Find The Product Of These Complex Numbers. { (2-3i)(1-i) =$}$A. ${ 5-5i\$} B. ${ 5+5i\$} C. { -1+5i$}$D. { -1-5i$}$
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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 mathematical and scientific applications. In this article, we will focus on finding the product of two 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 the part that is not multiplied by i, while the imaginary part is the part that is multiplied by i.
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 and then combine the results.
Example: Find the Product of (2-3i) and (1-i)
Let's use the distributive property to find the product of (2-3i) and (1-i).
Step 1: Multiply the Real Parts
First, we multiply the real parts of the two complex numbers:
2 × 1 = 2
Step 2: Multiply the Imaginary Parts
Next, we multiply the imaginary parts of the two complex numbers:
-3i × -i = 3
Step 3: Multiply the Real and Imaginary Parts
Now, we multiply the real part of the first complex number by the imaginary part of the second complex number and vice versa:
2 × -i = -2i -3i × 1 = -3i
Step 4: Combine the Results
Finally, we combine the results of the previous steps to find the product of the two complex numbers:
(2-3i)(1-i) = 2 + 3 - 2i - 3i = 5 - 5i
Conclusion
In this article, we have shown how to find the product of two complex numbers using the distributive property of multiplication over addition. We have also provided an example of how to find the product of (2-3i) and (1-i), which results in 5 - 5i.
Answer
The correct answer is A. 5 - 5i.
Tips and Tricks
- When multiplying complex numbers, make sure to use the distributive property to multiply each term in the first complex number by each term in the second complex number.
- When combining the results, make sure to add the real parts and the imaginary parts separately.
- Practice multiplying complex numbers to become more comfortable with the process.
Common Mistakes
- Forgetting to use the distributive property when multiplying complex numbers.
- Not adding the real parts and the imaginary parts separately when combining the results.
- Not checking the answer to make sure it is in the correct form (a + bi).
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 represent the behavior of control systems and analyze their stability.
Conclusion
In conclusion, finding the product of complex numbers is a fundamental operation in complex number arithmetic. By using the distributive property of multiplication over addition, we can find the product of two complex numbers. With practice, you can become more comfortable with this process and apply it to real-world problems.
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Introduction
Complex numbers are a fundamental concept in mathematics, and they have many real-world applications. In this article, we will answer some common questions about complex numbers, including their definition, properties, and operations.
Q: What is a complex number?
A: 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.
Q: What is the imaginary unit i?
A: The imaginary unit i is a mathematical concept that is used to extend the real numbers to the complex numbers. It is defined as the square root of -1, and it is denoted by the symbol i.
Q: What are the properties of complex numbers?
A: Complex numbers have several properties, including:
- Closure: The sum and product of two complex numbers are also complex numbers.
- Commutativity: The order of the complex numbers does not change the result of the operation.
- Associativity: The order in which we perform the operations does not change the result.
- Distributivity: The product of a complex number and a sum of two complex numbers is equal to the sum of the products of the complex number and each of the two complex numbers.
Q: How do we add complex numbers?
A: To add complex numbers, we simply add the real parts and the imaginary parts separately. For example, if we have two complex numbers 2 + 3i and 4 + 5i, their sum is (2 + 4) + (3 + 5)i = 6 + 8i.
Q: How do we multiply complex numbers?
A: To multiply complex numbers, we use the distributive property of multiplication over addition. We multiply each term in the first complex number by each term in the second complex number and then combine the results. For example, if we have two complex numbers 2 + 3i and 4 + 5i, their product is (2 × 4) + (2 × 5i) + (3i × 4) + (3i × 5i) = 8 + 10i + 12i - 15 = -7 + 22i.
Q: What is the conjugate of a complex number?
A: The conjugate of a complex number a + bi is a - bi. The conjugate of a complex number is used to simplify expressions and to find the absolute value of a complex number.
Q: How do we find the absolute value of a complex number?
A: To find the absolute value of a complex number, we use the formula |a + bi| = √(a^2 + b^2). This formula gives us the distance of the complex number from the origin in the complex plane.
Q: What are some common mistakes when working with complex numbers?
A: Some common mistakes when working with complex numbers include:
- Forgetting to use the distributive property when multiplying complex numbers.
- Not adding the real parts and the imaginary parts separately when combining the results.
- Not checking the answer to make sure it is in the correct form (a + bi).
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 represent the behavior of control systems and analyze their stability.
Conclusion
In conclusion, complex numbers are a fundamental concept in mathematics, and they have many real-world applications. By understanding the properties and operations of complex numbers, we can solve problems and analyze systems in a variety of fields.