Integer And Fractional Parts: Calculations & Examples
Hey guys! Today, we're diving into the exciting world of integer and fractional parts of numbers. We'll be tackling some calculations and exploring different examples to really nail down this concept. So, buckle up and let's get started!
Calculating Integer and Fractional Parts
Before we jump into the problems, let's quickly recap what integer and fractional parts are. The integer part of a number, denoted by [x], is the greatest integer less than or equal to x. Think of it as rounding down to the nearest whole number. The fractional part, denoted by {x}, is the difference between the number and its integer part, i.e., {x} = x - [x]. It's the decimal part of the number, always between 0 (inclusive) and 1 (exclusive).
Now, let's dive into the calculations!
Part A: Calculating Integer and Fractional Parts of Powers
In this section, we'll be calculating the integer and fractional parts of numbers raised to various powers. This involves a bit of arithmetic and a good understanding of how exponents work. Let's break it down step-by-step.
1. (2.1)^2
Let's begin with calculating (2.1)^2. This means 2.1 multiplied by itself. Doing the math, we get:
(2.1)^2 = 2.1 * 2.1 = 4.41
Now, to find the integer part, we look for the greatest integer less than or equal to 4.41. That's simply 4. So, [4.41] = 4. The integer part is 4.
Next, we calculate the fractional part by subtracting the integer part from the original number:
{4.41} = 4.41 - 4 = 0.41
Thus, the fractional part is 0.41. It's crucial to understand that the fractional part will always be a non-negative value less than 1.
2. (-1.1)^4
Moving on, let's tackle (-1.1)^4. Remember that raising a negative number to an even power results in a positive number. So, we have:
(-1.1)^4 = (-1.1) * (-1.1) * (-1.1) * (-1.1) = 1.4641
To find the integer part, we identify the largest integer that is less than or equal to 1.4641. That's 1. So, [1.4641] = 1. The integer part is 1.
Now, for the fractional part, we subtract the integer part from the original number:
{1.4641} = 1.4641 - 1 = 0.4641
The fractional part is 0.4641. It represents the decimal portion of the number.
3. (-3.2)^3
Next up is (-3.2)^3. Raising a negative number to an odd power results in a negative number. Let's calculate:
(-3.2)^3 = (-3.2) * (-3.2) * (-3.2) = -32.768
Finding the integer part of a negative number requires a bit of care. We need the greatest integer that is less than or equal to -32.768. That's -33 (not -32!). So, [-32.768] = -33. The integer part is -33.
To get the fractional part, we subtract the integer part from the original number:
{-32.768} = -32.768 - (-33) = -32.768 + 33 = 0.232
Therefore, the fractional part is 0.232. Always remember that the fractional part is always positive and less than 1.
Part B: Calculating Integer and Fractional Parts with Radicals
This part introduces radicals, which might seem a bit more challenging, but don't worry! We'll break it down just like before. We'll be working with square roots and their properties to find the integer and fractional parts.
1. (√2 + 1)^2
Let's start with (√2 + 1)^2. First, we need to expand the square using the formula (a + b)^2 = a^2 + 2ab + b^2:
(√2 + 1)^2 = (√2)^2 + 2 * √2 * 1 + 1^2 = 2 + 2√2 + 1 = 3 + 2√2
Now, we need to approximate the value of 2√2. We know that √2 is approximately 1.414. So,
2√2 ≈ 2 * 1.414 = 2.828
Adding this to 3, we get:
3 + 2√2 ≈ 3 + 2.828 = 5.828
The integer part of 5.828 is 5. So, [5.828] = 5. The integer part is 5.
The fractional part is the difference:
{5.828} = 5.828 - 5 = 0.828
Thus, the fractional part is approximately 0.828.
2. (√5 - √2)^2
Next, we have (√5 - √2)^2. Again, we expand the square, this time using the formula (a - b)^2 = a^2 - 2ab + b^2:
(√5 - √2)^2 = (√5)^2 - 2 * √5 * √2 + (√2)^2 = 5 - 2√10 + 2 = 7 - 2√10
Now, we approximate 2√10. Since √10 is approximately 3.162,
2√10 ≈ 2 * 3.162 = 6.324
So, we have:
7 - 2√10 ≈ 7 - 6.324 = 0.676
The integer part of 0.676 is 0. So, [0.676] = 0. The integer part is 0.
The fractional part is:
{0.676} = 0.676 - 0 = 0.676
The fractional part is approximately 0.676.
