Floor

The greatest integer less than or equal to x: $\lfloor x \rfloor$.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaFloor and ceiling functions↗Floor function↗MathWorldFloorFunction↗WikidataQ56860783↗FungrimFloor✓↗Fungrim entryWolfram LanguageFloor!↗

Floor(x)the greatest integer $\le x$, $\lfloor x \rfloor$.

SageMathenumeratio_floor(x)SymPyfloor(x)↗mpmathfloor(x)↗Oscarfloor(BigInt, x)Juliaenumeratio_floor(x)Rustfloor(x)

Floor(x, step)the greatest multiple of step at or below x.

Typed signatures

Type	Package	Applies
(x: real | signed_infinity | ~oo, step: real?) -> real | signed_infinity | ~oo	analyticreplaces compute-engine	always
(x: real | signed_infinity | ~oo, step: real?) -> real | signed_infinity | ~oo	compute-engine	always
(any, any?) -> any	structuresreplaces analytic	always

Domain: Arithmetic

Details

• The greatest integer $\le x$: $\lfloor x \rfloor$.
• Rounds toward $-\infty$, not toward 0 -- so $\lfloor -3.5 \rfloor = -4$, not $-3$.
• For a non-integer x, $\lceil x \rceil = \lfloor x \rfloor + 1$. See Ceil.
• $\lfloor -x \rfloor = -\lceil x \rceil$.
• Threads element-wise over a list.
• A second argument floors to the nearest multiple of it -- the step needn't be an integer.
• The second argument is a step, a multiple to floor to, as in Wolfram's Floor[x, a]: $\mathrm{Floor}(226, 10) = 220$. Round's second argument is a step too. Round breaks ties to even, as Wolfram does (see Round).

Examples

4 more kept as data, checked against the oracles

Basic #

live ↗

In$\lfloor\frac{7}{2}\rfloor$Out$3$

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In$\lfloor3.7\rfloor$Out$3$

Integers pass through unchanged

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In$\lfloor5\rfloor$Out$5$

Threads element-wise over a list

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In$\lfloor\bigl\lbrack1.2, -1.2, 3.7\bigr\rbrack\rfloor$Out$\bigl\lbrack1, -2, 3\bigr\rbrack$

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In$\lfloor2.4\rfloor$Out$2$

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In$\lfloor2.6\rfloor$Out$2$

Scope #

Stays symbolic

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In$\lfloor x\rfloor$Out$\lfloor x\rfloor$

Exact radicals are decided numerically

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In$\lfloor\sqrt{50}\rfloor$Out$7$

Evaluates at an exact constant: $\lfloor\pi\rfloor = 3$

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In$\lfloor π\rfloor$Out$3$

...and at its negative, $\lfloor -\pi \rfloor = -4$

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In$\lfloor-π\rfloor$Out$-4$

A second argument floors to a multiple of it -- the nearest multiple of 10 below 226

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In$\mathrm{Floor}(226, 10)$Out$220$

...the step needn't be an integer: $3.5 \cdot \lfloor -10.3/3.5 \rfloor = -10.5$

live ↗

In$\mathrm{Floor}(-10.3, 3.5)$Out$-10.5$

...nor rational: $2\pi - e \approx 3.57$ floors to the multiple $5/2$ of $5/4$

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In$\mathrm{Floor}(2π-e, \frac{5}{4})$Out$\frac{5}{2}$

Floors the real and imaginary parts separately

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In$\lfloor5.37-1.3i\rfloor$Out$5-2i$

Threads entrywise over a matrix

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In$\lfloor\bigl\lbrack\bigl\lbrack\frac{1}{2}, -1\bigr\rbrack, \bigl\lbrack0, -\frac{1}{2}\bigr\rbrack\bigr\rbrack\rfloor$Out$\bigl\lbrack\bigl\lbrack0, -1\bigr\rbrack, \bigl\lbrack0, -1\bigr\rbrack\bigr\rbrack$

As a matrix function it agrees with entrywise floor on a triangular matrix

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In$\mathrm{MatrixFunction}(\mathrm{Floor}, \bigl\lbrack\bigl\lbrack\frac{1}{2}, -1\bigr\rbrack, \bigl\lbrack0, -\frac{1}{2}\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack0, -1\bigr\rbrack, \bigl\lbrack0, -1\bigr\rbrack\bigr\rbrack$

An integer is its own floor

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In$\lfloor0\rfloor$Out$0$

Infinity passes through

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In$\lfloor∞\rfloor$Out$∞$

Two-argument floor to a multiple of 2: a branch per multiple, the zero stretch the default

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In$\mathrm{PiecewiseExpand}(\mathrm{Floor}(x, 2), 0\lt x\lt4)$Out$\mathrm{Piecewise}(\bigl\lbrack\bigl\lbrack2, 2≤x\bigr\rbrack\bigr\rbrack, 0)$

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In$\frac{\mathrm{d}\lfloor x\rfloor}{\mathrm{d}x}$Out$0$

Applications #

How many full 5-item packs fit in 17 items

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In$\lfloor\frac{17}{5}\rfloor$Out$3$

Legendre's formula: $100!$ ends in $\lfloor 100/5 \rfloor + \lfloor 100/25 \rfloor = 24$ zeros

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In$\lfloor\frac{100}{5}\rfloor+\lfloor\frac{100}{25}\rfloor$Out$24$

