Add The Following Complex Numbers: ( − 12 + 11 I ) + ( − 6 + 2 I (-12+11i) + (-6+2i ( − 12 + 11 I ) + ( − 6 + 2 I ]Give Your Answer In The Form A + B I A + Bi A + Bi .Provide Your Answer Below:

Introduction

Complex numbers are mathematical expressions that consist of a real number part and an imaginary number part. They are used to represent points in a two-dimensional plane and are essential in various fields, including algebra, geometry, and calculus. In this article, we will focus on adding complex numbers, which is a fundamental operation in complex number arithmetic.

What are 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 denoted by $a$, and the imaginary part is denoted by $b$. For example, the complex number $3 + 4i$ has a real part of $3$ and an imaginary part of $4$.

Adding Complex Numbers

To add complex numbers, we simply add the real parts and the imaginary parts separately. This is similar to adding real numbers, but we need to remember to keep the imaginary unit $i$ intact. Let's consider the example given in the problem: $(-12+11i) + (-6+2i)$.

Step-by-Step Solution

To add these complex numbers, we will follow these steps:

1. Add the real parts: We add the real parts of the two complex numbers, which are $-12$ and $-6$. This gives us $-12 + (-6) = -18$.
2. Add the imaginary parts: We add the imaginary parts of the two complex numbers, which are $11i$ and $2i$. This gives us $11i + 2i = 13i$.
3. Combine the real and imaginary parts: We combine the real part and the imaginary part to get the final result. This gives us $-18 + 13i$.

Conclusion

In this article, we have learned how to add complex numbers. We have seen that adding complex numbers is similar to adding real numbers, but we need to remember to keep the imaginary unit $i$ intact. We have also seen that the real part and the imaginary part are added separately. By following these steps, we can add complex numbers with ease.

Example Problems

Here are a few example problems to help you practice adding complex numbers:

• $(3+4i) + (2+5i)$
• $(-1+2i) + (3+4i)$
• $(5+6i) + (-2+3i)$

Tips and Tricks

Here are a few tips and tricks to help you add complex numbers:

• Make sure to add the real parts and the imaginary parts separately.
• Keep the imaginary unit $i$ intact.
• Use parentheses to group the real and imaginary parts.
• Simplify the expression by combining like terms.

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 signals and analyze their frequency content.
• Navigation: Complex numbers are used to represent positions and velocities in navigation systems.

Conclusion

Introduction

In our previous article, we discussed how to add complex numbers. However, we know that practice makes perfect, and there's no better way to practice than by answering questions. In this article, we will provide a Q&A guide on complex number addition, covering various scenarios and examples.

Q1: What is the sum of $(-3+4i) + (2-5i)$?

A1: To find the sum, we add the real parts and the imaginary parts separately. The real part is $-3 + 2 = -1$, and the imaginary part is $4i - 5i = -i$. Therefore, the sum is $-1 - i$.

Q2: What is the sum of $(5+2i) + (-3+4i)$?

A2: To find the sum, we add the real parts and the imaginary parts separately. The real part is $5 - 3 = 2$, and the imaginary part is $2i + 4i = 6i$. Therefore, the sum is $2 + 6i$.

Q3: What is the sum of $(-2+3i) + (1-2i)$?

A3: To find the sum, we add the real parts and the imaginary parts separately. The real part is $-2 + 1 = -1$, and the imaginary part is $3i - 2i = i$. Therefore, the sum is $-1 + i$.

Q4: What is the sum of $(4+5i) + (3-2i)$?

A4: To find the sum, we add the real parts and the imaginary parts separately. The real part is $4 + 3 = 7$, and the imaginary part is $5i - 2i = 3i$. Therefore, the sum is $7 + 3i$.

Q5: What is the sum of $(-1+2i) + (-3+4i)$?

A5: To find the sum, we add the real parts and the imaginary parts separately. The real part is $-1 - 3 = -4$, and the imaginary part is $2i + 4i = 6i$. Therefore, the sum is $-4 + 6i$.

Q6: What is the sum of $(2+3i) + (1+4i)$?

A6: To find the sum, we add the real parts and the imaginary parts separately. The real part is $2 + 1 = 3$, and the imaginary part is $3i + 4i = 7i$. Therefore, the sum is $3 + 7i$.

Q7: What is the sum of $(-4+5i) + (2-3i)$?

A7: To find the sum, we add the real parts and the imaginary parts separately. The real part is $-4 + 2 = -2$, and the imaginary part is $5i - 3i = 2i$. Therefore, the sum is $-2 + 2i$.

Q8: What is the sum of $(3+4i) + (5+6i)$?

A8: To find the sum, we add the real parts and the imaginary parts separately. The real part is $3 + 5 = 8$, and the imaginary part is $4i + 6i = 10i$. Therefore, the sum is $8 + 10i$.

Q9: What is the sum of $(-2+3i) + (-1+2i)$?

A9: To find the sum, we add the real parts and the imaginary parts separately. The real part is $-2 - 1 = -3$, and the imaginary part is $3i + 2i = 5i$. Therefore, the sum is $-3 + 5i$.

Q10: What is the sum of $(1+2i) + (3+4i)$?

A10: To find the sum, we add the real parts and the imaginary parts separately. The real part is $1 + 3 = 4$, and the imaginary part is $2i + 4i = 6i$. Therefore, the sum is $4 + 6i$.

Conclusion

In this Q&A guide, we have covered various scenarios and examples of complex number addition. We have seen that adding complex numbers is similar to adding real numbers, but we need to remember to keep the imaginary unit $i$ intact. By following the steps outlined in this guide, you can add complex numbers with ease and become proficient in this skill.