Calculator Inputs
Example Data Table
These sample inputs show how the greatest integer function handles positive and negative decimals.
| Input x | Floor ⌊x⌋ | Ceiling ⌈x⌉ | Fractional Part |
|---|---|---|---|
| -3.8 | -4 | -3 | 0.2 |
| -1.25 | -2 | -1 | 0.75 |
| 0 | 0 | 0 | 0 |
| 2.41 | 2 | 3 | 0.41 |
| 6.99 | 6 | 7 | 0.99 |
Formula Used
Transformed input: t = ax + b
Greatest integer function: ⌊t⌋ = largest integer less than or equal to t
Ceiling function: ⌈t⌉ = smallest integer greater than or equal to t
Fractional part: {t} = t - ⌊t⌋
Step interval: If n = ⌊t⌋, then n ≤ t < n + 1.
How to Use This Calculator
- Enter the main decimal or integer value in the x field.
- Select the function type, such as floor, ceiling, truncation, or nearest integer.
- Use coefficient a and shift b when you need a transformed expression.
- Add batch values to calculate many inputs at once.
- Set the graph range and step size for the visual plot.
- Press Calculate to view the result above the form.
- Use CSV or PDF buttons to export the calculated table.
Greater Integer Function Guide
What This Calculator Does
The greater integer function calculator helps you study step functions. It mainly evaluates the greatest integer function, also called the floor function. This function returns the largest integer that is less than or equal to a number. For example, the value of floor 4.8 is 4. The value of floor -4.8 is -5. Negative decimals often confuse students. This tool makes those steps clear.
Why Step Functions Matter
Step functions appear in algebra, calculus, number theory, and computing. They are useful when a value must move in whole units. Tickets, packages, floors, counters, and grouped charges often use this idea. A graph of the function does not form a smooth curve. It forms flat steps. Each step covers one interval. The jump happens at an integer point.
Advanced Input Options
This page can evaluate a direct value or a transformed value. The expression t equals ax plus b. You can change a and b to test shifts, stretches, and reflections. When a is negative, the step pattern reverses. When b changes, the graph moves left or right. The batch input lets you test many values in one run. That is helpful for homework tables and teaching notes.
Reading the Output
The result summary gives the selected function value. It also gives the transformed input. The fractional part shows the decimal distance after the previous integer. The interval column shows where the value belongs. For floor output n, the input lies from n up to n plus one. The left end is included. The right end is not included.
Using the Graph
The graph displays the selected integer function over your chosen range. A smaller plot step gives a smoother looking step chart. A larger step calculates faster. Use the graph to locate jumps. Compare it with the table to confirm each output. Export the table when you need a saved report.
FAQs
1. What is the greatest integer function?
It returns the largest integer less than or equal to the given number. It is also called the floor function and is written as ⌊x⌋.
2. Is floor 3.9 equal to 3 or 4?
Floor 3.9 equals 3 because 3 is the largest integer that does not exceed 3.9. It never rounds upward.
3. Why is floor -2.4 equal to -3?
For negative values, the floor moves toward the smaller integer. Since -3 is less than -2.4 and is the nearest lower integer, the answer is -3.
4. What is the difference between floor and ceiling?
Floor gives the largest integer not greater than the number. Ceiling gives the smallest integer not less than the number. They differ for most decimal values.
5. Can this calculator handle transformed functions?
Yes. It supports t = ax + b. You can adjust a and b to evaluate shifted, stretched, or reflected integer step functions.
6. What does the fractional part mean?
The fractional part is calculated as x minus floor x. It shows the decimal part left after removing the greatest lower integer.
7. Why does the graph look like stairs?
The output remains constant across each interval, then jumps at integer boundaries. That repeated flat pattern creates a staircase shaped graph.
8. Can I export my results?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a printable report with the current calculation table.