Floor (⌊) is a monadic scalar function which gives the floor of a real number, i.e. the greatest integer not exceeding the given value. This operation is also known as integral part, entier, and round down. Floor shares the glyph ⌊ with the dyadic arithmetic function Minimum.
Examples
Floor rounds down the given numbers to the nearest integers.
⌊2 2.8 ¯2 ¯2.8
2 2 ¯2 ¯3
Rounding to the nearest integer (rounding up on half) can be achieved by adding 0.5 before applying Floor.
⌊0.5+2 2.3 2.5 2.8
2 2 3 3
Integral quotient of division can be found with division followed by Floor.
⌊10 20 30÷3
3 6 10
Properties
The floor of any real number is an integer.
Floor is affected by comparison tolerance. If the given number is tolerantly equal to its ceiling, it is rounded to that number instead.
⎕PP←16
⊢v←1+0.6×⎕CTׯ2 ¯1 0
0.999999999999988 0.999999999999994 1
⌊v
0 1 1
Complex floor
- Main article: Complex Floor
Eugene McDonnell designed the domain extension of Floor to complex numbers.[1] Complex floor maps every complex number to a Gaussian integer. It has an important property that the magnitude of difference between any complex number Z and its floor is less than 1. This extension is currently implemented in Dyalog APL, J, and NARS2000.
v←1.8J2.5 2.2J2.5 2.5J2.2 2.5J1.8
⌊v
2J2 2J2 2J2 2J2
1>|v-⌊v
1 1 1 1
External links
Documentation
| APL built-ins [edit]
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| Primitives (Timeline) |
Functions
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| Scalar
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| Monadic
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Conjugate ∙ Negate ∙ Signum ∙ Reciprocal ∙ Magnitude ∙ Exponential ∙ Natural Logarithm ∙ Floor ∙ Ceiling ∙ Factorial ∙ Not ∙ Pi Times ∙ Roll ∙ Type ∙ Imaginary ∙ Square Root ∙ Round
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| Dyadic
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Add ∙ Subtract ∙ Times ∙ Divide ∙ Residue ∙ Power ∙ Logarithm ∙ Minimum ∙ Maximum ∙ Binomial ∙ Comparison functions ∙ Boolean functions (And, Or, Nand, Nor) ∙ GCD ∙ LCM ∙ Circular ∙ Complex ∙ Root
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| Non-Scalar
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| Structural
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Shape ∙ Reshape ∙ Tally ∙ Depth ∙ Ravel ∙ Enlist ∙ Table ∙ Catenate ∙ Reverse ∙ Rotate ∙ Transpose ∙ Raze ∙ Mix ∙ Split ∙ Enclose ∙ Nest ∙ Cut (K) ∙ Pair ∙ Link ∙ Partitioned Enclose ∙ Partition
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| Selection
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First ∙ Pick ∙ Take ∙ Drop ∙ Unique ∙ Identity ∙ Stop ∙ Select ∙ Replicate ∙ Expand ∙ Set functions (Intersection ∙ Union ∙ Without) ∙ Bracket indexing ∙ Index ∙ Cartesian Product ∙ Sort
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| Selector
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Index generator ∙ Grade ∙ Index Of ∙ Interval Index ∙ Indices ∙ Deal ∙ Prefix and suffix vectors
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| Computational
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Match ∙ Not Match ∙ Membership ∙ Find ∙ Nub Sieve ∙ Encode ∙ Decode ∙ Matrix Inverse ∙ Matrix Divide ∙ Format ∙ Execute ∙ Materialise ∙ Range
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| Operators |
Monadic
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Each ∙ Commute ∙ Constant ∙ Replicate ∙ Expand ∙ Reduce ∙ Windowed Reduce ∙ Scan ∙ Outer Product ∙ Key ∙ I-Beam ∙ Spawn ∙ Function axis ∙ Identity (Null, Ident)
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| Dyadic
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Bind ∙ Compositions (Compose, Reverse Compose, Beside, Withe, Atop, Over) ∙ Inner Product ∙ Determinant ∙ Power ∙ At ∙ Under ∙ Rank ∙ Depth ∙ Variant ∙ Stencil ∙ Cut ∙ Direct definition (operator) ∙ Identity (Lev, Dex)
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| Quad names
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Index origin ∙ Comparison tolerance ∙ Migration level ∙ Atomic vector
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