Which Subtraction Expression Has The Difference $1+4i$?A. $(-2+6i)-(1-2i$\] B. $(-2+6i)-(-1-2i$\] C. $(3+5i)-(2-i$\] D. $(3+5i)-(2+i$\]

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Introduction

In mathematics, subtraction is a fundamental operation that involves finding the difference between two numbers or expressions. When dealing with complex numbers, subtraction can be a bit more complex due to the presence of imaginary parts. In this article, we will explore which subtraction expression has the difference $1+4i$.

Understanding Complex Numbers

Before we dive into the problem, let's quickly 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 $b$. For example, $3+4i$ is a complex number with real part $3$ and imaginary part $4$.

Subtraction of Complex Numbers

To subtract two complex numbers, we simply subtract their real parts and imaginary parts separately. For example, to subtract $2+3i$ from $4+5i$, we get:

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

The Problem

Now, let's look at the problem at hand. We are given four subtraction expressions, and we need to determine which one has the difference $1+4i$. The expressions are:

A. $(-2+6i)-(1-2i)$ B. $(-2+6i)-(-1-2i)$ C. $(3+5i)-(2-i)$ D. $(3+5i)-(2+i)$

Analyzing Expression A

Let's start by analyzing expression A: $(-2+6i)-(1-2i)$. To evaluate this expression, we need to subtract the real parts and imaginary parts separately:

$(-2+6i)-(1-2i) = (-2-1)+(6i+2i) = -3+8i$

Analyzing Expression B

Next, let's analyze expression B: $(-2+6i)-(-1-2i)$. Again, we need to subtract the real parts and imaginary parts separately:

$(-2+6i)-(-1-2i) = (-2+1)+(6i-2i) = -1+4i$

Analyzing Expression C

Now, let's analyze expression C: $(3+5i)-(2-i)$. To evaluate this expression, we need to subtract the real parts and imaginary parts separately:

$(3+5i)-(2-i) = (3-2)+(5i+i) = 1+6i$

Analyzing Expression D

Finally, let's analyze expression D: $(3+5i)-(2+i)$. Again, we need to subtract the real parts and imaginary parts separately:

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

Conclusion

After analyzing all four expressions, we can see that only one of them has the difference $1+4i$. That expression is:

D. $(3+5i)-(2+i)$

Therefore, the correct answer is D.

Discussion

In this article, we explored which subtraction expression has the difference $1+4i$. We reviewed complex numbers and the process of subtracting them. We then analyzed each of the four given expressions and determined that only one of them has the difference $1+4i$. This problem requires a good understanding of complex numbers and the process of subtracting them. It also requires careful analysis and attention to detail.

Final Thoughts

In conclusion, subtraction of complex numbers is an important operation in mathematics. It requires a good understanding of complex numbers and the process of subtracting them. This problem is a good example of how to apply this operation to find the difference between two complex numbers. We hope that this article has provided a clear and concise explanation of the problem and its solution.

Introduction

In our previous article, we explored which subtraction expression has the difference $1+4i$. We reviewed complex numbers and the process of subtracting them. In this article, we will answer some frequently asked questions (FAQs) on subtraction of complex numbers.

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

A: The main difference between subtracting complex numbers and subtracting real numbers is that complex numbers have both real and imaginary parts. When subtracting complex numbers, we need to subtract both the real parts and the imaginary parts separately.

Q: How do I subtract complex numbers with negative real parts?

A: When subtracting complex numbers with negative real parts, we need to remember to change the sign of the real part when subtracting. For example, to subtract $-2+3i$ from $4+5i$, we get:

$(4+5i)-(-2+3i) = (4+2)+(5i-3i) = 6+2i$

Q: Can I subtract complex numbers with different imaginary parts?

A: Yes, you can subtract complex numbers with different imaginary parts. When subtracting complex numbers with different imaginary parts, we need to subtract the imaginary parts separately. For example, to subtract $2+4i$ from $3+5i$, we get:

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

Q: How do I subtract complex numbers with zero imaginary parts?

A: When subtracting complex numbers with zero imaginary parts, we only need to subtract the real parts. For example, to subtract $2+0i$ from $4+0i$, we get:

$(4+0i)-(2+0i) = (4-2)+(0i-0i) = 2+0i$

Q: Can I subtract complex numbers with negative imaginary parts?

A: Yes, you can subtract complex numbers with negative imaginary parts. When subtracting complex numbers with negative imaginary parts, we need to remember to change the sign of the imaginary part when subtracting. For example, to subtract $-2-3i$ from $4+5i$, we get:

$(4+5i)-(-2-3i) = (4+2)+(5i+3i) = 6+8i$

Q: How do I subtract complex numbers with different magnitudes?

A: When subtracting complex numbers with different magnitudes, we need to remember that the magnitude of a complex number is the square root of the sum of the squares of its real and imaginary parts. For example, to subtract $3+4i$ from $2+5i$, we get:

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

Q: Can I subtract complex numbers with zero real parts?

A: Yes, you can subtract complex numbers with zero real parts. When subtracting complex numbers with zero real parts, we only need to subtract the imaginary parts. For example, to subtract $0+3i$ from $0+4i$, we get:

$(0+4i)-(0+3i) = (0-0)+(4i-3i) = 0+i$

Conclusion

In this article, we answered some frequently asked questions (FAQs) on subtraction of complex numbers. We reviewed the process of subtracting complex numbers and provided examples to illustrate the concepts. We hope that this article has provided a clear and concise explanation of the FAQs on subtraction of complex numbers.

Final Thoughts

In conclusion, subtraction of complex numbers is an important operation in mathematics. It requires a good understanding of complex numbers and the process of subtracting them. We hope that this article has provided a helpful resource for students and teachers who are learning about complex numbers and their operations.