Example 1
- x: 3.75
0.75
What is {3.75}? 0.75.
Fractional Part Calculator
Enter a real number x. In analysis the fractional part is {x} = x − ⌊x⌋ and always lies in [0, 1). 3.75 leaves 0.75. 9 leaves 0. −1.3 leaves 0.7 — not −0.3 — because floor goes toward minus infinity.
Result
Enter a number to see the fractional part {x}.
One field: the number x. The result {x} lies in [0, 1). Negative inputs use floor toward −∞, following the same rule as the floor (integer part) calculator.
How it works
x−1.3
⌊x⌋−2
{x}0.7
The fractional part of a real number x, written with curly braces {x}, is defined in analysis and number theory via the floor function: {x} = x − ⌊x⌋. The symbol ⌊x⌋ denotes the greatest integer not greater than x — the same function computed by the floor (integer part) calculator. The result satisfies 0 ≤ {x} < 1. The identity x = ⌊x⌋ + {x} decomposes every real into an integer part and a fractional part in this sense. Graham, Knuth, and Patashnik give this definition in Concrete Mathematics; MathWorld (Fractional Part) matches it; NIST DLMF §1.4 covers floor and ceiling, from which {x} follows directly.
For nonnegative numbers the result matches the school intuition of “digits after the decimal”: {3.75} = 0.75, {9} = 0. For negatives that intuition diverges from the floor definition. ⌊−1.3⌋ = −2, so {−1.3} = −1.3 − (−2) = 0.7. The value −0.3 appears only under truncation toward zero or under the odd extension that uses ceiling. Wikipedia lists these three conventions side by side; this calculator uses the [0, 1) variant and shows the others in the result panel so conventions are not confused.
The calculator uses the single number you enter in the “Number x” field. The calculator computes ⌊x⌋ and subtracts that from x. A period and a comma are equivalent. For x = 3.75 the result is 0.75; for x = −2.3 the result is 0.7, because ⌊−2.3⌋ = −3.
Links to other integer-valued functions are immediate. Ceiling satisfies ⌈x⌉ = −⌊−x⌋. The nonnegative residue modulo 1 under this floor definition equals {x}. The distance to the next integer (when {x} ≠ 0) is 1 − {x}. The function {x} is periodic with period 1 and discontinuous at integers when approached from the left — hence the characteristic sawtooth graph.
In number theory the fractional part appears in sequences ({nα}), in uniform-distribution criteria, and in sums involving the Gauss bracket. In signal work, phase normalized to a period is often a value in [0, 1). This calculator does not solve equations in {x} and does not evaluate sums — it returns {x} for the entered x.
Check examples: {3.75} = 0.75; {9} = 0; {−2.3} = 0.7; {−1.3} = 0.7; {−1.25} = 0.75; {4.1} = 0.1; {0} = 0; {−0.1} = 0.9. The same numbers appear in the clickable examples and the FAQ.
Formula
{x} = x − ⌊x⌋, where x ∈ ℝ and 0 ≤ {x} < 1.
The symbol x is the entered number (the “Number x” field). The symbol ⌊x⌋ is floor: the greatest integer ≤ x. The symbol {x} is the fractional part in the analysis sense. Computation: x − ⌊x⌋.
Mathematical model: floor-based fractional part ({x} = x − ⌊x⌋), following Concrete Mathematics (Graham, Knuth, Patashnik) and the floor/ceiling discussion in NIST DLMF §1.4.
Theory, conventions, and relations
Definition
The fractional part of a real number x, written with curly braces {x}, is defined in analysis and number theory via the floor function: {x} = x − ⌊x⌋. It satisfies 0 ≤ {x} < 1. The identity x = ⌊x⌋ + {x} decomposes every real into an integer part and a fractional part in this sense. Graham, Knuth, and Patashnik give this definition in Concrete Mathematics; MathWorld (Fractional Part) matches it; NIST DLMF §1.4 covers floor and ceiling, from which {x} follows directly.
For nonnegative numbers the result matches the school intuition of “digits after the decimal”: {3.75} = 0.75, {9} = 0. For negatives that intuition diverges from the floor definition. ⌊−1.3⌋ = −2, so {−1.3} = −1.3 − (−2) = 0.7. The value −0.3 appears only under truncation toward zero or under other conventions — not in this model.
In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.
Historical and conceptual context
The fractional part is a basic tool in uniform distribution theory, in Fourier series of periodic functions (the sawtooth), and in hashing and pseudorandom generators. The notation {x} is sometimes also used for a singleton set — on this calculator it always means the fractional part.
