Multiply The Complex Numbers \[$(5 - 3i)\$\] And \[$(9 + 2i)\$\].A. \[$51 + 17i\$\]B. \[$51 - 17i\$\]C. \[$39 - 17i\$\]D. \[$-39 - 17i\$\]
Introduction
Complex numbers are a fundamental concept in mathematics, and they have numerous applications in various fields, including engineering, physics, and computer science. In this article, we will focus on multiplying complex numbers, which is a crucial operation in complex number arithmetic. We will use the given problem to illustrate the process of multiplying complex numbers.
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 denoted by a, and the imaginary part is denoted by bi.
The Problem
We are given two complex numbers: (5 - 3i) and (9 + 2i). Our task is to multiply these two complex numbers.
Multiplying Complex Numbers
To multiply two complex numbers, we can use the distributive property of multiplication over addition. We will multiply each term of the first complex number by each term of the second complex number and then combine the results.
Let's start by multiplying (5 - 3i) and (9 + 2i):
(5 - 3i) × (9 + 2i) = ?
Step 1: Multiply the Real Parts
We will multiply the real part of the first complex number (5) by the real part of the second complex number (9):
5 × 9 = 45
Step 2: Multiply the Imaginary Parts
We will multiply the imaginary part of the first complex number (-3i) by the real part of the second complex number (9):
-3i × 9 = -27i
Step 3: Multiply the Real Part by the Imaginary Part
We will multiply the real part of the first complex number (5) by the imaginary part of the second complex number (2i):
5 × 2i = 10i
Step 4: Multiply the Imaginary Part by the Imaginary Part
We will multiply the imaginary part of the first complex number (-3i) by the imaginary part of the second complex number (2i):
-3i × 2i = -6i^2
Since i^2 = -1, we can substitute this value into the expression:
-6i^2 = -6(-1) = 6
Step 5: Combine the Results
We will combine the results from the previous steps:
45 - 27i + 10i + 6
We can combine the real parts (45 and 6) and the imaginary parts (-27i and 10i):
45 + 6 = 51
-27i + 10i = -17i
Therefore, the product of (5 - 3i) and (9 + 2i) is:
51 - 17i
Conclusion
In this article, we have demonstrated how to multiply complex numbers using the distributive property of multiplication over addition. We have used the given problem to illustrate the process and have arrived at the correct solution: 51 - 17i.
Answer
The correct answer is:
A. 51 - 17i
Final Thoughts
Introduction
In our previous article, we demonstrated how to multiply complex numbers using the distributive property of multiplication over addition. In this article, we will provide a Q&A guide to help you understand the process of multiplying complex numbers.
Q: What is the formula for multiplying complex numbers?
A: The formula for multiplying complex numbers is:
(z1 × z2) = (a1 × a2) - (b1 × b2) + (a1 × b2)i - (b1 × a2)i
where z1 = a1 + b1i and z2 = a2 + b2i.
Q: How do I multiply complex numbers using the distributive property?
A: To multiply complex numbers using the distributive property, you need to multiply each term of the first complex number by each term of the second complex number and then combine the results.
Let's use the example (5 - 3i) and (9 + 2i) to illustrate the process:
(5 - 3i) × (9 + 2i) = ?
Step 1: Multiply the Real Parts
We will multiply the real part of the first complex number (5) by the real part of the second complex number (9):
5 × 9 = 45
Step 2: Multiply the Imaginary Parts
We will multiply the imaginary part of the first complex number (-3i) by the real part of the second complex number (9):
-3i × 9 = -27i
Step 3: Multiply the Real Part by the Imaginary Part
We will multiply the real part of the first complex number (5) by the imaginary part of the second complex number (2i):
5 × 2i = 10i
Step 4: Multiply the Imaginary Part by the Imaginary Part
We will multiply the imaginary part of the first complex number (-3i) by the imaginary part of the second complex number (2i):
-3i × 2i = -6i^2
Since i^2 = -1, we can substitute this value into the expression:
-6i^2 = -6(-1) = 6
Step 5: Combine the Results
We will combine the results from the previous steps:
45 - 27i + 10i + 6
We can combine the real parts (45 and 6) and the imaginary parts (-27i and 10i):
45 + 6 = 51
-27i + 10i = -17i
Therefore, the product of (5 - 3i) and (9 + 2i) is:
51 - 17i
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 of multiplication over addition
- Not combining the real and imaginary parts correctly
- Not using the correct formula for multiplying complex numbers
Q: How do I simplify complex numbers after multiplying them?
A: To simplify complex numbers after multiplying them, you need to combine the real and imaginary parts. You can do this by adding or subtracting the real parts and the imaginary parts separately.
For example, if you have the complex number 3 + 4i, you can simplify it by combining the real and imaginary parts:
3 + 4i = 3 + 4i
There is no need to simplify this complex number further.
Q: Can I multiply complex numbers with different magnitudes?
A: Yes, you can multiply complex numbers with different magnitudes. The magnitude of a complex number is the distance from the origin to the point in the complex plane.
For example, if you have the complex numbers 3 + 4i and 2 - 3i, you can multiply them:
(3 + 4i) × (2 - 3i) = ?
To multiply these complex numbers, you need to use the distributive property of multiplication over addition:
(3 + 4i) × (2 - 3i) = (3 × 2) - (3 × -3i) + (4i × 2) - (4i × -3i)
= 6 + 9i + 8i + 12i^2
Since i^2 = -1, we can substitute this value into the expression:
= 6 + 17i - 12
= -6 + 17i
Therefore, the product of (3 + 4i) and (2 - 3i) is:
-6 + 17i
Conclusion
In this article, we have provided a Q&A guide to help you understand the process of multiplying complex numbers. We have covered common mistakes to avoid, how to simplify complex numbers after multiplying them, and how to multiply complex numbers with different magnitudes. By following the steps outlined in this article, you can multiply complex numbers with ease and confidence.