Simple Math: (3 + I)(3 – I) Solved
Hey math enthusiasts, are you ready to dive into some seriously cool complex number action? Today, we're tackling a classic problem involving the imaginary unit, where i = √-1. We'll be simplifying the expression (3 + i)(3 – i), and trust me, it's way easier than it looks. We'll break down exactly how to get to the answer, exploring the magic behind complex numbers and why this particular expression simplifies so neatly. So, grab your calculators, your notebooks, or just your brilliant brains, and let's get this math party started! We're going to demystify complex numbers and show you that solving problems like this can be a piece of cake, or should I say, a piece of pi? Haha, get it? Anyway, let's get down to business and figure out what (3 + i)(3 – i) equals.
Understanding the Imaginary Unit: The Core of Complex Numbers
Alright guys, before we jump straight into solving (3 + i)(3 – i), let's get a solid grip on what i actually is. For ages, mathematicians were stumped by equations like x² = -1. There was no real number that, when multiplied by itself, would give you a negative result. Enter the imaginary unit, denoted by i. This groundbreaking concept, where i = √-1, essentially unlocked a whole new universe of numbers – the complex numbers. Think of it as a superpower for algebra! Complex numbers have the form a + bi, where a is the real part and b is the imaginary part. So, for example, 3 + 2i is a complex number. The introduction of i didn't just solve a mathematical puzzle; it opened doors to understanding phenomena in physics, engineering, signal processing, and so much more. It's a fundamental building block that allows us to work with square roots of negative numbers, which are pretty common in advanced math and science. The beauty of i is its predictable pattern when raised to powers: i¹ = i, i² = -1, i³ = -i, i⁴ = 1, and then it repeats. This cyclic nature is super useful for simplifying higher powers of i. So, when you see i in an equation like (3 + i)(3 – i), remember you're dealing with a number whose square is negative one. This fundamental property is key to simplifying expressions involving i, especially when we're multiplying complex numbers. The more you work with it, the more natural it becomes, and you'll start spotting these patterns and simplifications like a pro. It's all about embracing the idea that numbers can be more than just what we see on a number line!
The Multiplication Magic: Applying the Distributive Property
Now that we've got our heads around i, let's tackle the main event: multiplying (3 + i) by (3 – i). This is where the fun really begins! When you multiply two binomials like these, the go-to method is the distributive property, often remembered by the acronym FOIL (First, Outer, Inner, Last). Let's apply that here, guys.
First: Multiply the first terms in each binomial. That's 3 * 3, which equals 9.
Outer: Multiply the outer terms. That's 3 * (-i), which equals -3i.
Inner: Multiply the inner terms. That's i * 3, which equals +3i.
Last: Multiply the last terms in each binomial. That's i * (-i), which equals -i².
So, putting it all together, we get: 9 - 3i + 3i - i².
Notice something super cool here? The -3i and +3i terms cancel each other out! This is a direct result of multiplying a complex number by its conjugate (which we'll talk more about in a sec). So, our expression simplifies to just 9 - i².
This step is crucial because it shows how complex numbers can simplify in unexpected ways. The distributive property is a fundamental algebraic tool, and applying it correctly here is key. Don't be afraid to write out each step, especially when you're first learning. The more you practice, the faster you'll become at spotting these simplifications. Remember, math is like a puzzle, and each step brings you closer to the solution. And in this case, the cancellation of the imaginary terms is a big clue that we're on the right track to a much simpler answer. Keep that FOIL method handy – it’s a lifesaver for binomial multiplication!
The Conjugate Trick: Simplifying i²
We're almost there, folks! We've simplified (3 + i)(3 – i) down to 9 - i². Now, the final, and perhaps most magical, step is dealing with that i². Remember our earlier chat about the imaginary unit? We established that i = √-1. So, what happens when we square i?
i² = (√-1)²
When you square a square root, they cancel each other out, right? So, i² is simply -1. This is the golden rule when working with complex numbers!
Now, let's substitute this back into our simplified expression: 9 - i².
Since i² = -1, we replace i² with -1:
9 - (-1)
And what is 9 - (-1)? It's the same as 9 + 1, which equals 10.
Ta-da! So, (3 + i)(3 – i) equals 10. This brilliant simplification happens because we multiplied a complex number by its complex conjugate. The complex conjugate of a number a + bi is a – bi. When you multiply a complex number by its conjugate, the imaginary terms always cancel out, leaving you with a real number. Specifically, (a + bi)(a – bi) = a² - (bi)² = a² - b²i² = a² - b²(-1) = a² + b². In our case, a=3 and b=1, so 3² + 1² = 9 + 1 = 10. See? It all fits together perfectly! This conjugate property is super handy and appears a lot in more advanced math, like solving quadratic equations or in electrical engineering. It's a neat trick that makes complex number calculations much more manageable. Pretty neat, huh?
The Answer Revealed: Choosing the Right Option
So, after all that awesome mathematical exploration, we've arrived at our final answer for (3 + i)(3 – i). We used the distributive property (FOIL) to expand the expression, getting 9 - 3i + 3i - i². Then, we saw that the imaginary terms -3i and +3i canceled each other out, leaving us with 9 - i². Finally, we remembered the fundamental definition of the imaginary unit: i² = -1. Substituting that in, we got 9 - (-1), which simplifies to 9 + 1, giving us a grand total of 10.
Now, let's look back at the options provided:
(A) 8 (B) 9 (C) 10 (D) 9 - i
Our calculated answer is 10, which matches option (C) perfectly! This problem is a fantastic example of how complex numbers, despite seeming a bit abstract at first, follow consistent and elegant mathematical rules. The key takeaway here is understanding the definition of i and how to apply basic algebraic principles like the distributive property and the concept of conjugates. It's these building blocks that allow us to simplify complex expressions and arrive at straightforward answers. So, next time you see a problem like this, don't be intimidated! Just remember your roots (pun intended!) in algebra and the special properties of i, and you'll be solving them in no time. Keep practicing, and you'll master these concepts before you know it. High five!
Beyond the Problem: Why This Matters
Hey, you guys crushed it! We just solved (3 + i)(3 – i) and got 10. But why is this type of problem important, you ask? Well, understanding how to multiply complex numbers, especially using the conjugate trick, is super fundamental in a lot of cool areas. Think about electrical engineering – they use complex numbers constantly to describe things like AC circuits, impedance, and wave functions. The fact that multiplying by a conjugate gives you a real number is a massive simplification in these calculations. Another huge field is signal processing. Whether it's for audio, video, or telecommunications, understanding the frequency domain often involves complex numbers. Fourier transforms, which are used everywhere from compressing music files to analyzing brain waves, heavily rely on complex exponentials.
Furthermore, in quantum mechanics, the fundamental equations describing particles and their behavior are intrinsically complex-valued. The wave function, which tells you the probability of finding a particle in a certain place, is a complex function. Even in fractal geometry, like generating those mind-bending Mandelbrot sets, complex number arithmetic is the engine driving the visualizations. So, while (3 + i)(3 – i) = 10 might seem like a simple algebra exercise, it's actually a gateway to understanding much more complex and fascinating concepts in science and technology. It shows us that even seemingly abstract mathematical ideas have very real and practical applications. It’s all about building those foundational skills, guys, because you never know where they might lead you! Keep exploring, keep learning, and keep being curious about the amazing world of math!