Solving Complex Number Equations A Step-by-Step Guide To (-21 + 65i) + (-30 + 12i)

Hey guys! Let's dive into the fascinating world of complex numbers and tackle this equation: (-21 + 65i) + (-30 + 12i) = a + bi. Don't worry if it looks a bit intimidating at first. We'll break it down step by step and you'll see it's actually quite straightforward. Complex numbers, at their core, are simply numbers that have two parts: a real part and an imaginary part. The imaginary part is denoted by the symbol 'i,' which represents the square root of -1. Understanding complex numbers is crucial in various fields like electrical engineering, quantum mechanics, and even computer graphics. So, paying attention here can really open doors to some cool stuff later on. The general form of a complex number is a + bi, where 'a' is the real part and 'b' is the imaginary part. In our equation, we're essentially adding two complex numbers together, and our goal is to express the result in the same a + bi form. The beauty of complex number arithmetic lies in its simplicity. We treat 'i' as a variable, just like 'x' or 'y' in algebra, but with the added knowledge that i² = -1. This property is what gives complex numbers their unique flavor and allows us to perform operations that wouldn't be possible with real numbers alone. When adding complex numbers, we simply combine the real parts and the imaginary parts separately. Think of it like combining like terms in an algebraic expression. This means we add the real parts together and the imaginary parts together. This makes the process very intuitive and easy to remember. So, let’s get started and unlock the secrets of this equation together!

Step-by-Step Breakdown of the Complex Number Addition

Alright, let's get our hands dirty and solve this complex number addition. Remember, the key to success here is to treat the real and imaginary parts separately. Think of it like sorting apples and oranges – you wouldn't mix them, right? Similarly, we'll group the real numbers together and the imaginary numbers together. This simple principle makes the whole process much easier to manage and understand. So, in our equation, (-21 + 65i) + (-30 + 12i), we have two complex numbers: (-21 + 65i) and (-30 + 12i). The first step is to identify the real parts and the imaginary parts in each number. In the first complex number, -21 is the real part and 65i is the imaginary part. In the second complex number, -30 is the real part and 12i is the imaginary part. Now that we've identified the components, the next step is to group them together. We'll add the real parts together: -21 + (-30). And we'll add the imaginary parts together: 65i + 12i. It’s like organizing your toolbox – putting all the wrenches in one place and all the screwdrivers in another. Once we have them grouped, the addition becomes straightforward. We're simply adding two real numbers and two imaginary numbers. This is where the magic happens! We can now perform the arithmetic operations on the grouped terms. Adding the real parts, we get -21 + (-30) = -51. Adding the imaginary parts, we get 65i + 12i = 77i. See how clean and simple that is? It’s all about breaking down the problem into manageable chunks. Now, let's put these results together to form our final answer in the a + bi format. This will give us a clear and concise representation of the solution. So, stay with me, and let’s complete this journey together!

Expressing the Final Answer in a + bi Form

Okay, guys, we've done the heavy lifting! We've successfully added the real and imaginary parts separately. Now comes the satisfying part – putting it all together and expressing our final answer in the standard a + bi form. This is like the grand finale of our mathematical performance, where we showcase the beautiful result of our efforts. Remember, the a + bi form is the standard way of representing complex numbers, where 'a' represents the real part and 'b' represents the imaginary part. It's a clear and concise way to communicate complex number solutions, and it makes it easy to compare and manipulate complex numbers further. So, we've calculated that the sum of the real parts is -51, and the sum of the imaginary parts is 77i. To express this in a + bi form, we simply write it as -51 + 77i. This is our final answer! Isn't it elegant how everything comes together so neatly? We started with a seemingly complex equation, and through careful steps and understanding of the principles, we arrived at a clear and simple solution. This solution, -51 + 77i, tells us that the real part of the sum is -51, and the imaginary part is 77. This representation allows us to easily visualize the complex number on the complex plane, where the real part is plotted on the horizontal axis and the imaginary part is plotted on the vertical axis. So, in essence, we've not only solved the equation, but we've also gained a deeper understanding of how complex numbers work. This understanding is a valuable asset in many fields, and it’s something you can build upon as you explore more advanced mathematical concepts. Let's recap what we've done and solidify our understanding.

Recap and Key Takeaways from Solving Complex Number Equations

Alright, let's take a step back and recap what we've learned today. This is like reviewing your notes after a great lecture – it helps solidify the concepts in your mind and makes sure everything sticks. We started with the equation (-21 + 65i) + (-30 + 12i) = a + bi, which looked a bit daunting at first glance. But we broke it down into manageable steps and conquered it! The key takeaway here is that adding complex numbers is just like combining like terms in algebra. We treat the real and imaginary parts separately, adding the real parts together and the imaginary parts together. This simple principle makes the process surprisingly straightforward. We identified the real parts (-21 and -30) and added them to get -51. We then identified the imaginary parts (65i and 12i) and added them to get 77i. Finally, we expressed the result in the standard a + bi form, which gave us our final answer: -51 + 77i. This final form is crucial because it allows us to easily interpret the complex number. We know that -51 is the real component and 77 is the imaginary component. This representation is not just a mathematical formality; it has practical implications. For example, in electrical engineering, complex numbers are used to represent alternating currents and voltages. The real part represents the resistive component, and the imaginary part represents the reactive component. Understanding this representation is key to analyzing and designing electrical circuits. So, by solving this simple equation, we've not only honed our mathematical skills but also gained a glimpse into the real-world applications of complex numbers. Remember, the journey of learning mathematics is like building a house – each concept is a brick that contributes to the overall structure. Mastering the basics of complex number arithmetic is a crucial step in this journey. Keep practicing, keep exploring, and you'll be amazed at what you can achieve!