Which Addition Expression Has The Sum $8 - 3i$?A. $(9 + 2i) + (1 - I)$B. \$(9 + 4i) + (-1 - 7i)$[/tex\]C. $(7 + 2i) + (1 - I)$D. $(7 + 4i) + (-1 - 7i)$

Introduction

In mathematics, addition is a fundamental operation that combines two or more numbers to produce a new number. When dealing with complex numbers, addition involves combining the real and imaginary parts of each number. In this article, we will explore which addition expression has the sum $8 - 3i$.

Understanding Complex Numbers

Before we dive into the addition expressions, let's briefly review 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 $i^2 = -1$. The real part of a complex number is $a$, and the imaginary part is $b$.

Addition of Complex Numbers

To add two complex numbers, we add their real parts and their imaginary parts separately. For example, if we want to add $2 + 3i$ and $4 - 2i$, we add the real parts $2 + 4 = 6$ and the imaginary parts $3i - 2i = i$. Therefore, the sum is $6 + i$.

Evaluating the Addition Expressions

Now, let's evaluate each of the addition expressions given in the problem.

Option A: $(9 + 2i) + (1 - i)$

To evaluate this expression, we add the real parts $9 + 1 = 10$ and the imaginary parts $2i - i = i$. Therefore, the sum is $10 + i$.

Option B: $(9 + 4i) + (-1 - 7i)$

To evaluate this expression, we add the real parts $9 - 1 = 8$ and the imaginary parts $4i - 7i = -3i$. Therefore, the sum is $8 - 3i$.

Option C: $(7 + 2i) + (1 - i)$

To evaluate this expression, we add the real parts $7 + 1 = 8$ and the imaginary parts $2i - i = i$. Therefore, the sum is $8 + i$.

Option D: $(7 + 4i) + (-1 - 7i)$

To evaluate this expression, we add the real parts $7 - 1 = 6$ and the imaginary parts $4i - 7i = -3i$. Therefore, the sum is $6 - 3i$.

Conclusion

Based on our evaluation of each addition expression, we can see that only one expression has the sum $8 - 3i$. Therefore, the correct answer is:

B. $(9 + 4i) + (-1 - 7i)$

This expression has the sum $8 - 3i$, which is the desired result.

Final Thoughts

In this article, we explored which addition expression has the sum $8 - 3i$. We reviewed complex numbers and the process of adding complex numbers. We then evaluated each of the addition expressions given in the problem and determined that only one expression has the desired sum. This article provides a clear and concise explanation of the process of adding complex numbers and how to evaluate addition expressions.

Introduction

In our previous article, we explored which addition expression has the sum $8 - 3i$. We reviewed complex numbers and the process of adding complex numbers. In this article, we will answer some frequently asked questions about complex numbers and addition.

Q&A

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 $i^2 = -1$.

Q: What is the real part of a complex number?

A: The real part of a complex number is the number that is multiplied by $1$, i.e., the number that is not multiplied by $i$. For example, in the complex number $3 + 4i$, the real part is $3$.

Q: What is the imaginary part of a complex number?

A: The imaginary part of a complex number is the number that is multiplied by $i$. For example, in the complex number $3 + 4i$, the imaginary part is $4i$.

Q: How do you add complex numbers?

A: To add complex numbers, you add their real parts and their imaginary parts separately. For example, if you want to add $2 + 3i$ and $4 - 2i$, you add the real parts $2 + 4 = 6$ and the imaginary parts $3i - 2i = i$. Therefore, the sum is $6 + i$.

Q: Can you give an example of adding complex numbers?

A: Yes, let's say you want to add $5 + 2i$ and $3 - 4i$. To do this, you add the real parts $5 + 3 = 8$ and the imaginary parts $2i - 4i = -2i$. Therefore, the sum is $8 - 2i$.

Q: What is the difference between adding complex numbers and adding real numbers?

A: The main difference between adding complex numbers and adding real numbers is that complex numbers have an imaginary part, which is multiplied by $i$. When adding complex numbers, you must add the real parts and the imaginary parts separately.

Q: Can you give an example of a complex number that is not in the form $a + bi$?

A: Yes, let's say you have the complex number $3i$. This is not in the form $a + bi$ because there is no real part. However, you can rewrite it as $0 + 3i$, where $0$ is the real part.

Q: What is the conjugate of a complex number?

A: The conjugate of a complex number is the complex number with the opposite sign of the imaginary part. For example, the conjugate of $3 + 4i$ is $3 - 4i$.

Q: Can you give an example of finding the conjugate of a complex number?

A: Yes, let's say you want to find the conjugate of $2 - 3i$. To do this, you change the sign of the imaginary part, which gives you $2 + 3i$.

Conclusion

In this article, we answered some frequently asked questions about complex numbers and addition. We reviewed the basics of complex numbers and the process of adding complex numbers. We also provided examples to help illustrate the concepts. This article provides a clear and concise explanation of complex numbers and addition.

Final Thoughts

Complex numbers and addition are fundamental concepts in mathematics. Understanding these concepts is essential for solving problems in algebra, geometry, and other areas of mathematics. We hope that this article has helped you to better understand complex numbers and addition.