Solving Complex Equations A Step By Step Guide

Complex number multiplication can appear daunting at first, but understanding the core principles of commutativity and distributivity can unlock its secrets. In this detailed exploration, we will dissect the equation $(\sqrt{2} - i) - i(1 - \sqrt{2i}) = -2i$, revealing the step-by-step process of how to solve it using these fundamental properties. Our journey will involve manipulating complex numbers, simplifying expressions, and ultimately verifying the equality. By the end, you'll not only understand the solution but also grasp the underlying concepts of complex number arithmetic.

Breaking Down the Equation: A Step-by-Step Solution

Let's embark on a journey to dissect the complex equation $(\sqrt{2} - i) - i(1 - \sqrt{2i}) = -2i$. Our mission is to simplify and solve it, showcasing the power of commutative and distributive properties within the realm of complex numbers. The journey involves several key steps, each designed to unravel the complexity and bring clarity to the solution.

1. The Distributive Property: Unveiling Hidden Terms

At the heart of our equation lies the term $-i(1 - \sqrt{2i})$. This is where the distributive property comes into play, acting as our key to unlock hidden terms. The distributive property, a cornerstone of algebra, allows us to multiply a single term by each term within a set of parentheses. It is crucial for simplifying expressions and bringing them into a manageable form. In essence, the distributive property states that for any numbers a, b, and c, the equation a(b + c) = ab + ac holds true. This seemingly simple rule is a powerful tool, especially when dealing with complex numbers.

Applying this property, we multiply $-i$ by both $1$ and $-\sqrt{2i}$. This transforms the equation as follows:

$(\sqrt{2} - i) - i(1 - \sqrt{2i}) = (\sqrt{2} - i) - i + i\sqrt{2i}$

The application of the distributive property has effectively expanded our equation, revealing a new term, $i\sqrt{2i}$, which requires further simplification. This is a critical step as it sets the stage for subsequent operations. By distributing the $-i$, we've not only expanded the equation but also identified a new element that demands our attention. This is a common strategy in solving complex equations: breaking them down into smaller, more manageable parts. The distributive property serves as a powerful tool in this process, allowing us to systematically address each component of the equation.

2. Simplifying Radicals: Taming the Square Root of Complex Numbers

Our next challenge lies in simplifying the term $i\sqrt{2i}$. This involves dealing with the square root of a complex number, which requires a delicate touch. The square root of a complex number isn't as straightforward as the square root of a real number. To tackle this, we need to express $2i$ in its polar form. The polar form of a complex number provides a convenient way to represent complex numbers in terms of magnitude and angle, making operations like finding square roots more manageable.

The complex number $2i$ can be visualized as a point on the complex plane, located 2 units along the imaginary axis. This corresponds to a magnitude of 2 and an angle of $\frac{\pi}{2}$ radians (90 degrees). Therefore, the polar form of $2i$ is $2(\cos(\frac{\pi}{2}) + i\sin(\frac{\pi}{2}))$. Now, to find the square root, we utilize De Moivre's Theorem, a powerful tool for dealing with powers and roots of complex numbers.

De Moivre's Theorem states that for any complex number in polar form, $r(\cos(\theta) + i\sin(\theta))$, and any integer n, the following holds true:

$[r(\cos(\theta) + i\sin(\theta))]^n = r^n(\cos(n\theta) + i\sin(n\theta))$

In our case, we want to find the square root, which is equivalent to raising to the power of $\frac{1}{2}$. Applying De Moivre's Theorem, we get:

$\sqrt{2i} = [2(\cos(\frac{\pi}{2}) + i\sin(\frac{\pi}{2}))]^{\frac{1}{2}} = \sqrt{2}(\cos(\frac{\pi}{4}) + i\sin(\frac{\pi}{4}))$

Now we know that $\cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$ and $\sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$, so we can substitute these values:

$\sqrt{2i} = \sqrt{2}(\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}) = 1 + i$

This simplification is a major breakthrough. We've successfully transformed the square root of a complex number into a simple complex number, $1 + i$. This allows us to proceed with the equation with greater ease and clarity. The process of converting to polar form and applying De Moivre's Theorem highlights the elegance and power of these mathematical tools in simplifying complex expressions. This step underscores the importance of understanding the underlying principles of complex number representation and manipulation.

