Multiply Complex Numbers: A Step-by-Step Guide To (8-3i)(4+7i)
In the realm of mathematics, complex numbers hold a significant position, extending the familiar realm of real numbers by incorporating an imaginary unit, denoted as 'i'. This imaginary unit is defined as the square root of -1, a concept that expands the horizons of mathematical operations and problem-solving. Among the various operations that can be performed on complex numbers, multiplication stands out as a fundamental technique with wide-ranging applications.
Understanding Complex Number Multiplication
When it comes to multiplying complex numbers, we delve into a process that combines both real and imaginary components, revealing the intricate interplay between these two facets of the mathematical world. To fully grasp the essence of complex number multiplication, let's first establish a clear understanding of what complex numbers are and how they are represented.
A complex number is generally expressed in the form a + bi, where 'a' represents the real part and 'b' represents the imaginary part. The 'i' in this expression is the imaginary unit, the cornerstone of complex numbers, defined as the square root of -1. This seemingly simple addition of an imaginary component opens up a realm of mathematical possibilities beyond the scope of real numbers alone.
Multiplying complex numbers involves a systematic approach that leverages the distributive property, a cornerstone of algebraic manipulations. The distributive property, in essence, allows us to break down complex multiplications into simpler, manageable steps. When multiplying two complex numbers, we meticulously multiply each term of the first complex number by each term of the second complex number, ensuring that we account for all possible combinations. This meticulous approach forms the bedrock of accurate complex number multiplication.
To illustrate this process, let's consider the multiplication of two generic complex numbers, (a + bi) and (c + di). Applying the distributive property, we systematically multiply each term: (a + bi)(c + di) = a(c + di) + bi(c + di). Expanding further, we get ac + adi + bci + bdi². Here, we encounter the term bdi², which involves the square of the imaginary unit 'i'. This is where the unique nature of complex numbers comes into play.
Since i is defined as the square root of -1, i² is equal to -1. Substituting this value into our expression, we get ac + adi + bci - bd. Now, we can group the real and imaginary terms together, resulting in (ac - bd) + (ad + bc)i. This final form represents the product of the two complex numbers, expressed in the standard complex number format of a real part (ac - bd) and an imaginary part (ad + bc).
This methodical approach to multiplying complex numbers ensures that we accurately account for both the real and imaginary components, leading to a comprehensive understanding of the resulting complex number. By mastering this technique, we unlock the ability to solve a wide range of mathematical problems involving complex numbers, paving the way for further exploration in this fascinating area of mathematics.
Step-by-Step Solution for (8 - 3i)(4 + 7i)
In this section, we will delve into the step-by-step solution for multiplying the complex numbers (8 - 3i) and (4 + 7i). This practical example will solidify your understanding of the principles we discussed earlier and equip you with the skills to tackle similar problems with confidence. Let's embark on this mathematical journey, breaking down the process into manageable steps and revealing the elegant solution.
Our mission is to find the product of (8 - 3i) and (4 + 7i). To achieve this, we will employ the distributive property, the cornerstone of complex number multiplication. This property allows us to systematically multiply each term of the first complex number by each term of the second complex number, ensuring that we account for all possible interactions. By following this methodical approach, we can unravel the intricacies of the multiplication and arrive at the correct answer.
Step 1: Apply the Distributive Property
The first step in our journey is to apply the distributive property. We multiply each term in the first complex number (8 - 3i) by each term in the second complex number (4 + 7i). This gives us:
(8 - 3i)(4 + 7i) = 8(4 + 7i) - 3i(4 + 7i)
Here, we have carefully distributed the terms, setting the stage for the next step in our calculation. By breaking down the multiplication into smaller parts, we make the process more manageable and reduce the risk of errors. This methodical approach is key to success in complex number arithmetic.
Step 2: Expand the Expression
Now that we have applied the distributive property, we move on to expanding the expression. This involves multiplying each term within the parentheses:
8(4 + 7i) - 3i(4 + 7i) = 32 + 56i - 12i - 21i²
In this step, we have meticulously multiplied each term, paying close attention to the signs and coefficients. The resulting expression now contains both real and imaginary terms, as well as a term involving i², the square of the imaginary unit. This term holds the key to simplifying our expression further, as we will see in the next step.
Step 3: Simplify using i² = -1
The heart of complex number manipulation lies in the understanding that i², the square of the imaginary unit, is equal to -1. This fundamental identity allows us to bridge the gap between real and imaginary numbers, paving the way for simplification. Let's apply this knowledge to our expression:
32 + 56i - 12i - 21i² = 32 + 56i - 12i - 21(-1)
By substituting i² with -1, we have transformed the term -21i² into +21, effectively converting an imaginary component into a real component. This seemingly small step has a profound impact on the overall simplification process, bringing us closer to the final answer.
