Calculate with Floor
Your result will appear here.
What is a floor calculator?
Floor is directed rounding toward negative infinity. At a chosen place it returns the greatest target-place multiple less than or equal to the input. Positive inexact values move toward zero, but negative values move away from zero: floor −12.341 at hundredths is −12.35.
Important terms
- Negative infinity
- The decreasing direction on the number line.
- Greatest lower multiple
- The largest allowed target-place value that is not above the input.
- Floor function
- The integer floor ⌊z⌋ gives the greatest integer less than or equal to z.
- Target grid
- All multiples of the selected unit 10⁻ᵖ.
When this method is useful
- Finding lower grid bounds.
- Grouping a value into an interval indexed from below.
- Matching systems that specify mathematical floor.
- Distinguishing negative-infinity rounding from truncation.
Floor Calculator
Always move toward negative infinity
Formula and variables
- x
- the original value
- p
- the selected decimal-place count
- ⌊ · ⌋
- greatest integer less than or equal to the scaled value
- Fₚ(x)
- the selected-place floor
After scaling, integer floor selects the lower integer boundary toward negative infinity. Reversing the scale gives the greatest target-place multiple not exceeding the input.
Domain note: Places range from −20 through 50. Floor never increases numerical value. It differs from truncation for negative inexact values.
How the rule makes its decision
How to use this calculator
- 1
Enter the number and target place.
- 2
Scale the value by 10ᵖ.
- 3
Take the greatest integer less than or equal to the scaled value.
- 4
Divide by 10ᵖ.
- 5
Confirm that the result is less than or equal to the input.
Step-by-step examples
Example 1
Problem: Apply floor to −12.341 at two decimal places.
Substitution: −12.341 × 100 = −1234.1; floor(−1234.1) = −1235
Result: −12.35
Toward negative infinity makes the negative value more negative.
Example 2
Problem: Apply floor to 12.349 at two places.
Substitution: floor(1234.9) = 1234
Result: 12.34
The positive value moves down toward zero.
Example 3
Problem: Apply floor to −4,201 at hundreds.
Substitution: p = −2; floor(−42.01) = −43
Result: −4,300
−4,300 is the greatest hundred not above −4,201.
Worked rounding example
Rules, edge cases, and related ideas
Lower bound
Fₚ(x) ≤ x for every valid input.
Sign-dependent magnitude
Positive values lose magnitude; negative values gain magnitude.
Ceiling duality
floor(x) = −ceiling(−x) at corresponding precision.
Edge cases and limitations
- Exact inputs remain fixed.
- Zero stays zero.
- A tiny negative remainder moves to the next more-negative multiple.
- Negative-place floor can add trailing zeros to a larger-magnitude result.
How this differs from a related concept
Floor and truncation agree for positive values. For negative values floor moves toward negative infinity while truncation moves toward zero: −4.21 at tenths becomes −4.3 by floor and −4.2 by truncation.
Understand the result
The result is the greatest selected-place multiple no larger than the input. Its numerical value can stay the same or decrease, never increase.
Common mistakes
- Treating floor as removing decimal digits for negative values.
- Using a 5 threshold.
- Moving an exact multiple.
- Assuming floor always means smaller magnitude.
- Confusing floor with a minimum function.
Frequently asked questions
What is floor(−2.1)?
At whole numbers it is −3 because −3 is the greatest integer less than or equal to −2.1.
Why is it not −2?
−2 is greater than −2.1, so it is in the ceiling direction, not the floor direction.
Does floor inspect a guard digit?
No. Any nonzero remainder moves toward negative infinity.
Can floor operate at tenths?
Yes. Use one decimal place.
Is floor the same as Round Down?
Only for non-negative values; Round Down moves negative values toward zero.