Calculate The Product: { (4 - 6i)(-2 + 5i)$}$

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Introduction

In mathematics, complex numbers are numbers 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. Complex numbers have numerous applications in various fields, including algebra, geometry, and calculus. In this article, we will focus on calculating the product of two complex numbers, (4 - 6i) and (-2 + 5i).

Understanding Complex Number Multiplication

To multiply two complex numbers, we can use the distributive property, which states that for any complex numbers a, b, and c, a(b + c) = ab + ac. We can also use the fact that i^2 = -1 to simplify the multiplication process.

Step 1: Multiply the Real Parts

To calculate the product of (4 - 6i) and (-2 + 5i), we first multiply the real parts of the two complex numbers. The real part of the first complex number is 4, and the real part of the second complex number is -2. Therefore, we multiply 4 and -2 to get -8.

Step 2: Multiply the Imaginary Parts

Next, we multiply the imaginary parts of the two complex numbers. The imaginary part of the first complex number is -6i, and the imaginary part of the second complex number is 5i. Therefore, we multiply -6i and 5i to get -30i^2.

Step 3: Simplify the Imaginary Part

Since i^2 = -1, we can simplify the imaginary part of the product by replacing i^2 with -1. Therefore, -30i^2 becomes -30(-1), which equals 30.

Step 4: Combine the Real and Imaginary Parts

Now that we have calculated the real and imaginary parts of the product, we can combine them to get the final result. The real part of the product is -8, and the imaginary part is 30. Therefore, the product of (4 - 6i) and (-2 + 5i) is -8 + 30i.

Conclusion

In this article, we have calculated the product of two complex numbers, (4 - 6i) and (-2 + 5i). We used the distributive property and the fact that i^2 = -1 to simplify the multiplication process. The final result is -8 + 30i.

Example Use Cases

Complex number multiplication has numerous applications in various fields, including algebra, geometry, and calculus. Here are a few example use cases:

  • Electrical Engineering: Complex numbers are used to represent AC circuits, where the voltage and current are represented as complex numbers.
  • Signal Processing: Complex numbers are used to represent signals in signal processing, where the signal is represented as a complex number.
  • Navigation: Complex numbers are used in navigation systems, where the position and velocity of an object are represented as complex numbers.

Tips and Tricks

Here are a few tips and tricks to help you with complex number multiplication:

  • Use the Distributive Property: The distributive property is a powerful tool for multiplying complex numbers. It allows you to multiply the real and imaginary parts of the two complex numbers separately.
  • Simplify the Imaginary Part: Since i^2 = -1, you can simplify the imaginary part of the product by replacing i^2 with -1.
  • Combine the Real and Imaginary Parts: Once you have calculated the real and imaginary parts of the product, you can combine them to get the final result.

Conclusion

In conclusion, complex number multiplication is a powerful tool for solving problems in algebra, geometry, and calculus. By using the distributive property and simplifying the imaginary part, you can calculate the product of two complex numbers easily. We hope this article has helped you understand complex number multiplication and how to apply it in various fields.

Frequently Asked Questions

Here are a few frequently asked questions about complex number multiplication:

  • What is the product of (3 + 4i) and (2 - 5i)?
    • To calculate the product of (3 + 4i) and (2 - 5i), we can use the distributive property and simplify the imaginary part. The final result is -11 + 23i.
  • What is the product of (1 - 2i) and (3 + 4i)?
    • To calculate the product of (1 - 2i) and (3 + 4i), we can use the distributive property and simplify the imaginary part. The final result is -5 + 10i.
  • What is the product of (2 + 3i) and (4 - 5i)?
    • To calculate the product of (2 + 3i) and (4 - 5i), we can use the distributive property and simplify the imaginary part. The final result is -14 + 17i.

Conclusion

In conclusion, complex number multiplication is a powerful tool for solving problems in algebra, geometry, and calculus. By using the distributive property and simplifying the imaginary part, you can calculate the product of two complex numbers easily. We hope this article has helped you understand complex number multiplication and how to apply it in various fields.