Step 4: Combine Like Terms
The penultimate step in our journey involves combining like terms. This is a fundamental algebraic technique that allows us to group together real terms and imaginary terms, streamlining the expression and revealing its underlying structure:
32 + 56i - 12i + 21 = (32 + 21) + (56i - 12i)
By carefully grouping the real terms (32 and 21) and the imaginary terms (56i and -12i), we have set the stage for the final simplification. This step not only makes the expression more concise but also highlights the distinct real and imaginary components of the resulting complex number.
Step 5: Final Result
Finally, we arrive at the culmination of our efforts – the final result. By performing the addition and subtraction within the grouped terms, we obtain the simplified complex number:
(32 + 21) + (56i - 12i) = 53 + 44i
Therefore, the product of the complex numbers (8 - 3i) and (4 + 7i) is 53 + 44i. This final answer represents a complex number with a real part of 53 and an imaginary part of 44. By meticulously following the steps of distribution, expansion, simplification using i² = -1, and combining like terms, we have successfully navigated the intricacies of complex number multiplication and arrived at the solution.
Common Mistakes to Avoid
Navigating the realm of complex number multiplication can be a rewarding experience, but it's crucial to be aware of common pitfalls that can lead to errors. By understanding these potential missteps, you can fortify your approach and ensure accurate results. Let's delve into some of the most frequent mistakes encountered when multiplying complex numbers and equip you with the knowledge to avoid them.
Forgetting to Distribute Properly
The distributive property serves as the cornerstone of complex number multiplication, acting as the roadmap for accurately multiplying each term. Neglecting to distribute properly can lead to a cascade of errors, compromising the entire solution. Remember, each term in the first complex number must be multiplied by each term in the second complex number. For instance, in the example (8 - 3i)(4 + 7i), forgetting to multiply the -3i by the 4 or the 8 by the 7i would result in an incomplete and incorrect answer. To avoid this, meticulously apply the distributive property, ensuring that every term interacts with every other term.
Incorrectly Simplifying i²
The imaginary unit 'i' and its square, i², hold a unique position in complex number arithmetic. The fundamental identity i² = -1 is the key to simplifying expressions and bridging the gap between real and imaginary components. Mishandling this identity can lead to significant errors. For example, incorrectly simplifying -21i² as -21 instead of -21(-1) = 21 would disrupt the entire calculation. Always remember that i² is -1, and substitute it accordingly during simplification.
Combining Real and Imaginary Terms Prematurely
Complex numbers are composed of two distinct parts: the real part and the imaginary part. Maintaining the separation between these components is crucial for accurate calculations. Prematurely combining real and imaginary terms can lead to confusion and errors. For instance, attempting to add 32 and 56i before simplifying the i² term in the expression 32 + 56i - 12i - 21i² would be a misstep. Ensure that you simplify the expression completely, particularly the i² terms, before combining like terms. This methodical approach will help you maintain clarity and avoid errors.
Sign Errors
Sign errors are a common nemesis in mathematical calculations, and complex number multiplication is no exception. A seemingly small sign error can propagate through the entire solution, leading to an incorrect final answer. Pay meticulous attention to the signs of each term throughout the multiplication and simplification process. For example, when expanding -3i(4 + 7i), ensure that you correctly distribute the negative sign to both terms, resulting in -12i - 21i². A simple oversight, such as writing +21i² instead of -21i², can derail the entire solution. Double-check your signs at each step to minimize the risk of errors.
Forgetting the Order of Operations
The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), is a fundamental principle in mathematics. Adhering to the correct order of operations is essential for accurate calculations in complex number multiplication. For instance, in the expression 32 + 56i - 12i - 21i², you must simplify the i² term before combining like terms. Neglecting this order and attempting to add 56i and -12i before simplifying -21i² would violate the order of operations and lead to an incorrect result. Always prioritize operations according to PEMDAS to ensure accurate calculations.
By being mindful of these common mistakes and adopting a meticulous approach, you can navigate the intricacies of complex number multiplication with confidence. Remember to distribute carefully, simplify i² correctly, maintain the separation between real and imaginary terms, pay close attention to signs, and adhere to the order of operations. With these strategies in your arsenal, you'll be well-equipped to conquer complex number multiplication and achieve accurate results.