3. Substituting and Simplifying: Bringing it All Together

With $\sqrt{2i}$ simplified to $1 + i$, we can now substitute this value back into our equation. This substitution is a crucial step in piecing together the puzzle and moving closer to the final solution. It demonstrates the interconnectedness of the different parts of the equation and how simplifying one component can have a ripple effect, making the entire equation more tractable. Our equation now reads:

$(\sqrt{2} - i) - i + i\sqrt{2i} = (\sqrt{2} - i) - i + i(1 + i)$

Next, we apply the distributive property again to the term $i(1 + i)$. This reinforces the importance of this fundamental property in simplifying complex number expressions. Distributing the $i$, we get:

i(1 + i) = i + i^2

Recall that $i^2$ is defined as $-1$. This is a critical identity in complex number arithmetic and is essential for simplifying expressions involving the imaginary unit. Substituting $i^2$ with $-1$, we have:

i(1 + i) = i - 1

Now we substitute this back into our equation:

$(\sqrt{2} - i) - i + i(1 + i) = (\sqrt{2} - i) - i + i - 1$

At this point, we can combine like terms. The terms $-i$ and $+i$ cancel each other out, simplifying the equation further. This cancellation is a testament to the elegance of mathematical operations and how terms can interact to simplify expressions. Our equation now becomes:

$\sqrt{2} - i - i + i - 1 = \sqrt{2} - 1 - i$

This step highlights the power of careful substitution and simplification. By breaking down the equation into smaller parts and applying fundamental properties and identities, we've transformed a seemingly complex equation into a much simpler form. This streamlined equation is now ready for the final comparison with the right-hand side of our original equation.

4. Verification: Does it Match?

Our final step is to compare our simplified expression, $\sqrt{2} - 1 - i$, with the right-hand side of the original equation, which is $-2i$. This comparison is the moment of truth, where we determine if our step-by-step simplification has led us to the correct solution. It's a critical step in any mathematical problem-solving process, ensuring the accuracy and validity of our results.

We need to check if $\sqrt{2} - 1 - i$ is equal to $-2i$. A quick glance reveals that these two expressions are not equal. This means that the initial equation, $(\sqrt{2} - i) - i(1 - \sqrt{2i}) = -2i$, is not true.

This outcome is a valuable lesson in the importance of verification. Even with careful step-by-step simplification, it's crucial to verify the final result to ensure accuracy. The discrepancy between our simplified expression and the right-hand side of the equation indicates either an error in our calculations or that the original equation is simply false. In this case, it appears the original equation is incorrect.

It's important to emphasize that encountering a false equation is not a failure. It's an opportunity for learning and critical thinking. It encourages us to re-examine our steps, identify potential errors, and deepen our understanding of the underlying concepts. The process of verification is not just about confirming a solution; it's about strengthening our mathematical reasoning and problem-solving skills. It reinforces the importance of precision and attention to detail in mathematical calculations.

Commutative and Distributive Properties in Action

Throughout our solution, we've prominently used the distributive property. Let's take a moment to appreciate how the commutative property plays a role in complex number multiplication as well. The commutative property states that the order of multiplication doesn't affect the result. In other words, for any complex numbers u and v, uv = vu. This property might seem obvious, but it's fundamental to how we manipulate complex numbers.

For example, if we had encountered a term like $(\sqrt{2} - i)i$, we could rewrite it as $i(\sqrt{2} - i)$ thanks to the commutative property. This allows us to then apply the distributive property, simplifying the expression. While not directly used in the main steps of this particular equation's solution, the commutative property is an ever-present principle in complex number arithmetic, enabling us to rearrange terms and simplify expressions more effectively. It provides flexibility in our calculations and allows us to approach problems from different angles.

The commutative property, alongside the distributive property, forms the backbone of complex number multiplication. They are the fundamental tools that allow us to break down complex expressions, rearrange terms, and ultimately solve equations. Understanding and mastering these properties is essential for anyone venturing into the world of complex numbers. They are not just abstract rules; they are the keys to unlocking the complexities of this fascinating branch of mathematics.

Conclusion: The Power of Perseverance and Verification

Our journey through the equation $(\sqrt{2} - i) - i(1 - \sqrt{2i}) = -2i$ has demonstrated the importance of breaking down complex problems into manageable steps. We've seen how the distributive property and the concept of simplifying radicals, particularly using De Moivre's Theorem, are crucial tools in our arsenal. We also highlighted the ever-present role of the commutative property in complex number multiplication. Most importantly, we've learned the vital lesson of verification. Even with meticulous calculations, it's essential to check our results to ensure accuracy.

While the original equation turned out to be false, the process of solving it provided valuable insights into the world of complex numbers. We honed our skills in manipulating complex expressions, applying fundamental properties, and simplifying radicals. We also reinforced the importance of critical thinking and perseverance in problem-solving. The ability to identify errors, learn from them, and adapt our approach is a key skill in mathematics and beyond.

The world of complex numbers can be both challenging and rewarding. By mastering the fundamental principles and embracing a systematic approach to problem-solving, we can unlock its secrets and appreciate its elegance. The journey may not always lead to the expected destination, but the knowledge and skills gained along the way are invaluable. So, embrace the challenge, persevere through the complexities, and never underestimate the power of verification.