What Is The Quotient Of The Complex Numbers Below? ( 8 − 6 I ) ÷ ( 3 − I (8-6i) \div (3-i ( 8 − 6 I ) ÷ ( 3 − I ]A. 3 − I 3-i 3 − I B. 2 − I 2-i 2 − I C. 2 + I 2+i 2 + I D. 3 + I 3+i 3 + I

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Introduction to Complex Numbers

Complex numbers are mathematical expressions that consist of a real number and an imaginary number. They are used to represent points in a two-dimensional plane, where the real number is the x-coordinate and the imaginary number is the y-coordinate. Complex numbers are denoted by the letter 'z' and can be written in the form z = a + bi, where 'a' is the real part and 'b' is the imaginary part.

The Quotient of Complex Numbers

The quotient of two complex numbers is the result of dividing one complex number by another. To divide two complex numbers, we multiply the numerator and denominator by the conjugate of the denominator. The conjugate of a complex number is obtained by changing the sign of the imaginary part.

The Problem

We are given the complex numbers (8-6i) and (3-i). We need to find the quotient of these two complex numbers.

Step 1: Multiply the Numerator and Denominator by the Conjugate of the Denominator

To divide the complex numbers (8-6i) and (3-i), we multiply the numerator and denominator by the conjugate of the denominator, which is (3+i).

Step 2: Simplify the Expression

We multiply the numerator and denominator by (3+i):

(8-6i) / (3-i) = ((8-6i) * (3+i)) / ((3-i) * (3+i))

Step 3: Expand the Numerator and Denominator

We expand the numerator and denominator:

((8-6i) * (3+i)) = (24 + 8i - 18i - 6i^2) ((3-i) * (3+i)) = (9 - i^2)

Step 4: Simplify the Expression

We simplify the expression by using the fact that i^2 = -1:

(24 + 8i - 18i + 6) / (9 + 1) = (30 - 10i) / 10

Step 5: Simplify the Expression

We simplify the expression by dividing the numerator and denominator by 10:

(30 - 10i) / 10 = 3 - i

Conclusion

The quotient of the complex numbers (8-6i) and (3-i) is 3 - i.

Final Answer

The final answer is 3 - i.

Comparison with the Options

We compare the final answer with the options:

A. 3 - i B. 2 - i C. 2 + i D. 3 + i

The final answer matches option A.

Discussion

The quotient of complex numbers is an important concept in mathematics, particularly in algebra and geometry. It is used to solve equations and inequalities involving complex numbers. In this problem, we used the concept of conjugates to simplify the expression and find the quotient of the complex numbers.

Real-World Applications

The concept of complex numbers and their quotients has many real-world applications, such as:

  • Electrical engineering: Complex numbers are used to represent AC circuits and analyze their behavior.
  • Signal processing: Complex numbers are used to represent signals and analyze their frequency content.
  • Control systems: Complex numbers are used to analyze and design control systems.

Conclusion

In conclusion, the quotient of complex numbers is an important concept in mathematics that has many real-world applications. We used the concept of conjugates to simplify the expression and find the quotient of the complex numbers (8-6i) and (3-i). The final answer is 3 - i.

Introduction

Complex numbers are a fundamental concept in mathematics, and their quotients are used to solve equations and inequalities involving complex numbers. In this article, we will answer some frequently asked questions about complex numbers quotients.

Q: What is the quotient of two complex numbers?

A: The quotient of two complex numbers is the result of dividing one complex number by another. To divide two complex numbers, we multiply the numerator and denominator by the conjugate of the denominator.

Q: How do I find the conjugate of a complex number?

A: The conjugate of a complex number is obtained by changing the sign of the imaginary part. For example, the conjugate of 3 + 4i is 3 - 4i.

Q: What is the formula for dividing complex numbers?

A: The formula for dividing complex numbers is:

(z1 / z2) = (z1 * z2̄) / (z2 * z2̄)

where z1 and z2 are complex numbers, and z2̄ is the conjugate of z2.

Q: How do I simplify a complex number quotient?

A: To simplify a complex number quotient, we multiply the numerator and denominator by the conjugate of the denominator and then simplify the expression.

Q: What is the difference between a complex number and a real number?

A: A complex number is a number that has both a real part and an imaginary part, while a real number is a number that has only a real part.

Q: Can complex numbers be used to represent points in a two-dimensional plane?

A: Yes, complex numbers can be used to represent points in a two-dimensional plane, where the real part of the complex number is the x-coordinate and the imaginary part is the y-coordinate.

Q: What are some real-world applications of complex numbers and their quotients?

A: Complex numbers and their quotients have many real-world applications, such as:

  • Electrical engineering: Complex numbers are used to represent AC circuits and analyze their behavior.
  • Signal processing: Complex numbers are used to represent signals and analyze their frequency content.
  • Control systems: Complex numbers are used to analyze and design control systems.

Q: How do I know if a complex number quotient is in its simplest form?

A: A complex number quotient is in its simplest form if the numerator and denominator have no common factors.

Q: Can complex numbers be used to solve equations and inequalities?

A: Yes, complex numbers can be used to solve equations and inequalities involving complex numbers.

Q: What is the difference between a complex number quotient and a complex number product?

A: A complex number quotient is the result of dividing one complex number by another, while a complex number product is the result of multiplying two complex numbers.

Q: How do I find the quotient of a complex number and a real number?

A: To find the quotient of a complex number and a real number, we divide the complex number by the real number.

Q: Can complex numbers be used to represent points in a three-dimensional space?

A: Yes, complex numbers can be used to represent points in a three-dimensional space, where the real part of the complex number is the x-coordinate, the imaginary part is the y-coordinate, and the third dimension is represented by the magnitude of the complex number.

Conclusion

In conclusion, complex numbers and their quotients are an important concept in mathematics that has many real-world applications. We have answered some frequently asked questions about complex numbers quotients and provided examples and explanations to help you understand the concept.

Final Tips

  • Make sure to simplify complex number quotients by multiplying the numerator and denominator by the conjugate of the denominator.
  • Use the formula for dividing complex numbers to find the quotient of two complex numbers.
  • Practice solving equations and inequalities involving complex numbers to become proficient in using complex numbers and their quotients.

Resources

  • Khan Academy: Complex Numbers
  • MIT OpenCourseWare: Complex Analysis
  • Wolfram MathWorld: Complex Numbers

Further Reading

  • "Complex Analysis" by Serge Lang
  • "Complex Numbers and Geometry" by John H. Conway
  • "Complex Variables and Applications" by James W. Brown

Note: The resources and further reading list are provided for additional learning and reference.