Practice Problems
To solidify your understanding of complex number multiplication, working through practice problems is essential. These exercises provide an opportunity to apply the concepts and techniques we've discussed, reinforcing your skills and building confidence. Let's delve into a series of practice problems that will challenge your abilities and enhance your mastery of complex number multiplication.
Problem 1: Multiply (2 + 5i) by (3 - 2i).
Solution:
(2 + 5i)(3 - 2i) = 2(3 - 2i) + 5i(3 - 2i)
= 6 - 4i + 15i - 10i²
= 6 - 4i + 15i - 10(-1)
= 6 - 4i + 15i + 10
= (6 + 10) + (-4i + 15i)
= 16 + 11i
Therefore, the product of (2 + 5i) and (3 - 2i) is 16 + 11i.
Problem 2: Multiply (-1 - i) by (4 + 3i).
Solution:
(-1 - i)(4 + 3i) = -1(4 + 3i) - i(4 + 3i)
= -4 - 3i - 4i - 3i²
= -4 - 3i - 4i - 3(-1)
= -4 - 3i - 4i + 3
= (-4 + 3) + (-3i - 4i)
= -1 - 7i
Thus, the product of (-1 - i) and (4 + 3i) is -1 - 7i.
Problem 3: Multiply (7 - i) by (7 + i).
Solution:
(7 - i)(7 + i) = 7(7 + i) - i(7 + i)
= 49 + 7i - 7i - i²
= 49 + 7i - 7i - (-1)
= 49 + 7i - 7i + 1
= (49 + 1) + (7i - 7i)
= 50 + 0i
= 50
Therefore, the product of (7 - i) and (7 + i) is 50.
Problem 4: Multiply (3 + 4i) by its complex conjugate.
Solution:
The complex conjugate of (3 + 4i) is (3 - 4i).
(3 + 4i)(3 - 4i) = 3(3 - 4i) + 4i(3 - 4i)
= 9 - 12i + 12i - 16i²
= 9 - 12i + 12i - 16(-1)
= 9 - 12i + 12i + 16
= (9 + 16) + (-12i + 12i)
= 25 + 0i
= 25
Hence, the product of (3 + 4i) and its complex conjugate is 25.
These practice problems provide a spectrum of scenarios, allowing you to hone your skills in multiplying complex numbers. Remember to apply the distributive property meticulously, simplify i² correctly, combine like terms carefully, and pay attention to signs. By working through these problems and similar exercises, you'll solidify your understanding of complex number multiplication and gain the confidence to tackle more complex mathematical challenges.
Conclusion
In this comprehensive guide, we have embarked on a journey to unravel the intricacies of multiplying complex numbers. We began by establishing a solid foundation, defining complex numbers and understanding their unique structure, which includes both real and imaginary components. We then delved into the heart of complex number multiplication, exploring the distributive property, the cornerstone of this operation, and how it enables us to systematically multiply complex numbers.
We meticulously walked through a step-by-step solution for multiplying (8 - 3i) by (4 + 7i), illustrating the practical application of the distributive property, the simplification process using i² = -1, and the combination of like terms. This detailed example provided a clear roadmap for tackling similar problems, empowering you to approach complex number multiplication with confidence.
Furthermore, we addressed common mistakes to avoid, shedding light on potential pitfalls such as forgetting to distribute properly, incorrectly simplifying i², combining real and imaginary terms prematurely, making sign errors, and neglecting the order of operations. By understanding these common errors, you can proactively safeguard your calculations and ensure accuracy.
To reinforce your learning and solidify your skills, we presented a series of practice problems, ranging from basic multiplications to scenarios involving complex conjugates. These exercises provided valuable opportunities to apply the concepts and techniques we discussed, enhancing your mastery of complex number multiplication.
Multiplying complex numbers is a fundamental operation in mathematics, with applications extending far beyond the classroom. It serves as a building block for more advanced concepts in algebra, calculus, and other areas of mathematics. By mastering complex number multiplication, you unlock a gateway to a deeper understanding of mathematical principles and their applications in various fields.
As you continue your mathematical journey, remember the key takeaways from this guide: apply the distributive property meticulously, simplify i² correctly, combine like terms carefully, pay close attention to signs, and adhere to the order of operations. With these principles as your compass, you'll navigate the world of complex numbers with confidence and skill.
So, embrace the challenge, practice diligently, and continue to explore the fascinating realm of complex numbers. Your newfound expertise in complex number multiplication will serve as a valuable asset in your mathematical endeavors, opening doors to new discoveries and insights.
Multiply the complex numbers (8 - 3i) and (4 + 7i).
Multiply Complex Numbers A Step-by-Step Guide to (8-3i)(4+7i)
