Select The Correct Answer.Write { (9+6i)-(1+3i)$}$ As A Complex Number In Standard Form.A. ${ 10+9i\$} B. ${ 8-3\$} C. ${ 8+3i\$} D. ${ 7-12i\$}

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Introduction

Complex numbers are an extension of the real number system, allowing us to represent and work with numbers that have both real and imaginary parts. In this article, we will focus on solving complex number operations, specifically the subtraction of two complex numbers. We will use the given problem as a case study to demonstrate the steps involved in solving complex number operations.

Understanding Complex Numbers

A complex number is a number 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 part without the imaginary unit, while the imaginary part is the part with the imaginary unit.

The Problem

The problem asks us to write the expression {(9+6i)-(1+3i)$}$ as a complex number in standard form. To solve this problem, we need to follow the order of operations (PEMDAS) and perform the subtraction of the two complex numbers.

Step 1: Distribute the Negative Sign

The first step in solving the problem is to distribute the negative sign to the second complex number. This means that we need to multiply the second complex number by -1.

{(9+6i)-(1+3i) = 9+6i-1-3i$}$

Step 2: Combine Like Terms

The next step is to combine like terms. In this case, we have two real parts (9 and -1) and two imaginary parts (6i and -3i). We can combine the real parts by adding them together, and the imaginary parts by adding them together.

${9+6i-1-3i = (9-1) + (6i-3i)\$}

Step 3: Simplify the Expression

Now that we have combined like terms, we can simplify the expression by performing the arithmetic operations.

{(9-1) + (6i-3i) = 8 + 3i$}$

Conclusion

In conclusion, the correct answer to the problem is ${8+3i\$}. This is the standard form of a complex number, where the real part is 8 and the imaginary part is 3i.

Answer Key

The correct answer is:

  • C. ${8+3i\$}

Discussion

This problem requires the student to understand the concept of complex numbers and how to perform operations on them. The student needs to be able to distribute the negative sign, combine like terms, and simplify the expression. This problem is a good example of how complex numbers can be used to represent and work with numbers that have both real and imaginary parts.

Tips and Tricks

  • When working with complex numbers, it is essential to follow the order of operations (PEMDAS).
  • When distributing the negative sign, make sure to multiply each term by -1.
  • When combining like terms, make sure to add or subtract the coefficients of the real and imaginary parts.
  • When simplifying the expression, make sure to perform the arithmetic operations.

Real-World Applications

Complex numbers have many real-world applications, including:

  • Electrical engineering: Complex numbers are used to represent AC circuits and analyze their behavior.
  • Signal processing: Complex numbers are used to represent and analyze signals in various fields, including audio and image processing.
  • Control systems: Complex numbers are used to analyze and design control systems.

Conclusion

Introduction

Complex numbers are a fundamental concept in mathematics, and they have many real-world applications. However, they can be challenging to understand and work with, especially for beginners. In this article, we will answer some frequently asked questions about complex numbers, covering topics such as their definition, operations, and applications.

Q: What is a complex number?

A: A complex number is a number 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.

Q: What is the imaginary unit i?

A: The imaginary unit i is a mathematical concept that is used to extend the real number system. It is defined as the square root of -1, denoted by i = √(-1). This means that i^2 = -1.

Q: How do I add complex numbers?

A: To add complex numbers, you need to add the real parts and the imaginary parts separately. For example, if you have two complex numbers 3 + 4i and 2 + 5i, you can add them as follows:

(3 + 4i) + (2 + 5i) = (3 + 2) + (4i + 5i) = 5 + 9i

Q: How do I subtract complex numbers?

A: To subtract complex numbers, you need to subtract the real parts and the imaginary parts separately. For example, if you have two complex numbers 3 + 4i and 2 + 5i, you can subtract them as follows:

(3 + 4i) - (2 + 5i) = (3 - 2) + (4i - 5i) = 1 - i

Q: How do I multiply complex numbers?

A: To multiply complex numbers, you need to use the distributive property and the fact that i^2 = -1. For example, if you have two complex numbers 3 + 4i and 2 + 5i, you can multiply them as follows:

(3 + 4i) × (2 + 5i) = (3 × 2) + (3 × 5i) + (4i × 2) + (4i × 5i) = 6 + 15i + 8i + 20i^2 = 6 + 23i - 20 = -14 + 23i

Q: How do I divide complex numbers?

A: To divide complex numbers, you need to multiply the numerator and the denominator by the conjugate of the denominator. For example, if you have two complex numbers 3 + 4i and 2 + 5i, you can divide them as follows:

(3 + 4i) ÷ (2 + 5i) = (3 + 4i) × (2 - 5i) ÷ (2 + 5i) × (2 - 5i) = ((3 × 2) + (3 × -5i) + (4i × 2) + (4i × -5i)) ÷ ((2 × 2) + (2 × -5i) + (5i × 2) + (5i × -5i)) = (6 - 15i + 8i - 20i^2) ÷ (4 - 10i + 10i - 25i^2) = (6 - 7i + 20) ÷ (4 + 25) = 26 - 7i ÷ 29 = (26 ÷ 29) - (7i ÷ 29) = 26/29 - 7i/29

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

A: Complex numbers have many real-world applications, including:

  • Electrical engineering: Complex numbers are used to represent AC circuits and analyze their behavior.
  • Signal processing: Complex numbers are used to represent and analyze signals in various fields, including audio and image processing.
  • Control systems: Complex numbers are used to analyze and design control systems.
  • Quantum mechanics: Complex numbers are used to describe the behavior of particles in quantum mechanics.

Conclusion

In conclusion, complex numbers are a fundamental concept in mathematics, and they have many real-world applications. This article has answered some frequently asked questions about complex numbers, covering topics such as their definition, operations, and applications. We hope that this article has been helpful in understanding complex numbers and their uses.