Which Of The Following Is The Product Of The Complex Numbers Below? { (3 - 2i)(5 + I)$}$A. ${ 17 + 13i\$} B. ${ 13 - 7i\$} C. ${ 17 - 7i\$} D. ${ 13 + 13i\$}
Complex Numbers: Understanding the Product of (3 - 2i)(5 + i)
Introduction to Complex Numbers
Complex numbers are mathematical expressions that consist of a real number and an imaginary number. They are used to represent points in a two-dimensional plane, known as the complex plane. The real number is denoted by the variable 'a', and the imaginary number is denoted by the variable 'bi', where 'b' is the coefficient of the imaginary unit 'i'. The imaginary unit 'i' is defined as the square root of -1.
Understanding the Product of Complex Numbers
To find the product of 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. The product of two complex numbers is also a complex number.
Step-by-Step Solution
To find the product of (3 - 2i)(5 + i), we can follow these steps:
- Multiply the first term in the first complex number (3) by each term in the second complex number (5 + i).
- Multiply the second term in the first complex number (-2i) by each term in the second complex number (5 + i).
- Combine the results of steps 1 and 2 to get the final product.
Step 1: Multiply 3 by 5 and 3 by i
To multiply 3 by 5, we get 15. To multiply 3 by i, we get 3i.
Step 2: Multiply -2i by 5 and -2i by i
To multiply -2i by 5, we get -10i. To multiply -2i by i, we get -2i^2. Since i^2 is equal to -1, we can simplify -2i^2 to 2.
Step 3: Combine the Results
Now, we can combine the results of steps 1 and 2 to get the final product:
(3 - 2i)(5 + i) = (3)(5) + (3)(i) + (-2i)(5) + (-2i)(i) = 15 + 3i - 10i + 2 = 17 - 7i
Conclusion
Therefore, the product of (3 - 2i)(5 + i) is 17 - 7i.
Answer
The correct answer is C. [17 - 7i]
Why is this Important?
Understanding the product of complex numbers is crucial in various mathematical applications, such as algebra, geometry, and calculus. It is also essential in physics, engineering, and computer science, where complex numbers are used to represent electrical circuits, signal processing, and quantum mechanics.
Real-World Applications
Complex numbers have numerous real-world applications, including:
- Electrical engineering: Complex numbers are used to represent electrical circuits, impedance, and admittance.
- Signal processing: Complex numbers are used to represent signals, filters, and Fourier transforms.
- Quantum mechanics: Complex numbers are used to represent wave functions, operators, and eigenvalues.
- Computer science: Complex numbers are used in algorithms, data structures, and numerical analysis.
Conclusion
In conclusion, the product of (3 - 2i)(5 + i) is 17 - 7i. Understanding complex numbers and their products is essential in various mathematical and real-world applications.
Complex Numbers: A Q&A Guide
Introduction
Complex numbers are a fundamental concept in mathematics, and they have numerous real-world applications. In our previous article, we discussed the product of complex numbers and how to find the product of (3 - 2i)(5 + i). In this article, we will answer some frequently asked questions about complex numbers.
Q: What is a complex number?
A: A complex number is a mathematical expression that consists of a real number and an imaginary number. It is denoted by the variable 'a + bi', where 'a' is the real part and 'b' is the imaginary part.
Q: What is the imaginary unit 'i'?
A: The imaginary unit 'i' is defined as the square root of -1. It is used to represent the imaginary part of a complex number.
Q: How do I add complex numbers?
A: To add complex numbers, we simply add the real parts and the imaginary parts separately. For example, (3 + 4i) + (2 + 5i) = (3 + 2) + (4i + 5i) = 5 + 9i.
Q: How do I subtract complex numbers?
A: To subtract complex numbers, we simply subtract the real parts and the imaginary parts separately. For example, (3 + 4i) - (2 + 5i) = (3 - 2) + (4i - 5i) = 1 - i.
Q: How do I multiply complex numbers?
A: To multiply 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. For example, (3 + 4i)(2 + 5i) = (3)(2) + (3)(5i) + (4i)(2) + (4i)(5i) = 6 + 15i + 8i + 20i^2. Since i^2 is equal to -1, we can simplify this to 6 + 23i - 20 = -14 + 23i.
Q: How do I divide complex numbers?
A: To divide complex numbers, we can use the conjugate of the denominator. The conjugate of a complex number is obtained by changing the sign of the imaginary part. For example, to divide (3 + 4i) by (2 + 5i), we can multiply the numerator and the denominator by the conjugate of the denominator, which is (2 - 5i). This gives us (3 + 4i)(2 - 5i) / (2 + 5i)(2 - 5i) = (6 - 15i + 8i - 20i^2) / (4 - 25i^2) = (6 - 7i + 20) / (4 + 25) = 26 - 7i / 29.
Q: What are some common complex numbers?
A: Some common complex numbers include:
- i: the imaginary unit
- -i: the negative of the imaginary unit
- 1 + i: a complex number with a real part of 1 and an imaginary part of 1
- 1 - i: a complex number with a real part of 1 and an imaginary part of -1
- i^2: the square of the imaginary unit, which is equal to -1
Q: What are some real-world applications of complex numbers?
A: Complex numbers have numerous real-world applications, including:
- Electrical engineering: Complex numbers are used to represent electrical circuits, impedance, and admittance.
- Signal processing: Complex numbers are used to represent signals, filters, and Fourier transforms.
- Quantum mechanics: Complex numbers are used to represent wave functions, operators, and eigenvalues.
- Computer science: Complex numbers are used in algorithms, data structures, and numerical analysis.
Conclusion
In conclusion, complex numbers are a fundamental concept in mathematics, and they have numerous real-world applications. We hope that this Q&A guide has helped to answer some of your questions about complex numbers.