Multiply The Following Complex Numbers:\[$(5 + 3i)(4 + 2i)\$\]A. \[$26 + 22i\$\] B. \[$26 - 22i\$\] C. \[$14 - 22i\$\] D. \[$14 + 22i\$\]

Introduction

Complex numbers are a fundamental concept in mathematics, and they have numerous applications in various fields, including algebra, geometry, and calculus. In this article, we will focus on multiplying complex numbers, which is a crucial operation in complex number arithmetic. We will use the given example to demonstrate the process of multiplying complex numbers and provide a step-by-step guide on how to perform this operation.

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 the part that is not multiplied by the imaginary unit, while the imaginary part is the part that is multiplied by the imaginary unit.

Multiplying Complex Numbers

To multiply two complex numbers, we can use the distributive property of multiplication over addition. This means that we can multiply each term in the first complex number by each term in the second complex number and then combine the results.

Let's consider the given example: (5 + 3i)(4 + 2i). To multiply these two complex numbers, we can use the distributive property as follows:

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

Expanding the Expression

Now, let's expand the expression by multiplying each term in the first complex number by each term in the second complex number:

5(4 + 2i) = 20 + 10i 3i(4 + 2i) = 12i + 6i^2

Simplifying the Expression

Now, let's simplify the expression by combining like terms and using the fact that i^2 = -1:

20 + 10i + 12i + 6i^2 = 20 + 22i - 6

Final Result

Now, let's simplify the expression further by combining the real and imaginary parts:

20 - 6 + 22i = 14 + 22i

Therefore, the final result of multiplying the complex numbers (5 + 3i)(4 + 2i) is 14 + 22i.

Conclusion

In this article, we have demonstrated the process of multiplying complex numbers using the distributive property of multiplication over addition. We have also provided a step-by-step guide on how to perform this operation and have used the given example to illustrate the process. By following these steps, you can multiply complex numbers with ease and accuracy.

Common Mistakes to Avoid

When multiplying complex numbers, it's essential to avoid common mistakes, such as:

• Forgetting to use the distributive property of multiplication over addition
• Not combining like terms correctly
• Not using the fact that i^2 = -1

By avoiding these common mistakes, you can ensure that your calculations are accurate and reliable.

Real-World Applications

Multiplying complex numbers has numerous 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 represent the behavior of control systems and analyze their stability.

Practice Problems

To practice multiplying complex numbers, try the following problems:

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

By practicing these problems, you can develop your skills and become proficient in multiplying complex numbers.

Conclusion

Introduction

In our previous article, we discussed the process of multiplying complex numbers using the distributive property of multiplication over addition. We also provided a step-by-step guide on how to perform this operation and used the given example to illustrate the process. In this article, we will answer some frequently asked questions about multiplying complex numbers.

Q: What is the formula for multiplying complex numbers?

A: The formula for multiplying complex numbers is:

(z1 + w1)(z2 + w2) = (z1z2 - w1w2) + (z1w2 + z2w1)i

where z1 and z2 are the real parts of the complex numbers, and w1 and w2 are the imaginary parts.

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

A: When multiplying complex numbers with different signs, you need to remember that i^2 = -1. This means that if you have a complex number with a positive real part and a negative imaginary part, you need to multiply the real part by the negative imaginary part and vice versa.

Q: Can I multiply complex numbers with zero?

A: Yes, you can multiply complex numbers with zero. When you multiply a complex number by zero, the result is always zero.

Q: How do I multiply complex numbers with fractions?

A: When multiplying complex numbers with fractions, you need to multiply the numerators and denominators separately. For example, if you have the complex number (3/4 + 5/6)i, you need to multiply the numerators (3 and 5) and denominators (4 and 6) separately.

Q: Can I multiply complex numbers with decimals?

A: Yes, you can multiply complex numbers with decimals. When multiplying complex numbers with decimals, you need to multiply the decimal parts separately.

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

A: When multiplying complex numbers with negative real parts, you need to remember that i^2 = -1. This means that if you have a complex number with a negative real part and a positive imaginary part, you need to multiply the real part by the negative imaginary part and vice versa.

Q: Can I multiply complex numbers with complex conjugates?

A: Yes, you can multiply complex numbers with complex conjugates. When you multiply a complex number with its complex conjugate, the result is always a real number.

Q: How do I multiply complex numbers with complex numbers in polar form?

A: When multiplying complex numbers in polar form, you need to multiply the magnitudes and add the angles. For example, if you have the complex numbers z1 = 3(cosθ1 + isinθ1) and z2 = 4(cosθ2 + isinθ2), you need to multiply the magnitudes (3 and 4) and add the angles (θ1 and θ2).

Conclusion

In conclusion, multiplying complex numbers is a crucial operation in complex number arithmetic. By following the steps outlined in this article and answering the frequently asked questions, you can become proficient in multiplying complex numbers and apply this knowledge to real-world problems.

Practice Problems

To practice multiplying complex numbers, try the following problems:

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

By practicing these problems, you can develop your skills and become proficient in multiplying complex numbers.

Real-World Applications

Multiplying complex numbers has numerous 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 represent the behavior of control systems and analyze their stability.

Conclusion

In conclusion, multiplying complex numbers is a crucial operation in complex number arithmetic. By following the steps outlined in this article and answering the frequently asked questions, you can become proficient in multiplying complex numbers and apply this knowledge to real-world problems.