The number of decimal digits of n is $\lfloor \log_{10} n \rfloor + 1$

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In$\lfloor\log_{10}(12\,345)\rfloor+1$Out$5$

The first four digits of $e$

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In$\lfloor1\,000e\rfloor$Out$2\,718$

Extracts a digit of the decimal expansion of a fraction via modular power

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In$\lfloor\frac{1}{997}×10(\mathrm{PowerMod}(10, 999\,999, 997)\bmod997)\rfloor$Out$2$

Properties #

For a non-integer x, $\lceil x \rceil = \lfloor x \rfloor + 1$. See Ceil

live ↗

In$\lfloor\frac{7}{2}\rfloor+1=\lceil\frac{7}{2}\rceil$Out$\mathrm{True}$

$\lfloor -x \rfloor = -\lceil x \rceil$

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In$\lfloor-\frac{7}{2}\rfloor=-\lceil\frac{7}{2}\rceil$Out$\mathrm{True}$

Idempotent: $\lfloor\lfloor x \rfloor\rfloor = \lfloor x \rfloor$, since the inner value is already an integer

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In$\lfloor\lfloor x\rfloor\rfloor$Out$\lfloor x\rfloor$

Largest multiple of 3 not above 16

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In$\mathrm{Floor}(16, 3)$Out$15$

$\lfloor x^2\rfloor$ on $(0, 2)$: a branch per integer, bounded by its square root

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In$\mathrm{PiecewiseExpand}(\lfloor x^2\rfloor, 0\lt x\lt2)$Out$\mathrm{Piecewise}(\bigl\lbrack\bigl\lbrack1, 1≤x\lt2^{1/2}\bigr\rbrack, \bigl\lbrack2, 2^{1/2}≤x\lt3^{1/2}\bigr\rbrack, \bigl\lbrack3, 3^{1/2}≤x\bigr\rbrack\bigr\rbrack, 0)$

Nested floors: the inner one is expanded first, then the outer one over its pieces

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In$\mathrm{PiecewiseExpand}(\lfloor x+\frac{1}{3}+\lfloor1-x/2\rfloor^3\rfloor, -1\lt x\lt1)$Out$\mathrm{Piecewise}(\bigl\lbrack\bigl\lbrack1, 2/3≤x∨-1/3≤x≤0\bigr\rbrack\bigr\rbrack, 0)$

Integers are fixed points

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In$\lfloor1\rfloor$Out$1$

Known value · Wolfram MathematicalFunctionData, Floor

A negative integer is unchanged

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In$\lfloor-1\rfloor$Out$-1$

Known value · Wolfram MathematicalFunctionData, Floor

The imaginary part $-1$ is already an integer

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In$\lfloor-i\rfloor$Out$-i$

Known value · Wolfram MathematicalFunctionData, Floor

Rounds down: $\lfloor 2.3\rfloor = 2$

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In$\lfloor\frac{23}{10}\rfloor$Out$2$

Known value · Wolfram MathematicalFunctionData, Floor

Negative integers are fixed

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In$\lfloor-3\rfloor$Out$-3$

Known value · Wolfram MathematicalFunctionData, Floor

Negatives round away from zero: $\lfloor -2.7\rfloor=-3$

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In$\lfloor-\frac{27}{10}\rfloor$Out$-3$

Known value · Wolfram MathematicalFunctionData, Floor

Complex input: each part rounds down, $2-3i$

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In$\lfloor\frac{23}{10}-ie\rfloor$Out$2-3i$

Known value · Wolfram MathematicalFunctionData, Floor

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In$2\lfloor\frac{2}{3}\rfloor+\lfloor1\rfloor$Out$1$

Known value · Wolfram MathematicalFunctionData, Floor, FunctionalEquations

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In$\lfloor\frac{1}{2}+\frac{\sqrt{10}}{2}\rfloor$Out$2$

Known value · Wolfram MathematicalFunctionData, Floor, FunctionalEquations

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In$\lfloor\sqrt{2}+\sqrt{3}\rfloor$Out$3$

Known value · Wolfram MathematicalFunctionData, Floor, FunctionalEquations

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In$\lfloor1\rfloor+\lfloor\frac{5}{6}\rfloor+\lfloor\frac{7}{6}\rfloor$Out$2$

Known value · Wolfram MathematicalFunctionData, Floor, FunctionalEquations

Possible issues #

Rounds toward $-\infty$, not toward 0: floor(-3.2) is -4, not -3

live ↗

In$\lfloor-3.2\rfloor$Out$-4$

Infinities pass through unchanged

live ↗

In$\lfloor-∞\rfloor$Out$-∞$

About

Examples	48
Checked against	Wolfram Language 46 agree2 differSageMath 38 agree1 unevaluated1 errorSymPy 39 agree1 differ1 unevaluatedmpmath 21 agree1 errorOscar 19 agreeJulia 32 agree4 errorRust 29 agree2 error
Crosswalk	Wikipedia ×2MathWorldWikidataFungrimFungrim entry ×12Wolfram LanguageSageMathSymPympmathOscarJuliaRust

Implementation

mappedoscarpackages/oracle/src/mappings.ts

mappedwolframpackages/oracle/src/mappings.ts

mappedsympypackages/oracle/src/mappings.ts

mappedmpmathpackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedrustpackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

See also: Ceil, Round