Finding a whole number between two given numbers depends on whether you're looking for a number strictly between them or a whole number that lies between them (inclusive). The method also depends on whether you're working with integers or real numbers. Based on the reference, it seems we're addressing a specific, potentially flawed, method. Let's analyze that first and then provide a more complete explanation.
Analyzing the Provided Method
The provided reference suggests the following to find a whole number between two numbers:
- Step 1: Subtract the numbers.
- Step 2: Subtract 1 from the obtained result.
This method is incorrect and will generally not yield a correct answer or even a whole number. The result of this calculation doesn't inherently relate to a whole number lying between the two original numbers.
A More Accurate Approach
Here's a more accurate and comprehensive way to find a whole number between two numbers:
1. Understanding the Goal
First, clarify what "between" means. Do you want a number strictly between the two given numbers (exclusive), or are you including the possibility that one of the given numbers is a whole number (inclusive)?
2. Identifying Whole Numbers
Check if the given numbers are whole numbers themselves. If they are, the process is simpler.
3. Finding a Whole Number Between Two Integers (Exclusive)
If you have two integers and want to find an integer strictly between them:
- Check the difference: If the larger number minus the smaller number equals 1, there is NO integer strictly between them.
- Example: Between 5 and 6, there are no integers.
- If the difference is greater than 1: The integer immediately after the smaller integer is a whole number between them.
- Example: Between 5 and 8, the integers 6 and 7 are between them.
- Formula: If a < b are integers, and b - a > 1, then a + 1 is an integer between a and b.
4. Finding a Whole Number Between Two Real Numbers
If you're dealing with real numbers (numbers that can have decimal parts):
- Find the integers immediately above and below:
- Find the smallest integer greater than the smaller number using the ceiling function (usually written as ⌈x⌉).
- Find the largest integer less than the larger number using the floor function (usually written as ⌊x⌋).
- Example:
- Numbers: 3.2 and 7.8
- Ceiling of 3.2: ⌈3.2⌉ = 4
- Floor of 7.8: ⌊7.8⌋ = 7
- Whole numbers between 3.2 and 7.8 are: 4, 5, 6, and 7.
- Check for Existence: Ensure that the integer found above (ceiling of the smaller number) is strictly less than the integer found below (floor of the larger number). If they are equal or the ceiling is greater than the floor, there are no whole numbers strictly between the given real numbers.
5. Inclusive Case
If you're looking for a whole number that lies between them (inclusive), then you would include the original numbers themselves if they happened to be whole numbers.
- Example: Between 3 and 5.7, the whole numbers lying between them (inclusive) are 3, 4, and 5.
Example Scenarios
Here are some examples to illustrate the different scenarios:
| Numbers | "Between" Meaning | Whole Number(s) Found |
|---|---|---|
| 2 and 5 | Strictly between | 3, 4 |
| 2 and 5 | Inclusive | 2, 3, 4, 5 |
| 2.3 and 5.8 | Strictly between | 3, 4, 5 |
| 2.3 and 5.8 | Inclusive | No change from above. Includes only integers. |
| 4 and 5 | Strictly between | None |
| 4 and 5 | Inclusive | 4, 5 |
| 4.1 and 4.9 | Strictly between | None |
| 4.1 and 5.0 | Strictly between | None |
| 4.1 and 5.0 | Inclusive | None |
Table: Illustrative Examples
| Number 1 | Number 2 | Whole Number Strictly Between? | Whole Number Inclusive? |
|---|---|---|---|
| 2 | 5 | 3, 4 | 2, 3, 4, 5 |
| 2.3 | 5.8 | 3, 4, 5 | 3, 4, 5 |
| 4 | 5 | None | 4, 5 |
| 4.1 | 4.9 | None | None |
