Input values and options
How to use this calculator
- Enter a single real number in the first field.
- Optionally, provide additional numbers in the list text area.
- Use the sequence generator when exploring evenly spaced ranges.
- Select an output mode for extra detail on each input.
- Press the calculate button to generate and review the results.
- Download CSV or PDF to store, share, or analyze later.
Calculation results
No calculations yet. Enter at least one value and press the button.
The export buttons use the latest table above. Ensure your inputs and options are correct before downloading.
Formula used for the greatest integer function
The greatest integer function is also called the floor function. For a real number
x, its greatest integer value is written as ⌊x⌋.
Formally, ⌊x⌋ is the largest integer n such that n ≤ x.
This means ⌊x⌋ is always an integer and never greater than x.
Examples: ⌊3.7⌋ = 3, ⌊-1.2⌋ = -2, and ⌊5⌋ = 5.
The function steps down at each integer, creating a staircase style graph.
Advanced options explained
- Greatest integer only: Shows the core floor value ⌊x⌋ for each input without extra annotations.
-
Greatest integer and fractional part: Decomposes each real number as
x = ⌊x⌋ + fractional part, where the fractional part lies between 0 and 1. -
Inequality interval: Expresses the result as
n ≤ x < n + 1, highlighting the closed-open interval where each value lies. - Greatest integer with ceiling: Shows both floor and ceiling to compare rounding direction above and below the real number.
- Sequence generator: Automatically builds evenly spaced inputs using start, end, and step, then applies the selected option to every generated value.
Worked examples using the greatest integer calculator
Example 1: Single value with inequality interval
Enter 3.75 in the single real value field and select
the output mode “Inequality interval n ≤ x < n + 1”.
Press the calculate button to generate the result.
The table shows item “Single value”, input 3.75,
greatest integer 3, and additional details
3 ≤ x < 4.
This confirms that 3 is the largest integer not exceeding 3.75.
Example 2: Sequence from -2 to 2 with step 0.5
In the sequence generator, set start -2, end 2,
and step 0.5.
Choose the “Greatest integer and fractional part” mode.
The calculator creates items -2.0, -1.5, -1.0, …, 1.5, 2.0.
For each value, the table lists ⌊x⌋ and a decomposition
such as x = -2 + 0.5 for -1.5.
This illustrates the staircase structure of the function.
Example data table for greatest integer values
| Input x | ⌊x⌋ (greatest integer) | Fractional part x − ⌊x⌋ | Inequality form |
|---|---|---|---|
| -3.7 | -4 | 0.3 | -4 ≤ x < -3 |
| -1.0 | -1 | 0.0 | -1 ≤ x < 0 |
| -0.25 | -1 | 0.75 | -1 ≤ x < 0 |
| 0 | 0 | 0.0 | 0 ≤ x < 1 |
| 1.2 | 1 | 0.2 | 1 ≤ x < 2 |
| 2.9 | 2 | 0.9 | 2 ≤ x < 3 |
| 5 | 5 | 0.0 | 5 ≤ x < 6 |
Use these examples to verify your own manual computations or classroom exercises.
Additional insights about greatest integer calculations
1. Behavior across different input ranges
For x between two consecutive integers n and n + 1, the greatest integer value stays equal to n. Only when x passes the next integer does ⌊x⌋ jump. This step pattern is central to many discrete problems.
2. Relationship with modular arithmetic
When x is written as n + r with integer n and 0 ≤ r < 1, the value n is ⌊x⌋ and r can be seen as the remainder. This viewpoint connects floor calculations to modular arithmetic reasoning.
3. Applications in summations and series
Many number theory and analysis exercises use ⌊x⌋ inside sums, such as counting lattice points or grouping terms by integer parts. Understanding the floor function helps interpret these counting arguments precisely.
Frequently asked questions
1. What does the greatest integer function represent?
It maps any real number to the largest integer less than or equal to it. For example, values between 3 and 4 all have greatest integer value 3.
2. How is this different from rounding to the nearest integer?
Rounding may move a value up or down to the nearest integer. The greatest integer function never exceeds the original value, it always steps downward or stays equal.
3. Can the greatest integer of a whole number change?
No. When the input is already an integer, the greatest integer equals that same number exactly, because it is not necessary to step down to a smaller integer.
4. Why do negative numbers floor to more negative integers?
For negative values, the greatest integer must still be less than or equal to the value. That means stepping to the next smaller integer, such as -2 for -1.2.
5. Where is the greatest integer function used in practice?
It appears in discrete mathematics, algorithms, scheduling, indexing, and summations. Whenever continuous values must map into integer slots, steps, or bins, the greatest integer function is useful.