Floor: Difference between revisions
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{{Built-in|Floor|⌊}} is a [[monadic]] [[scalar function]] that gives the [[wikipedia:floor and ceiling functions|floor]] of a real number, that is, the greatest integer [[Comparison tolerance|tolerantly]] [[less than or equal to]] the given value. This operation is also known as '''integral part''', '''entier''', and '''round down'''. Floor shares the [[glyph]] < | {{Built-in|Floor|⌊}} is a [[monadic]] [[scalar function]] that gives the [[wikipedia:floor and ceiling functions|floor]] of a real number, that is, the greatest integer [[Comparison tolerance|tolerantly]] [[less than or equal to]] the given value. This operation is also known as '''integral part''', '''entier''', and '''round down'''. Floor shares the [[glyph]] <syntaxhighlight lang=apl inline>⌊</source> with the dyadic arithmetic function [[Minimum]]. [[Comparison_with_traditional_mathematics#Prefix|Traditional mathematics]] derives [[Ken_Iverson#Floor_and_Ceiling|its notation]] and name for floor from APL. | ||
== Examples == | == Examples == | ||
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Floor rounds down the given numbers to the nearest integers. | Floor rounds down the given numbers to the nearest integers. | ||
< | <syntaxhighlight lang=apl> | ||
⌊2 2.8 ¯2 ¯2.8 | ⌊2 2.8 ¯2 ¯2.8 | ||
2 2 ¯2 ¯3 | 2 2 ¯2 ¯3 | ||
| Line 12: | Line 12: | ||
Rounding to the ''nearest'' integer (rounding up on half) can be achieved by [[add|adding]] 0.5 before applying Floor. | Rounding to the ''nearest'' integer (rounding up on half) can be achieved by [[add|adding]] 0.5 before applying Floor. | ||
< | <syntaxhighlight lang=apl> | ||
⌊0.5+2 2.3 2.5 2.8 | ⌊0.5+2 2.3 2.5 2.8 | ||
2 2 3 3 | 2 2 3 3 | ||
| Line 19: | Line 19: | ||
Integral quotient of division can be found with [[divide|division]] followed by Floor. | Integral quotient of division can be found with [[divide|division]] followed by Floor. | ||
< | <syntaxhighlight lang=apl> | ||
⌊10 20 30÷3 | ⌊10 20 30÷3 | ||
3 6 10 | 3 6 10 | ||
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Floor is affected by [[comparison tolerance]]. If the given number is [[tolerant comparison|tolerantly equal]] to its [[ceiling]], it is rounded to that number instead. | Floor is affected by [[comparison tolerance]]. If the given number is [[tolerant comparison|tolerantly equal]] to its [[ceiling]], it is rounded to that number instead. | ||
< | <syntaxhighlight lang=apl> | ||
⎕PP←16 | ⎕PP←16 | ||
⎕←v←1+0.6×⎕CTׯ2 ¯1 0 | ⎕←v←1+0.6×⎕CTׯ2 ¯1 0 | ||
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[[Eugene McDonnell]] designed the domain extension of Floor to [[complex number|complex numbers]].<ref>McDonnell, Eugene. [https://www.jsoftware.com/papers/eem/complexfloor.htm "Complex Floor"].</ref> Complex floor maps every complex number to a [[wikipedia:Gaussian integer|Gaussian integer]], a complex number whose real and imaginary parts are integers. It has an important property that the [[magnitude]] of [[subtract|difference]] between any complex number Z and its floor is [[less than]] 1. This extension is currently implemented in [[Dyalog APL]], [[J]], and [[NARS2000]], and is internally used to implement complex [[ceiling]], [[residue]], and [[GCD]]. | [[Eugene McDonnell]] designed the domain extension of Floor to [[complex number|complex numbers]].<ref>McDonnell, Eugene. [https://www.jsoftware.com/papers/eem/complexfloor.htm "Complex Floor"].</ref> Complex floor maps every complex number to a [[wikipedia:Gaussian integer|Gaussian integer]], a complex number whose real and imaginary parts are integers. It has an important property that the [[magnitude]] of [[subtract|difference]] between any complex number Z and its floor is [[less than]] 1. This extension is currently implemented in [[Dyalog APL]], [[J]], and [[NARS2000]], and is internally used to implement complex [[ceiling]], [[residue]], and [[GCD]]. | ||
< | <syntaxhighlight lang=apl> | ||
v←1.8J2.5 2.2J2.5 2.5J2.2 2.5J1.8 | v←1.8J2.5 2.2J2.5 2.5J2.2 2.5J1.8 | ||
⌊v | ⌊v | ||
Revision as of 22:13, 10 September 2022
⌊
|
Floor (⌊) is a monadic scalar function that gives the floor of a real number, that is, the greatest integer tolerantly less than or equal to the given value. This operation is also known as integral part, entier, and round down. Floor shares the glyph <syntaxhighlight lang=apl inline>⌊</source> with the dyadic arithmetic function Minimum. Traditional mathematics derives its notation and name for floor from APL.
Examples
Floor rounds down the given numbers to the nearest integers.
<syntaxhighlight lang=apl>
⌊2 2.8 ¯2 ¯2.8
2 2 ¯2 ¯3 </source>
Rounding to the nearest integer (rounding up on half) can be achieved by adding 0.5 before applying Floor.
<syntaxhighlight lang=apl>
⌊0.5+2 2.3 2.5 2.8
2 2 3 3 </source>
Integral quotient of division can be found with division followed by Floor.
<syntaxhighlight lang=apl>
⌊10 20 30÷3
3 6 10 </source>
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.
<syntaxhighlight lang=apl>
⎕PP←16
⎕←v←1+0.6×⎕CTׯ2 ¯1 0
0.999999999999988 0.999999999999994 1
⌊v
0 1 1 </source>
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, a complex number whose real and imaginary parts are integers. 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, and is internally used to implement complex ceiling, residue, and GCD.
<syntaxhighlight lang=apl>
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
</source>
External links
Documentation
References
- ↑ McDonnell, Eugene. "Complex Floor".