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Introduction

In our previous article, we discussed how to calculate the product of two complex numbers using the distributive property and simplifying the imaginary part. In this article, we will provide a Q&A guide to help you understand complex number multiplication better.

Q: What is the product of (3 + 4i) and (2 - 5i)?

A: To calculate the product of (3 + 4i) and (2 - 5i), we can use the distributive property and simplify the imaginary part. The final result is -11 + 23i.

Q: What is the product of (1 - 2i) and (3 + 4i)?

A: To calculate the product of (1 - 2i) and (3 + 4i), we can use the distributive property and simplify the imaginary part. The final result is -5 + 10i.

Q: What is the product of (2 + 3i) and (4 - 5i)?

A: To calculate the product of (2 + 3i) and (4 - 5i), we can use the distributive property and simplify the imaginary part. The final result is -14 + 17i.

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

A: To multiply complex numbers with different magnitudes, you can use the distributive property and simplify the imaginary part. For example, to calculate the product of (3 + 4i) and (2 - 5i), you can multiply the real and imaginary parts separately and then combine them.

Q: Can I use complex number multiplication to solve real-world problems?

A: Yes, complex number multiplication has numerous applications in various fields, including electrical engineering, signal processing, and navigation. By using complex number multiplication, you can solve problems that involve AC circuits, signals, and navigation systems.

Q: What are some common mistakes to avoid when multiplying complex numbers?

A: Some common mistakes to avoid when multiplying complex numbers include:

  • Not using the distributive property: Make sure to use the distributive property to multiply the real and imaginary parts of the two complex numbers.
  • Not simplifying the imaginary part: Make sure to simplify the imaginary part by replacing i^2 with -1.
  • Not combining the real and imaginary parts: Make sure to combine the real and imaginary parts to get the final result.

Q: Can I use complex number multiplication to solve problems with complex numbers in polar form?

A: Yes, you can use complex number multiplication to solve problems with complex numbers in polar form. To do this, you can convert the complex numbers from polar form to rectangular form and then multiply them using the distributive property.

Q: What are some real-world applications of complex number multiplication?

A: Some real-world applications of complex number multiplication include:

  • Electrical Engineering: Complex number multiplication is used to represent AC circuits, where the voltage and current are represented as complex numbers.
  • Signal Processing: Complex number multiplication is used to represent signals in signal processing, where the signal is represented as a complex number.
  • Navigation: Complex number multiplication is used in navigation systems, where the position and velocity of an object are represented as complex numbers.

Conclusion

In conclusion, complex number multiplication is a powerful tool for solving problems in algebra, geometry, and calculus. By using the distributive property and simplifying the imaginary part, you can calculate the product of two complex numbers easily. We hope this Q&A guide has helped you understand complex number multiplication better and how to apply it in various fields.

Frequently Asked Questions

Here are a few frequently asked questions about complex number multiplication:

  • What is the product of (5 + 6i) and (3 - 4i)?
    • To calculate the product of (5 + 6i) and (3 - 4i), we can use the distributive property and simplify the imaginary part. The final result is -7 + 22i.
  • What is the product of (2 - 3i) and (4 + 5i)?
    • To calculate the product of (2 - 3i) and (4 + 5i), we can use the distributive property and simplify the imaginary part. The final result is -14 + 17i.
  • What is the product of (1 + 2i) and (3 - 4i)?
    • To calculate the product of (1 + 2i) and (3 - 4i), we can use the distributive property and simplify the imaginary part. The final result is -5 + 10i.

Conclusion

In conclusion, complex number multiplication is a powerful tool for solving problems in algebra, geometry, and calculus. By using the distributive property and simplifying the imaginary part, you can calculate the product of two complex numbers easily. We hope this Q&A guide has helped you understand complex number multiplication better and how to apply it in various fields.