Conceptually {x} is not a “remainder on division by 1” in the signed-truncation sense. It is the distance from ⌊x⌋ up the axis, always in [0, 1). A separate floor calculator computes ⌊x⌋; here the primary result is {x}.
In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.
Formal properties
Domain: ℝ. Codomain: [0, 1). The map is periodic with period 1 in the sense that {x + n} = {x} for n ∈ ℤ. On each [n, n + 1) it is a shifted identity: {x} = x − n.
Identities: {x} = 0 ⇔ x ∈ ℤ; {−x} = 0 when x ∈ ℤ, otherwise {−x} = 1 − {x}. Sums: {x} + {y} may exceed 1; then {x + y} = { {x} + {y} }.
Computation: {x} = x − ⌊x⌋. The graph is the classic sawtooth.
In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.
Full derivation
Step 1. From the floor definition: ⌊x⌋ ≤ x < ⌊x⌋ + 1. Subtracting ⌊x⌋: 0 ≤ x − ⌊x⌋ < 1.
Step 2. Set {x} := x − ⌊x⌋. Then x = ⌊x⌋ + {x} with uniqueness under the constraint on {x}.
Step 3 (period 1). ⌊x + n⌋ = ⌊x⌋ + n for n ∈ ℤ, so {x + n} = x + n − ⌊x⌋ − n = {x}.
Step 4 (negatives). For x = −1.3: ⌊x⌋ = −2, {x} = 0.7. “Signed digits after the decimal” (−0.3) is a different operator — the panel may show it for comparison, but the primary result remains 0.7.
In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.
Special cases
Examples: {3.75} = 0.75; {9} = 0; {0.01} = 0.01; {−0.1} = 0.9; {−1.3} = 0.7; {−3} = 0. The distance to the next integer is 1 − {x} when {x} ≠ 0.
When x is an integer, the sawtooth returns to 0 (a jump from values near 1⁻). In floating-point arithmetic, very large |x| may not have an exact fractional part.
In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.
Numerical and computational notes
Computation: x - ⌊x⌋ in floating-point arithmetic. Comma and period are equivalent. The panel shows ceil, the reconstruction ⌊x⌋ + {x}, and competing interpretations for negatives (signed / absolute-value style) — the primary result is always {x} ∈ [0, 1).
The calculator does not compute a remainder on division by an arbitrary m (modulo) — only the fractional part relative to 1.
In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.
Related concepts and authorities
Related: floor, decimal-to-fraction, min–max. For a mod m use dedicated tools.
Authorities: Concrete Mathematics (Knuth et al.); NIST DLMF §1.4; MathWorld — Fractional Part; Wikipedia — Fractional part. Educational tool.
In classroom practice it helps to separate the mathematical formula from floating-point limits: this calculator reports the browser’s floating-point computation (about 15–16 significant digits), not a symbolic identity on paper. When inputs are extreme, check the domain first, then the floating-point range.
How to use
- Enter x in the Number x field, for example 3.75 or −1.3.
- Click Calculate or wait for the live result. 3.75 gives 0.75. 9 gives 0. −1.3 gives 0.7.
- Check the panel: floor, ceil, the reconstruction ⌊x⌋+{x}, and two competing fraction interpretations for negatives.
- The sawtooth diagram marks the point (x, {x}) and the segment from ⌊x⌋ to x on the line.
- Compute absolute value on its own page. Here you split x into integer and fractional parts via floor.
Symbol glossary
The fractional part {x} = x − ⌊x⌋ lies in [0, 1). For −1.3 the result is 0.7, because floor is −2.
- {x}
- Fractional part relative to floor. Always in [0, 1). {3.75} = 0.75; {−1.3} = 0.7.
- x
- Argument in the “Number x” field. Any real, including negatives.
- ⌊x⌋
- Floor: greatest integer ≤ x. ⌊3.75⌋ = 3; ⌊−1.3⌋ = −2.
- ⌈x⌉
- Ceiling: least integer ≥ x. Shown in the panel for comparison.
Examples
Example 2
- x: 9
0
What is {9}? 0.
Example 3
- x: −2.3
0.7
What is {−2.3}? 0.7.
Example 4
- x: −1.3
0.7
What is {−1.3}? 0.7 (not −0.3).
Example 5
- x: −1.25
0.75
What is {−1.25}? 0.75.
Example 6
- x: 4.1
0.1
What is {4.1}? 0.1.
Example 7
- x: 0
0
What is {0}? 0.
Example 8
- x: −0.1
0.9
What is {−0.1}? 0.9.
