Simplifying Complex Numbers: Finding The Value Of I^97 - I

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What is the value of $i^{97}-i$?

Hey everyone, let's dive into a fun little problem involving complex numbers! We're tasked with figuring out the value of i^{97} - i. Don't worry if complex numbers sound a bit intimidating; we'll break it down step-by-step and make it super clear. This is a classic math problem that really tests your understanding of imaginary numbers. So, buckle up, and let's get started!

Understanding the Basics: What's 'i' Anyway?

Okay, before we jump into the main problem, let's quickly recap what 'i' actually is. In the world of complex numbers, i is the imaginary unit. It's defined as the square root of -1 (√-1). This might seem a little strange at first because you can't find a real number that, when multiplied by itself, gives you a negative number. That's where 'i' comes to the rescue! It lets us work with the square roots of negative numbers and opens up a whole new realm of mathematical possibilities. This is important because it is a fundamental principle in how we'll solve i^{97} - i

The powers of 'i' follow a cyclical pattern, which is the key to solving this type of problem. Here's how it works:

  • i^1 = i
  • i^2 = -1 (because i * i = √-1 * √-1 = -1)
  • i^3 = -i (because i^2 * i = -1 * i = -i)
  • i^4 = 1 (because i^2 * i^2 = -1 * -1 = 1)

And then the cycle repeats! i^5 is the same as i^1, i^6 is the same as i^2, and so on. This cyclical nature is super important because it means we don't have to calculate every single power of 'i' from scratch. We can use the remainder when we divide the exponent by 4 to figure out the value.

Why Understanding the Cycle of 'i' Matters

Knowing the cycle of 'i' (i, -1, -i, 1) is absolutely crucial for simplifying expressions like i^{97} - i. Without this understanding, you'd be stuck trying to figure out what i raised to the power of 97 actually is. The cyclical pattern allows us to reduce the exponent and find an equivalent value that's much easier to work with. This method is not only helpful in this specific problem but also for any complex number problem you might encounter in the future. Once you grasp this pattern, working with complex numbers becomes significantly easier and less daunting. It's like having a secret code to unlock the solution! This is also why this question is a popular one for tests because it tests your ability to think critically about math.

Simplifying i^{97}

Now, let's focus on simplifying i^{97}. Remember the cyclical pattern? We can use that to make this easier. Here’s what you do: divide the exponent (97) by 4. When you divide 97 by 4, you get a quotient of 24 with a remainder of 1. So, we can write i^{97} as i^{(4 * 24 + 1)}. Because i^4 = 1, any multiple of 4 in the exponent will simplify to 1. This means i^{(4 * 24)} is just 1. Therefore, i^{97} simplifies to i^1, which is just 'i'.

So, we've successfully simplified i^{97}. This step is about applying the modular arithmetic principles to reduce the complexity of the calculation. Essentially, we’re saying that the power of 'i' behaves in a predictable cycle of four values, and we can use the remainder of the division by 4 to determine where we land in that cycle. This saves us a ton of work and makes the problem manageable. Now, let's use the result to calculate i^{97} - i!

The Importance of Remainder in Simplifying Powers of 'i'

The remainder when dividing the exponent by 4 is the most crucial part of simplifying powers of 'i'. As we've seen, it determines which value in the cycle (i, -1, -i, 1) the expression will equal. If the remainder is 0, the result is 1; if it's 1, the result is 'i'; if it's 2, the result is -1; and if it's 3, the result is -i. This simple concept drastically simplifies complex expressions, transforming what might seem like a daunting calculation into a straightforward process. Knowing this principle allows you to solve a wide variety of problems involving imaginary numbers with ease and confidence.

Putting It All Together: Calculating i^{97} - i

Alright, we've done the hard work of simplifying i^{97}. We found out that i^{97} = i. Now, we just need to subtract 'i' from it. So, the expression becomes i - i. And what does i - i equal? It equals 0! So the answer to i^{97} - i is 0. Easy peasy, right?

Let’s recap: We broke down the problem into smaller, manageable steps. We started by understanding the fundamental properties of 'i', and then we used the cyclical pattern to simplify i^{97}. Finally, we performed the simple subtraction. By taking it one step at a time, we arrived at the correct answer without any trouble.

The Final Calculation: A Step-by-Step Summary

Let's summarize the final calculation to ensure everything is crystal clear:

  1. Simplify i^{97}: Divide 97 by 4. The remainder is 1. Therefore, i^{97} = i^1 = i.
  2. Substitute and Calculate: Replace i^{97} with 'i' in the original expression: i - i.
  3. Final Result: i - i = 0.

So, the value of i^{97} - i is 0. This entire process demonstrates how understanding the core principles of complex numbers, particularly the cyclical nature of powers of 'i', can simplify complex calculations into something manageable and easy to solve. Congrats, guys! You did it!

The Answer and Why It Matters

So, the correct answer is B. 0. This kind of problem is really important because it tests your basic understanding of complex numbers and your ability to apply the rules. It shows that you can think critically and solve problems step-by-step. The whole point of learning this is to build a strong foundation for more advanced math concepts. This is like the building block for all other math lessons.

Conclusion: Mastering Complex Number Problems

So, there you have it, folks! We've successfully solved i^{97} - i. Remember that the key is understanding the properties of 'i' and its cyclical pattern. This approach isn't just useful for this specific problem; it's a valuable skill for tackling other complex number problems. Keep practicing, and you'll become a pro in no time! Complex numbers might seem tricky at first, but with a little practice and a good understanding of the rules, you'll be able to solve them with confidence. Now go out there and show off your math skills!