Calculator Inputs
Enter one main number for the detailed result. Add optional batch values and graph settings for a broader study view.
Example Data Table
| Input x | Greatest Integer ⌊x⌋ | Ceiling ⌈x⌉ | Fractional Part {x} | Interval |
|---|---|---|---|---|
| -4.90 | -5 | -4 | 0.10 | -5 ≤ x < -4 |
| -2.00 | -2 | -2 | 0.00 | -2 ≤ x < -1 |
| -0.20 | -1 | 0 | 0.80 | -1 ≤ x < 0 |
| 0.00 | 0 | 0 | 0.00 | 0 ≤ x < 1 |
| 1.70 | 1 | 2 | 0.70 | 1 ≤ x < 2 |
| 5.99 | 5 | 6 | 0.99 | 5 ≤ x < 6 |
Formula Used
The greatest integer function returns the largest integer less than or equal to a real number x.
⌊x⌋ = greatest integer less than or equal to x
If n = ⌊x⌋, then the input lies in the interval:
n ≤ x < n + 1
The fractional part is computed as:
{x} = x - ⌊x⌋
This calculator also shows the ceiling, truncation toward zero, and the exact interval that contains the input. These extra outputs help compare floor behavior with other rounding styles, especially for negative values.
How to Use This Calculator
- Enter the main number in the first field.
- Choose graph start, end, and step values.
- Optionally paste several values in the batch box.
- Click Calculate Greatest Integer.
- Read the result summary above the form.
- Review the graph, batch table, and interval statement.
- Use the CSV or PDF buttons to export results.
Frequently Asked Questions
1) What is the greatest integer function?
It is the floor function. It returns the largest integer that is less than or equal to a given real number. For 4.8, the result is 4. For -2.1, the result is -3.
2) Is the greatest integer the same as rounding down?
Yes, but only in the mathematical floor sense. It always moves to the next lower integer on the number line. For negative decimals, that means the result becomes more negative, not closer to zero.
3) Why does ⌊-1.2⌋ equal -2?
Because -2 is the largest integer that is still less than or equal to -1.2. The value -1 is greater than -1.2, so it cannot be the greatest integer in this case.
4) What happens when the input is already an integer?
The greatest integer is the same as the input. For example, ⌊7⌋ = 7 and ⌊-3⌋ = -3. The fractional part becomes zero for whole-number inputs.
5) What is the difference between floor and ceiling?
The floor gives the greatest integer less than or equal to x. The ceiling gives the smallest integer greater than or equal to x. They match only when the input is already an integer.
6) What does the interval result mean?
It shows the exact integer interval containing your input. If ⌊x⌋ = n, then x must lie between n and n + 1, including n but excluding n + 1.
7) What is the fractional part shown here?
It is the difference between the original number and its greatest integer value. The calculator uses {x} = x - ⌊x⌋, which always gives a value from 0 up to, but not including, 1.
8) Why is the graph shaped like steps?
The floor function stays constant between consecutive integers, then drops or rises abruptly at each integer boundary. That creates horizontal segments with jump points, which is why the graph looks like a staircase.