, simplify the expression by combining like terms, and write the result in the form a + bi, where a and b are real numbers.
  5. Yes, you can use complex number division to solve equations.
  6. To divide complex numbers with a complex denominator, multiply the numerator and denominator by the conjugate of the denominator, simplify the expression by combining like terms, and write the result in the form a + bi, where a and b are real numbers.
  7. Yes, you can use complex number division to find the inverse of a complex number.
  8. To divide complex numbers with a large or small denominator, multiply the numerator and denominator by the conjugate of the denominator, simplify the expression by combining like terms, and write the result in the form a + bi, where a and b are real numbers.
  9. Yes, you can use complex number division to solve systems of equations.