3. (√2 - 1)^3
Finally, let's calculate (√2 - 1)^3. We'll expand this using the binomial theorem or by multiplying (√2 - 1)^2 by (√2 - 1). We already know (√2 - 1)^2 = (√2)^2 - 2√2 + 1 = 3 - 2√2.
So, (√2 - 1)^3 = (3 - 2√2)(√2 - 1) = 3√2 - 3 - 4 + 2√2 = 5√2 - 7
Approximating 5√2, we get:
5√2 ≈ 5 * 1.414 = 7.07
Therefore:
5√2 - 7 ≈ 7.07 - 7 = 0.07
The integer part of 0.07 is 0. So, [0.07] = 0. The integer part is 0.
The fractional part is:
{0.07} = 0.07 - 0 = 0.07
Thus, the fractional part is approximately 0.07.
Solving Equations with Integer and Fractional Parts
Now, let's move on to solving equations involving integer and fractional parts. This requires a solid understanding of the properties of these functions and some algebraic manipulation.
Part A: Determining [x], {x}, and [5x]
We're given the equation: x - [√3] + {1.1(6)} = √3 + {1/2} + {4.(3)}
First, let's simplify the equation. We know that:
- [√3] = 1 (since √3 is approximately 1.732)
- {1.1(6)} = 0.1666... which is 1/6
- {1/2} = 1/2 = 0.5
- {4.(3)} = 0.3333... which is 1/3
- √3 ≈ 1.732
Substituting these values into the equation, we get:
x - 1 + 1/6 = 1.732 + 0.5 + 1/3
To solve for x, we'll first combine the constants on the right side:
- 5 + 1/3 = 1/2 + 1/3 = 3/6 + 2/6 = 5/6
So, the equation becomes:
x - 1 + 1/6 = 1.732 + 5/6
Now, convert 1/6 to its decimal approximation, which is about 0.1667, and 5/6, which is about 0.8333. The equation then looks like this:
x - 1 + 0.1667 = 1.732 + 0.8333
Combine like terms:
x - 0.8333 = 2.5653
Now, add 0.8333 to both sides to solve for x:
x = 2.5653 + 0.8333 x = 3.3986
Now that we have x, let's find [x], {x}, and [5x].
- [x] is the integer part of x, so [3.3986] = 3. Therefore, the integer part of x is 3.
- x} is the fractional part of x, which we find by subtracting the integer part from x = 3.3986 - 3 = 0.3986. So, the fractional part of x is approximately 0.3986.
- Now, we need to find [5x]. Multiply x by 5: 5 * 3.3986 = 16.993. The integer part of 16.993 is 16, so [5x] = 16. Hence, [5x] is 16.
Part B: Solving for y
We're given the equation: y + √5 - {2.8(3)}
Unfortunately, the equation seems incomplete. To solve for 'y', we need the full equation, including what it is equal to. The expression "y + √5 - {2.8(3)}" by itself doesn't allow us to isolate and solve for 'y'.
However, let's break down the part we have and simplify it as much as we can. The key here is to understand the fractional part notation and simplify the term {2.8(3)}.
First, let's recognize that 2.8(3) means 2.8333..., where the 3 repeats infinitely. The fractional part, {2.8(3)}, refers to the decimal portion of this number.
To find the fractional part, we take the decimal portion directly:
{2.8(3)} = 0.8333...
This repeating decimal can be converted into a fraction. We recognize 0.8333... as 5/6. So, we have:
{2.8(3)} = 5/6
Now, let's rewrite the given expression with this simplified fractional part:
y + √5 - {2.8(3)} = y + √5 - 5/6
Without knowing what this expression equals, we cannot solve for 'y'. If, for example, the complete equation was y + √5 - {2.8(3)} = 0, we could then solve for 'y'. Let’s assume, for the sake of demonstration, that the full equation is:
y + √5 - 5/6 = 0
In this case, we can isolate 'y' by moving the other terms to the right side of the equation:
y = -√5 + 5/6
We can approximate √5 as 2.236. So the value of 'y' is approximately:
y = -2.236 + 5/6 y = -2.236 + 0.8333... y ≈ -1.4027
However, without the complete original equation, this is just an example. If you have the full equation, just use this method of simplifying fractional parts and isolating 'y' to find the solution.
Conclusion
And there you have it! We've tackled the calculation of integer and fractional parts, including those involving powers and radicals. We've also delved into solving equations with these concepts. Remember, the key is to break down each problem into smaller, manageable steps, and always double-check your work. Keep practicing, and you'll master these skills in no time! High five, guys! You've learned something amazing today!
