Math Mania: Solving [65] + {√3} + [√3] - √3

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Hey math enthusiasts! Today, we're diving into a cool mathematical problem: calculating the value of the expression [65] + {√3} + [√3] - √3. Sounds a bit intimidating, right? Don't sweat it! We'll break it down step by step, making it super easy to understand. This problem is a fantastic way to sharpen your understanding of number theory, particularly the concepts of the integer part and the fractional part of a real number. We'll go through each component of the expression, explaining what it means and how to calculate it. By the end of this guide, you'll be able to solve similar problems with confidence. Let's get started and unlock the secrets hidden within this expression. Get ready to boost your math skills and have some fun along the way!

Understanding the Components

Alright guys, before we jump into the solution, let's decode what each part of the expression actually means. Understanding the notation is key to solving the problem. We've got a few key players here:

  1. [65]: This is the integer part of 65. But wait, 65 is already an integer, so what does that mean? Well, the integer part, denoted by the square brackets [ ], essentially means the largest integer that is less than or equal to the number inside. In this case, since 65 is already a whole number, its integer part is simply 65. Easy peasy, right?

  2. **√3}** This is the fractional part of the square root of 3 (√3). The fractional part, indicated by the curly braces `{ `, represents the difference between the number and its integer part. To find this, we first need to know the value of √3. The square root of 3 is approximately 1.732. So, we need to figure out the fractional part of 1.732. The integer part of 1.732 is 1. Therefore, the fractional part is 1.732 - 1 = 0.732. Got it?

  3. [√3]: Here, we have the integer part of √3 again. As we already know, √3 is approximately 1.732. The integer part of 1.732 is 1. This is similar to what we discussed earlier about the integer part.

  4. √3: This is simply the square root of 3, which, as we mentioned, is about 1.732. This is a real number and we need to include its value directly in our final calculation.

So, to recap, we've got the integer part of 65, the fractional part of √3, the integer part of √3, and the square root of 3 itself. Each component plays a role in our calculation, so understanding what they represent is crucial. Ready to put it all together?

Breaking Down the Calculation

Now that we understand what each part of the expression represents, let's calculate the value step by step. Don't worry, it's going to be simpler than it looks! We're going to substitute the values we found for each component into the original expression and then perform the arithmetic.

  1. [65]: As we discussed, the integer part of 65 is 65.

  2. {√3}: The fractional part of √3 is approximately 0.732 (since √3 ≈ 1.732 and 1.732 - 1 = 0.732).

  3. [√3]: The integer part of √3 is 1 (because √3 ≈ 1.732 and the integer part of 1.732 is 1).

  4. √3: The square root of 3 is approximately 1.732.

Now, let's plug these values into the expression: [65] + {√3} + [√3] - √3 becomes 65 + 0.732 + 1 - 1.732. You see, the equation is just composed of plus and minus operations, so it is really easy to work on. Simplify the above expression we have: 65 + 0.732 + 1 - 1.732. Combining the numbers, we get 65 + 1 + (0.732 - 1.732). First, adding the integers, we obtain 66 + (0.732 - 1.732). Then, 66 + (-1), which equals 65. So, the value of the entire expression is 65. Wow, that's it! We've successfully solved the problem. Wasn't that cool? We started with a seemingly complex expression, but by breaking it down into its components and understanding the notation, we managed to arrive at a simple, elegant solution. This kind of approach is the heart of problem-solving in mathematics – understanding the parts to see the whole.

Step-by-Step Solution: The Complete Breakdown

Let's go through the entire calculation from start to finish. This is especially helpful if you're the type of person who likes to see every single step laid out clearly. Here's how to do it, nice and easy:

  1. Identify the Components: The expression is [65] + {√3} + [√3] - √3. We need to find the values of the integer part of 65, the fractional part of √3, the integer part of √3, and the value of √3.

  2. Calculate the Integer Part of 65: The integer part of 65, denoted as [65], is 65 because 65 is an integer. So, [65] = 65.

  3. Calculate the Fractional Part of √3: The square root of 3 (√3) is approximately 1.732. The integer part of 1.732 is 1. The fractional part, {√3}, is the difference between √3 and its integer part: 1.732 - 1 = 0.732. Thus, {√3} = 0.732.

  4. Calculate the Integer Part of √3: As we know, √3 ≈ 1.732. The integer part of 1.732, denoted as [√3], is 1. Therefore, [√3] = 1.

  5. Substitute and Simplify: Now, substitute these values back into the original expression: [65] + {√3} + [√3] - √3 = 65 + 0.732 + 1 - 1.732. Now, the equation is just composed of plus and minus operations, so it is really easy to work on. You can rearrange the terms to make the calculation easier, such as the following: 65 + 1 + 0.732 - 1.732 = 66 + (0.732 - 1.732). Then simplify the numbers inside the parenthesis: 0.732 - 1.732 = -1. Hence, we have 66 + (-1) = 65.

  6. Final Answer: The solution to the expression is 65. Congratulations, you have solved the problem! You've taken a complex-looking expression and turned it into something manageable and easy to solve. This approach to mathematics, breaking problems down into their parts, and understanding the notation, will help you solve countless other problems in the future!

Key Takeaways and Tips

Alright, let's wrap things up with some key takeaways and tips to help you become a math master. Remember, practice makes perfect, so the more you work with these concepts, the more comfortable you'll become. Here are a few things to keep in mind:

  • Understand the Notation: Make sure you clearly understand what each symbol means: [ ] for the integer part, { } for the fractional part, and the square root symbol (√). It's like learning a new language – once you know the alphabet, you can read anything!
  • Break It Down: Always break down the expression into smaller, manageable parts. Don't try to solve the entire problem at once. Focus on one component at a time.
  • Know Your Approximations: For square roots and other irrational numbers, knowing their approximate values is helpful. For example, knowing that √3 is about 1.732 is crucial for finding the fractional part. You don't need to memorize everything, but having a general idea helps.
  • Practice, Practice, Practice: The best way to improve is by solving similar problems. Try variations of this problem, changing the numbers and expressions. The more you practice, the better you'll get at recognizing patterns and applying the concepts.
  • Review and Reflect: After solving a problem, take a moment to review your steps. Ask yourself: