Add The Following Complex Numbers:\[$(3-9i) + (6-2i)\$\]A. \[$9-11i\$\] B. \[$9-7i\$\] C. \[$-3-11i\$\] D. \[$-3-7i\$\]

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Introduction

Complex numbers are mathematical expressions that consist of a real part and an imaginary part. They are used to represent points in a two-dimensional plane, where the real part is the x-coordinate and the imaginary part is the y-coordinate. In this article, we will focus on adding complex numbers, which is a fundamental operation in mathematics.

What are Complex Numbers?

A complex number is a number that can be expressed in the form a + bi, where a is the real part and bi is the imaginary part. The imaginary part is denoted by the letter i, which is defined as the square root of -1. Complex numbers can be represented graphically on a complex plane, where the real part is the x-coordinate and the imaginary part is the y-coordinate.

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 carry over any negative signs.

Example: Adding Complex Numbers

Let's consider the complex numbers (3 - 9i) and (6 - 2i). To add these numbers, we simply add the real parts and the imaginary parts separately.

(3 - 9i) + (6 - 2i) = ?

Step 1: Add the Real Parts

The real part of the first complex number is 3, and the real part of the second complex number is 6. We add these two numbers together:

3 + 6 = 9

Step 2: Add the Imaginary Parts

The imaginary part of the first complex number is -9i, and the imaginary part of the second complex number is -2i. We add these two numbers together:

-9i + (-2i) = -11i

Step 3: Combine the Real and Imaginary Parts

Now that we have added the real and imaginary parts separately, we can combine them to get the final result:

(3 - 9i) + (6 - 2i) = 9 - 11i

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 carry over any negative signs. We have also seen that the real and imaginary parts are added separately, and then combined to get the final result.

Common Mistakes to Avoid

When adding complex numbers, it is easy to make mistakes. Here are some common mistakes to avoid:

  • Forgetting to carry over negative signs: When adding complex numbers, it is easy to forget to carry over negative signs. Make sure to carry over any negative signs when adding the real and imaginary parts.
  • Adding the real and imaginary parts incorrectly: When adding the real and imaginary parts, make sure to add them separately and then combine them to get the final result.
  • Not using the correct notation: When writing complex numbers, make sure to use the correct notation. The real part should be written first, followed by the imaginary part.

Practice Problems

Here are some practice problems to help you practice adding complex numbers:

  1. Add the complex numbers (4 + 2i) and (3 - 5i).
  2. Add the complex numbers (2 - 3i) and (1 + 4i).
  3. Add the complex numbers (5 + i) and (2 - 3i).

Answer Key

Here are the answers to the practice problems:

  1. (4 + 2i) + (3 - 5i) = 7 - 3i
  2. (2 - 3i) + (1 + 4i) = 3 + i
  3. (5 + i) + (2 - 3i) = 7 - 2i

Conclusion

Introduction

In our previous article, we discussed how to add complex numbers. In this article, we will answer some frequently asked questions about complex number addition.

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 denoted by the letter i. When adding complex numbers, we need to remember to carry over any negative signs and add the real and imaginary parts separately.

Q: How do I add complex numbers with different signs?

A: When adding complex numbers with different signs, you need to remember to carry over the negative sign. For example, if you are adding (3 - 9i) and (-6 + 2i), you would add the real parts and the imaginary parts separately, like this:

(3 - 9i) + (-6 + 2i) = ?

Step 1: Add the Real Parts

The real part of the first complex number is 3, and the real part of the second complex number is -6. We add these two numbers together:

3 + (-6) = -3

Step 2: Add the Imaginary Parts

The imaginary part of the first complex number is -9i, and the imaginary part of the second complex number is 2i. We add these two numbers together:

-9i + 2i = -7i

Step 3: Combine the Real and Imaginary Parts

Now that we have added the real and imaginary parts separately, we can combine them to get the final result:

(3 - 9i) + (-6 + 2i) = -3 - 7i

Q: How do I add complex numbers with the same sign?

A: When adding complex numbers with the same sign, you can simply add the real parts and the imaginary parts separately. For example, if you are adding (3 + 9i) and (6 + 2i), you would add the real parts and the imaginary parts separately, like this:

(3 + 9i) + (6 + 2i) = ?

Step 1: Add the Real Parts

The real part of the first complex number is 3, and the real part of the second complex number is 6. We add these two numbers together:

3 + 6 = 9

Step 2: Add the Imaginary Parts

The imaginary part of the first complex number is 9i, and the imaginary part of the second complex number is 2i. We add these two numbers together:

9i + 2i = 11i

Step 3: Combine the Real and Imaginary Parts

Now that we have added the real and imaginary parts separately, we can combine them to get the final result:

(3 + 9i) + (6 + 2i) = 9 + 11i

Q: Can I add complex numbers with zero?

A: Yes, you can add complex numbers with zero. When adding a complex number to zero, the result is the original complex number. For example, if you are adding (3 + 9i) to zero, the result is (3 + 9i).

Q: Can I add complex numbers with a negative real part?

A: Yes, you can add complex numbers with a negative real part. When adding complex numbers with a negative real part, you need to remember to carry over the negative sign. For example, if you are adding (-3 - 9i) and (6 + 2i), you would add the real parts and the imaginary parts separately, like this:

(-3 - 9i) + (6 + 2i) = ?

Step 1: Add the Real Parts

The real part of the first complex number is -3, and the real part of the second complex number is 6. We add these two numbers together:

-3 + 6 = 3

Step 2: Add the Imaginary Parts

The imaginary part of the first complex number is -9i, and the imaginary part of the second complex number is 2i. We add these two numbers together:

-9i + 2i = -7i

Step 3: Combine the Real and Imaginary Parts

Now that we have added the real and imaginary parts separately, we can combine them to get the final result:

(-3 - 9i) + (6 + 2i) = 3 - 7i

Conclusion

In this article, we have answered some frequently asked questions about complex number addition. We have seen that adding complex numbers is similar to adding real numbers, but we need to remember to carry over any negative signs and add the real and imaginary parts separately. With practice, you will become more comfortable adding complex numbers and will be able to solve more complex problems.