Let z = [y] and y = [x] − x, where [.] is the greatest integer function. If x is not an integer but positive, then what is the value of z ?
−1
Finding the Value of z using the Greatest Integer Function
The problem asks us to find the value of z, where z is defined in terms of y, and y is defined in terms of x. We are given that \(z = [y]\) and \(y = [x] - x\). We are also told that x is a positive number but not an integer, and \( [.] \) represents the greatest integer function.
Let's break down the problem step by step to understand each part.
Understanding the Concepts: Greatest Integer Function
The greatest integer function, denoted by \( [u] \), gives the largest integer that is less than or equal to u.
- If u is an integer, \( [u] = u \). For example, \( [5] = 5 \), \( [-3] = -3 \).
- If u is not an integer, \( [u] \) is the integer immediately to the left of u on the number line. For example, \( [4.7] = 4 \), \( [-2.3] = -3 \).
Analyzing the Definition of y
We are given \( y = [x] - x \).
We know that x is a positive number but not an integer. This is a crucial piece of information. Since x is not an integer, \( [x] \) is the integer just below x.
Let's think about the difference \( [x] - x \). Since \( [x] \leq x \), the value of \( [x] - x \) will always be less than or equal to 0. Specifically, if x is an integer, \( [x] = x \), so \( [x] - x = 0 \). However, we are told that x is not an integer.
If x is not an integer, then \( [x] < x \). This means that \( [x] - x \) will be a negative number. The difference \( x - [x] \) is the fractional part of x, which is always between 0 (inclusive) and 1 (exclusive) for positive x. Let the fractional part be \( \{x\} \), so \( x = [x] + \{x\} \), where \( 0 \leq \{x\} < 1 \).
Since x is not an integer, \( \{x\} ≠ 0 \). Thus, \( 0 < \{x\} < 1 \).
Now substitute this into the expression for y:
\( y = [x] - x \)
\( y = [x] - ([x] + \{x\}) \)
\( y = [x] - [x] - \{x\} \)
\( y = -\{x\} \)
Since \( 0 < \{x\} < 1 \), multiplying by -1 reverses the inequality signs:
\( -1 < -\{x\} < 0 \)
So, we have \( -1 < y < 0 \).
Finding the Value of z
We are given \( z = [y] \).
From the previous step, we found that \( y \) is a number between -1 and 0 (not including -1 or 0). For any number u such that \( -1 < u < 0 \), the greatest integer less than or equal to u is -1.
For example:
- If \( y = -0.5 \), \( [y] = [-0.5] = -1 \).
- If \( y = -0.1 \), \( [y] = [-0.1] = -1 \).
- If \( y = -0.99 \), \( [y] = [-0.99] = -1 \).
Since \( -1 < y < 0 \), the greatest integer less than or equal to y is always -1.
Therefore, \( z = [y] = -1 \).
Summary of Steps
- Understand the definition of the greatest integer function \( [x] \).
- Use the condition that x is a positive non-integer.
- Rewrite \( y = [x] - x \) using the property \( x = [x] + \{x\} \), where \( 0 < \{x\} < 1 \) for non-integer x.
- Determine the range of possible values for y.
- Use the range of y to find \( z = [y] \).
Result
Based on our analysis, the value of z is -1.
| Variable | Definition | Property for positive non-integer x | Derived Value/Range |
|---|---|---|---|
| x | Input variable | Positive, not an integer | \(x = [x] + \{x\}\), where \(0 < \{x\} < 1\) |
| y | \( [x] - x \) | Result of subtraction | \(y = -\{x\}\), leading to \(-1 < y < 0\) |
| z | \( [y] \) | Greatest integer less than or equal to y | \(z = [-(\{x\})] = -1\) because \(-1 < -\{x\} < 0\) |
Revision Table: Greatest Integer Function Properties
| Property | Description | Example |
|---|---|---|
| Definition | \( [x] \leq x < [x] + 1 \) | If \( x = 3.7 \), \( [x] = 3 \), then \( 3 \leq 3.7 < 3 + 1 = 4 \) |
| Non-integer x | \( [x] < x \) | If \( x = 5.2 \), \( [x] = 5 \), so \( 5 < 5.2 \) |
| Integer x | \( [x] = x \) | If \( x = 7 \), \( [x] = 7 \) |
| Fractional Part | \( \{x\} = x - [x] \), where \( 0 \leq \{x\} < 1 \) | If \( x = 4.1 \), \( \{x\} = 4.1 - [4.1] = 4.1 - 4 = 0.1 \) |
Additional Information: Floor and Ceiling Functions
The greatest integer function \( [x] \) is also known as the floor function, often denoted as \( \lfloor x \rfloor \). It rounds a number down to the nearest integer.
Another related function is the ceiling function, denoted as \( \lceil x \rceil \), which gives the smallest integer greater than or equal to x. It rounds a number up to the nearest integer.
- For an integer x, \( [x] = \lfloor x \rfloor = \lceil x \rceil = x \).
- For a non-integer x, \( \lfloor x \rfloor < x < \lceil x \rceil \).
- Also for non-integer x, \( \lceil x \rceil = \lfloor x \rfloor + 1 = [x] + 1 \).
In this problem, the expression \( [x] - x \) for a non-integer x is equal to \( \lfloor x \rfloor - x \). This value is always the negative of the fractional part of x, which is between -1 and 0.
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