Related calculators
Frequently asked questions
What is the fractional part {x}?
The definition used here is {x} = x − ⌊x⌋, where ⌊x⌋ is the greatest integer not greater than x (floor toward −∞). The result always lies in [0, 1). For 3.75: ⌊3.75⌋ = 3, so {3.75} = 0.75.
Why is {−1.3} equal to 0.7, not −0.3?
Because floor goes toward minus infinity: ⌊−1.3⌋ = −2, not −1. Hence {−1.3} = −1.3 − (−2) = 0.7. The value −0.3 is a different convention: signed digits after the decimal (x − trunc(x) toward zero). This calculator follows Concrete Mathematics (Graham, Knuth, Patashnik).
What range is the result in?
Always in [0, 1): zero inclusive, one exclusive. An integer gives 0. For 0.99 the result is 0.99; for 1.0 the result is 0, because one is already the next integer.
How does {x} relate to floor, ceil, and mod 1?
We have x = ⌊x⌋ + {x}. Ceiling satisfies ⌈x⌉ = −⌊−x⌋. Under this definition {x} equals the nonnegative residue of x modulo 1. The result panel shows ⌊x⌋, ⌈x⌉, and the reconstruction ⌊x⌋+{x}.
Is this rounding?
No. Rounding 3.75 to the nearest integer may give 4. The fractional part keeps 0.75 and shows the integer part 3 separately. The calculator splits x into ⌊x⌋ and {x}; it does not round.
How is this different from absolute value?
Absolute value |−2.3| = 2.3 measures distance from zero and flips the sign. The floor-based fractional part gives {−2.3} = 0.7. Different formula, different meaning.
What are the fractional parts of 3.75, 9, and −2.3?
3.75 leaves 0.75. 9 leaves 0, because nine is an integer. −2.3 leaves 0.7, because ⌊−2.3⌋ = −3. These match the Fill this example buttons on the page.
What do −1.25, 4.1, 0, and −0.1 give?
−1.25 gives 0.75 (⌊−1.25⌋ = −2). 4.1 gives 0.1. Zero gives 0. −0.1 gives 0.9, because ⌊−0.1⌋ = −1.
Does a comma in 3,75 work?
Yes. 3.75 and 3,75 are the same number. The calculator accepts a period or a comma.
Do the currency and unit settings in the page header affect the result?
No. The result depends only on the entered number x. The currency and unit settings in the page header do not affect this result.
What other definitions exist for negative numbers?
Wikipedia lists three extensions: (1) x − ⌊x⌋ ∈ [0,1) — used here; (2) |x| − ⌊|x|⌋ (digits after the point without sign); (3) the odd function x − ⌈x⌉ for x negative, giving a negative result (e.g. −0.3 for −1.3). The panel shows variants (2) and (3) beside the primary result so you can compare conventions.
What is the sawtooth graph for?
The function {x} is periodic with period 1 and rises linearly from 0 toward 1 on each [n, n+1), hence the sawtooth shape. The live diagram marks the point (x, {x}) and the segment from ⌊x⌋ to x on the number line.
How do I calculate the fractional part?
Subtract floor from the number: {x} = x − ⌊x⌋. The result lies in [0, 1). For 3.75 you get 0.75; for −1.3 you get 0.7, because ⌊−1.3⌋ = −2.
What is the formula for {x}?
{x} = x − ⌊x⌋, where ⌊x⌋ is the greatest integer ≤ x. Equivalently x = ⌊x⌋ + {x}. That is the analysis definition (Graham–Knuth–Patashnik), not signed “digits after the decimal”.
What is the fractional part of π?
About 0.141593, because ⌊π⌋ = 3 and π − 3 ≈ 0.141593. Enter 3.141593 to see the result in the panel.
Can {x} be negative?
Not under this definition. The range is [0, 1). Negative x still yields a non-negative result, because floor goes to −∞ — hence {−1.3} = 0.7.
How does the fractional part differ from the integer part?
The integer part is ⌊x⌋ (floor). The fractional part is the remainder {x} = x − ⌊x⌋. Together they rebuild x. Floor lives on the neighboring integer-part page.
Sources
The definition, the [0, 1) range, and the negative-number conventions come from recognized references.
- Graham, R. L.; Knuth, D. E.; Patashnik, O., Concrete Mathematics, Addison-Wesley — definition {x} = x − ⌊x⌋.
- NIST DLMF §1.4 — Elementary special functions (floor, ceiling)
- MathWorld — Fractional Part
- Wikipedia — Fractional part (three conventions for negatives)
Page Updated November 2, 2026.