Simplify Complex Number Multiplication: (4i)(2i)

Welcome, math enthusiasts! Today, we're diving into the fascinating world of complex numbers, specifically focusing on how to perform multiplication. Our mission is to simplify the expression $(4i)(2i)$ and present the answer in the standard complex number form, which is $u + bi$, where '$u$' represents the real part and '$b$' represents the imaginary part. Complex numbers might seem a bit abstract at first, but they have incredibly important applications in fields like electrical engineering, quantum mechanics, and signal processing. Understanding how to manipulate them, including performing basic operations like multiplication, is a foundational skill that opens doors to more advanced mathematical concepts.

Let's break down the expression $(4i)(2i)$. You'll notice that both numbers are purely imaginary, meaning they only have an imaginary component and no real component (the real part is zero). The '$i$' in these expressions represents the imaginary unit, which is defined as the square root of -1 ($\sqrt{-1}$). This concept is what allows us to work with the square roots of negative numbers, a limitation in the realm of real numbers. When we multiply complex numbers, we follow the same rules of algebra as we do with real numbers, including the distributive property. So, let's get started with the calculation itself. We're going to meticulously go through each step to ensure clarity and accuracy, making sure you grasp the process behind simplifying this type of expression. Prepare to see how these seemingly simple terms combine to reveal a fundamental property of imaginary numbers when multiplied together. This exercise is not just about solving a problem; it's about building your confidence and understanding with complex number arithmetic, a skill that will serve you well in your mathematical journey. We'll be using the properties of exponents and the definition of '$i$' to arrive at our final answer, presented in the requested $u+bi$ format. Stay tuned as we unravel the mystery of $(4i)(2i)$!

Understanding the Building Blocks: The Imaginary Unit '$i$'

Before we tackle the multiplication of $(4i)(2i)$, it's crucial to have a solid grasp of the imaginary unit '$i$'. This is the cornerstone of all complex numbers. Remember, '$i$' is defined as the square root of -1, or $i = \sqrt{-1}$. A key property that arises from this definition is that when you square '$i$', you get -1: $i^2 = (\sqrt{-1})^2 = -1$. This property is absolutely fundamental to simplifying expressions involving imaginary numbers and is the secret sauce that allows us to convert results that initially look like they only contain '$i$' into the standard $u+bi$ form. Without understanding that $i^2 = -1$, multiplying complex numbers would be a bewildering process. It's this relationship that bridges the gap between the purely imaginary realm and the real number line, allowing for a unified number system that can solve a wider array of mathematical problems. Think of '$i$' as an extension of our number system, much like negative numbers extended the concept of counting positive quantities. It enables us to solve equations that were previously unsolvable, such as $x^2 + 1 = 0$. The solution to this equation is $x = \pm i$, demonstrating the necessity and power of '$i$'. When we encounter '$i$' in our expression $(4i)(2i)$, we're dealing with quantities that are 'multiples' of this fundamental imaginary unit. The coefficients, '4' and '2' in this case, are real numbers that scale the imaginary unit. So, $4i$ means 4 times the imaginary unit, and $2i$ means 2 times the imaginary unit. As we proceed to multiply these, we'll be leveraging the commutative and associative properties of multiplication, just like with regular numbers, but with the added twist of how '$i$' behaves, especially when squared. This foundational knowledge of '$i$' and its properties will make the subsequent steps of our calculation straightforward and intuitive.

Step-by-Step Calculation of $(4 i)(2 i)$

Now, let's get down to the business of multiplying $(4i)(2i)$. We'll approach this systematically, applying the rules of algebra and the properties of the imaginary unit we just discussed. First, we can use the commutative property of multiplication, which states that the order of multiplication does not affect the result (a * b = b * a). This means we can rearrange the terms: $(4i)(2i) = 4 \times i \times 2 \times i$. We can also use the associative property of multiplication, which allows us to group the factors in any way we like $((a \times b) \times c = a \times (b \times c))$. So, let's group the real coefficients together and the imaginary units together:

$(4 \times 2) \times (i \times i)$

This step is crucial because it separates the real numerical part from the imaginary part, making the calculation more manageable. We know how to multiply real numbers, and we know the special property of '$i$' when multiplied by itself. Let's perform the first part of the multiplication: the real coefficients.

