Simplify: \[$(4 - 5i)(4 + I)\$\]
Introduction
In mathematics, the process of simplifying complex expressions is a crucial skill that helps us solve problems efficiently. One of the most common operations in algebra is multiplication, and when it comes to complex numbers, it can be a bit tricky. In this article, we will simplify the expression {(4 - 5i)(4 + i)$}$, which involves multiplying two complex numbers.
What are Complex Numbers?
Before we dive into the simplification process, let's quickly review what complex numbers are. 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 without the imaginary unit, while the imaginary part is the part with the imaginary unit.
Multiplying Complex Numbers
To multiply two complex numbers, we follow the same rules as multiplying two binomials. We multiply each term in the first complex number by each term in the second complex number and then combine like terms. Let's apply this rule to the given expression.
Step 1: Multiply the Real Parts
The first step is to multiply the real parts of the two complex numbers. In this case, we have 4 and 4, so we multiply them together: 4 * 4 = 16.
Step 2: Multiply the Imaginary Parts
Next, we multiply the imaginary parts of the two complex numbers. We have -5i and i, so we multiply them together: -5i * i = -5i^2.
Step 3: Simplify the Imaginary Part
Now, we simplify the imaginary part by substituting i^2 with -1: -5i^2 = -5(-1) = 5.
Step 4: Combine the Real and Imaginary Parts
Finally, we combine the real and imaginary parts by adding them together: 16 + 5i.
Conclusion
In conclusion, the simplified expression for {(4 - 5i)(4 + i)$}$ is 16 + 5i. This result can be verified by plugging the expression back into the original equation and simplifying it.
Tips and Tricks
Here are some tips and tricks to help you simplify complex expressions:
- Always start by multiplying the real parts together.
- Then, multiply the imaginary parts together.
- Simplify the imaginary part by substituting i^2 with -1.
- Finally, combine the real and imaginary parts by adding them together.
Practice Problems
Here are some practice problems to help you reinforce your skills:
- Simplify the expression {(3 + 4i)(2 - 3i)$}$.
- Simplify the expression {(2 - i)(3 + 2i)$}$.
- Simplify the expression {(1 + 2i)(1 - 2i)$}$.
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, simplifying complex expressions is an essential skill in mathematics that has many real-world applications. By following the steps outlined in this article, you can simplify complex expressions and solve problems efficiently. Remember to always start by multiplying the real parts together, then multiply the imaginary parts together, simplify the imaginary part, and finally combine the real and imaginary parts by adding them together. With practice and patience, you can become proficient in simplifying complex expressions and tackle even the most challenging problems.
Introduction
In our previous article, we simplified the expression {(4 - 5i)(4 + i)$}$ and provided some tips and tricks to help you simplify complex expressions. In this article, we will answer some frequently asked questions about simplifying complex expressions.
Q&A
Q: What is the difference between a real number and a complex number?
A: A real number is a number that can be expressed without any imaginary unit, such as 3, 4, or 5. A complex number, on the other hand, 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.
Q: How do I multiply two complex numbers?
A: To multiply two complex numbers, you follow the same rules as multiplying two binomials. You multiply each term in the first complex number by each term in the second complex number and then combine like terms.
Q: What is the imaginary unit, i?
A: The imaginary unit, i, is a mathematical concept that satisfies the equation i^2 = -1. It is used to represent the square root of -1.
Q: How do I simplify a complex expression?
A: To simplify a complex expression, you follow these steps:
- Multiply the real parts together.
- Multiply the imaginary parts together.
- Simplify the imaginary part by substituting i^2 with -1.
- Combine the real and imaginary parts by adding them together.
Q: What are some common mistakes to avoid when simplifying complex expressions?
A: Some common mistakes to avoid when simplifying complex expressions include:
- Not multiplying the real parts together first.
- Not multiplying the imaginary parts together first.
- Not simplifying the imaginary part correctly.
- Not combining the real and imaginary parts correctly.
Q: How do I know if a complex expression is simplified?
A: A complex expression is simplified when it is in the form a + bi, where a and b are real numbers, and i is the imaginary unit.
Q: Can I use a calculator to simplify complex expressions?
A: Yes, you can use a calculator to simplify complex expressions. However, it's always a good idea to double-check your work by hand to make sure you understand the process.
Q: What are some real-world applications of complex numbers?
A: 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, simplifying complex expressions is an essential skill in mathematics that has many real-world applications. By following the steps outlined in this article and avoiding common mistakes, you can simplify complex expressions and solve problems efficiently. Remember to always start by multiplying the real parts together, then multiply the imaginary parts together, simplify the imaginary part, and finally combine the real and imaginary parts by adding them together. With practice and patience, you can become proficient in simplifying complex expressions and tackle even the most challenging problems.
Additional Resources
- For more information on complex numbers, visit the Khan Academy website.
- For more practice problems, visit the Mathway website.
- For more real-world applications of complex numbers, visit the Wolfram Alpha website.
Final Tips
- Always start by multiplying the real parts together.
- Then, multiply the imaginary parts together.
- Simplify the imaginary part by substituting i^2 with -1.
- Finally, combine the real and imaginary parts by adding them together.
- Practice, practice, practice! The more you practice, the more comfortable you will become with simplifying complex expressions.