Add The Following Complex Numbers: \[$(8+12i) + (9-7i)\$\].Give Your Answer In Complex Standard Form \[$a + Bi\$\].Provide Your Answer Below:

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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 and are essential in various fields, including mathematics, physics, and engineering. 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 bi.

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 imaginary parts.

Example: Adding Complex Numbers

Let's consider the complex numbers (8 + 12i) and (9 - 7i). To add these numbers, we simply add the real parts and the imaginary parts separately.

Step 1: Add the Real Parts

The real part of the first complex number is 8, and the real part of the second complex number is 9. To add these numbers, we simply add them together:

8 + 9 = 17

Step 2: Add the Imaginary Parts

The imaginary part of the first complex number is 12i, and the imaginary part of the second complex number is -7i. To add these numbers, we simply add them together:

12i + (-7i) = 5i

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:

17 + 5i

Conclusion

In this article, we have learned how to add complex numbers by adding the real parts and the imaginary parts separately. We have also seen an example of adding complex numbers, (8 + 12i) and (9 - 7i), and have obtained the final result in complex standard form.

Tips and Tricks

  • When adding complex numbers, always remember to add the real parts and the imaginary parts separately.
  • Use the imaginary unit i to represent the imaginary part of a complex number.
  • When combining the real and imaginary parts, make sure to use the correct notation, a + bi.

Common Mistakes

  • Adding the real and imaginary parts together without separating them.
  • Forgetting to use the imaginary unit i to represent the imaginary part of a complex number.
  • Not using the correct notation, a + bi, when combining the real and imaginary parts.

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.
  • Control systems: Complex numbers are used to analyze the stability and performance of control systems.

Conclusion

Introduction

In our previous article, we discussed how to add complex numbers by adding the real parts and the imaginary parts separately. In this article, we will provide a Q&A guide to help you understand complex number addition better.

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 represented by the letter i. When adding complex numbers, we need to add the real parts and the imaginary parts separately.

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

A: When adding complex numbers with different signs, we need to remember to subtract the imaginary parts. For example, if we have the complex numbers (3 + 4i) and (2 - 5i), we would add the real parts and subtract the imaginary parts:

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

Q: Can I add complex numbers with zero imaginary part?

A: Yes, you can add complex numbers with zero imaginary part. For example, if we have the complex numbers (3 + 0i) and (2 + 0i), we would add the real parts:

(3 + 0i) + (2 + 0i) = 3 + 2 = 5

Q: How do I add complex numbers with negative real part?

A: When adding complex numbers with negative real part, we need to remember to subtract the real parts. For example, if we have the complex numbers (-3 + 4i) and (2 - 5i), we would subtract the real parts and add the imaginary parts:

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

Q: Can I add complex numbers with different magnitudes?

A: Yes, you can add complex numbers with different magnitudes. For example, if we have the complex numbers (3 + 4i) and (5 - 2i), we would add the real parts and the imaginary parts separately:

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

Q: How do I add complex numbers with complex conjugates?

A: When adding complex numbers with complex conjugates, we need to remember that the imaginary parts will cancel each other out. For example, if we have the complex numbers (3 + 4i) and (3 - 4i), we would add the real parts and subtract the imaginary parts:

(3 + 4i) + (3 - 4i) = (3 + 3) + (4i - 4i) = 6 + 0i = 6

Conclusion

In conclusion, adding complex numbers is a fundamental operation in complex number arithmetic. By following the steps outlined in this article and the Q&A guide, you should be able to add complex numbers with ease and obtain the final result in complex standard form.

Tips and Tricks

  • When adding complex numbers, always remember to add the real parts and the imaginary parts separately.
  • Use the imaginary unit i to represent the imaginary part of a complex number.
  • When combining the real and imaginary parts, make sure to use the correct notation, a + bi.

Common Mistakes

  • Adding the real and imaginary parts together without separating them.
  • Forgetting to use the imaginary unit i to represent the imaginary part of a complex number.
  • Not using the correct notation, a + bi, when combining the real and imaginary parts.

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.
  • Control systems: Complex numbers are used to analyze the stability and performance of control systems.

Conclusion

In conclusion, adding complex numbers is a fundamental operation in complex number arithmetic. By following the steps outlined in this article and the Q&A guide, you should be able to add complex numbers with ease and obtain the final result in complex standard form.