$4 \times 2 = 8$

So, our expression now looks like this:

$8 \times (i \times i)$

Now, we turn our attention to the second part: $(i \times i)$. As we established earlier, '$i$' is the square root of -1, and when we multiply '$i$' by itself, we get $i^2$. And the most important property of $i^2$ is that it equals -1:

$i \times i = i^2 = -1$

This is the pivotal moment in simplifying this expression. We substitute -1 for $i^2$ in our ongoing calculation:

$8 \times (-1)$

Finally, we perform the last multiplication, which is straightforward:

$8 \times (-1) = -8$

So, the result of multiplying $(4i)(2i)$ is -8. We have successfully simplified the expression. However, the prompt requires us to write the answer in the form $u + bi$. Let's see how we can express -8 in this standard complex number format. Remember, '$u$' is the real part and '$b$' is the coefficient of the imaginary part. In our result, -8, the real part is clearly -8. What about the imaginary part? Since there is no '$i$' term explicitly written, it means the coefficient of '$i$' is zero. Therefore, we can write -8 as:

$-8 + 0i$

This representation perfectly fits the $u + bi$ form, where $u = -8$ and $b = 0$. This shows that -8 is a real number, which is a subset of complex numbers where the imaginary component is zero. This step-by-step process demonstrates how the fundamental properties of imaginary numbers, particularly $i^2 = -1$, are used to simplify expressions and present them in their standard forms. It highlights the elegance and consistency of the complex number system.

Expressing the Answer in the Standard Form $u+bi$

We've successfully performed the multiplication $(4i)(2i)$ and arrived at the result $-8$. The final, and essential, step is to ensure our answer is presented in the required standard complex number form, which is $u + bi$. This format is universally recognized and is crucial for consistency when working with complex numbers in various mathematical and scientific contexts. In this form, '$u$' denotes the real part of the complex number, and '$b$' denotes the coefficient of the imaginary part. The '$i$' signifies the imaginary unit.

Our calculated result is $-8$. To express this in the $u+bi$ form, we need to identify the real part and the imaginary part. In the number $-8$, the real part is unequivocally $-8$. This is the component that exists on the real number line. Now, we consider the imaginary part. Since the term $-8$ has no '$i$' appended to it, it implies that the coefficient of the imaginary unit is zero. In other words, the imaginary component contributes nothing to the value of the number. Therefore, we can write $-8$ as $-8 + 0i$. Here, we can clearly see that:

• The real part, $u$, is $-8$.
• The coefficient of the imaginary part, $b$, is $0$.

This representation, $-8 + 0i$, satisfies the $u + bi$ format precisely. It clearly separates the real and imaginary components, even when one of them is zero. This is important because, in complex number arithmetic, a result might sometimes be purely real (like in this case), purely imaginary (e.g., $5i$, which can be written as $0 + 5i$), or a combination of both.

Understanding this standard form is key. When you encounter a complex number problem, the instruction to write the answer in $u+bi$ form is a directive to ensure your final answer is fully simplified and explicitly shows both its real and imaginary components. For instance, if you were asked to add $(2+3i)$ and $(1-5i)$, you would combine the real parts $(2+1=3)$ and the imaginary parts $(3i-5i=-2i)$ to get $3-2i$. This result is already in the $u+bi$ form, with $u=3$ and $b=-2$. In our specific case of $(4i)(2i)$, the multiplication resulted in a value that is entirely real. By writing it as $-8 + 0i$, we are simply making this explicit, adhering to the conventions of complex number representation. This standard form is not just a stylistic choice; it's essential for operations like addition, subtraction, and multiplication where you need to align real parts with real parts and imaginary parts with imaginary parts. Thus, $-8 + 0i$ is the complete and correctly formatted answer to the problem $(4i)(2i)$.

Conclusion: Mastering Complex Number Multiplication

In conclusion, we have successfully navigated the process of multiplying complex numbers, specifically tackling the expression $(4i)(2i)$. By applying the fundamental rules of algebra and leveraging the unique properties of the imaginary unit '$i$', we arrived at the simplified result. The key insight was recognizing that $i^2 = -1$, which allowed us to convert the product of two imaginary numbers into a real number. We meticulously performed the steps: first, grouping the real coefficients $(4 \times 2 = 8)$, and then multiplying the imaginary units $(i \times i = i^2 = -1)$. The combination of these steps yielded $8 \times -1 = -8$. Crucially, we then expressed this real result in the standard complex number form $u + bi$. Since $-8$ is purely real, its imaginary component is zero, leading to the final answer of $-8 + 0i$. This adheres strictly to the requested format, with $u = -8$ and $b = 0$.

Mastering operations like this is fundamental to building a strong foundation in complex number theory. Complex numbers, while initially seeming abstract, are indispensable tools in various branches of science and engineering, from analyzing alternating current circuits to solving differential equations and processing signals. The ability to confidently perform addition, subtraction, multiplication, and division with complex numbers, and to express the results in their standard forms, opens up a wider range of problem-solving capabilities. Remember, the $i^2 = -1$ property is your most powerful ally when dealing with imaginary and complex number multiplications. Keep practicing these operations, and you'll find yourself becoming more adept at handling even more complex mathematical challenges.

For further exploration and a deeper understanding of complex numbers and their applications, I highly recommend visiting resources like Khan Academy's section on complex numbers. They offer excellent tutorials and practice problems that can reinforce what we've discussed today. Additionally, exploring Wikipedia's page on complex numbers can provide a comprehensive overview of their history, properties, and extensive applications across various fields.