Reciprocal: Difference between revisions

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{{Built-in|Reciprocal|÷}} is a [[monadic]] [[scalar function]] which gives the [[wikipedia:Multiplicative inverse|multiplicative inverse]] of a real or [[complex]] number. Reciprocal shares the [[glyph]] <source lang=apl inline>÷</source> with the dyadic arithmetic function [[Divide]].
{{Built-in|Reciprocal|÷}} is a [[monadic]] [[scalar function]] which gives the [[wikipedia:Multiplicative inverse|multiplicative inverse]] of a real or [[complex]] number. Reciprocal shares the [[glyph]] <syntaxhighlight lang=apl inline>÷</syntaxhighlight> with the dyadic arithmetic function [[Divide]].


== Examples ==
== Examples ==
<source lang=apl>
<syntaxhighlight lang=apl>
       ÷1 2 3 4 5
       ÷1 2 3 4 5
1 0.5 0.3333333333 0.25 0.2
1 0.5 0.3333333333 0.25 0.2
Line 17: Line 17:
       ÷0
       ÷0
0
0
</source>
</syntaxhighlight>


== Properties ==
== Properties ==


The reciprocal of any real or complex number is equal to 1 [[divide]]d by that number. Therefore the monadic <source lang=apl inline>÷</source> can be seen as dyadic <source lang=apl inline>÷</source> with default left argument of 1. This applies even to the reciprocal of 0; <source lang=apl inline>÷0</source> and <source lang=apl inline>1÷0</source> show identical behavior for both <source lang=apl inline>⎕DIV←0</source> (raising [[DOMAIN ERROR]]) and <source lang=apl inline>⎕DIV←1</source> (returning 0).
The reciprocal of any real or complex number is equal to 1 [[divide]]d by that number. Therefore the monadic <syntaxhighlight lang=apl inline>÷</syntaxhighlight> can be seen as dyadic <syntaxhighlight lang=apl inline>÷</syntaxhighlight> with default left argument of 1. This applies even to the reciprocal of 0; <syntaxhighlight lang=apl inline>÷0</syntaxhighlight> and <syntaxhighlight lang=apl inline>1÷0</syntaxhighlight> show identical behavior for both <syntaxhighlight lang=apl inline>⎕DIV←0</syntaxhighlight> (raising [[DOMAIN ERROR]]) and <syntaxhighlight lang=apl inline>⎕DIV←1</syntaxhighlight> (returning 0).


<source lang=apl>
<syntaxhighlight lang=apl>
       ÷1 2 3 4 5
       ÷1 2 3 4 5
1 0.5 0.3333333333 0.25 0.2
1 0.5 0.3333333333 0.25 0.2
Line 29: Line 29:
       1÷1 2 3 4 5
       1÷1 2 3 4 5
1 0.5 0.3333333333 0.25 0.2
1 0.5 0.3333333333 0.25 0.2
</source>
</syntaxhighlight>


For any non-zero real or complex numbers, the [[signum]] of reciprocal is equal to the [[conjugate]] of signum, and the [[magnitude]] of reciprocal is equal to the reciprocal of magnitude.
For any non-zero real or complex numbers, the [[signum]] of reciprocal is equal to the [[conjugate]] of signum, and the [[magnitude]] of reciprocal is equal to the reciprocal of magnitude.


<source lang=apl>
<syntaxhighlight lang=apl>
       (×∘÷ ≡ +∘×)1 2 3 ¯2 0.5 1J2
       (×∘÷ ≡ +∘×)1 2 3 ¯2 0.5 1J2
1
1
Line 39: Line 39:
       (|∘÷ ≡ ÷∘|)1 2 3 ¯2 0.5 1J2
       (|∘÷ ≡ ÷∘|)1 2 3 ¯2 0.5 1J2
1
1
</source>{{Works in|[[Dyalog APL]]}}
</syntaxhighlight>{{Works in|[[Dyalog APL]]}}


== See also ==
== See also ==
Line 49: Line 49:
=== Documentation ===
=== Documentation ===


* [http://help.dyalog.com/17.1/#Language/Primitive%20Functions/Reciprocal.htm Dyalog]
* [https://help.dyalog.com/17.1/#Language/Primitive%20Functions/Reciprocal.htm Dyalog]
* [https://www.jsoftware.com/help/dictionary/d130.htm J Dictionary], [https://code.jsoftware.com/wiki/Vocabulary/percent NuVoc]
* [https://www.jsoftware.com/help/dictionary/d130.htm J Dictionary], [https://code.jsoftware.com/wiki/Vocabulary/percent NuVoc]
* [https://mlochbaum.github.io/BQN/doc/arithmetic.html#basic-arithmetic BQN]
{{APL built-ins}}[[Category:Primitive functions]][[Category:Scalar monadic functions]]
{{APL built-ins}}[[Category:Primitive functions]][[Category:Scalar monadic functions]]

Latest revision as of 21:56, 10 September 2022

÷

Reciprocal (÷) is a monadic scalar function which gives the multiplicative inverse of a real or complex number. Reciprocal shares the glyph ÷ with the dyadic arithmetic function Divide.

Examples

      ÷1 2 3 4 5
1 0.5 0.3333333333 0.25 0.2
 
      ÷¯2 0.5 1J2
¯0.5 2 0.2J¯0.4

      ÷0
DOMAIN ERROR: Divide by zero
 ÷0                         
 ∧
 
      ⎕DIV←1  ⍝ this sets division by 0 to always return 0
      ÷0
0

Properties

The reciprocal of any real or complex number is equal to 1 divided by that number. Therefore the monadic ÷ can be seen as dyadic ÷ with default left argument of 1. This applies even to the reciprocal of 0; ÷0 and 1÷0 show identical behavior for both ⎕DIV←0 (raising DOMAIN ERROR) and ⎕DIV←1 (returning 0). 

      ÷1 2 3 4 5
1 0.5 0.3333333333 0.25 0.2

      1÷1 2 3 4 5
1 0.5 0.3333333333 0.25 0.2

For any non-zero real or complex numbers, the signum of reciprocal is equal to the conjugate of signum, and the magnitude of reciprocal is equal to the reciprocal of magnitude. 

      (×∘÷ ≡ +∘×)1 2 3 ¯2 0.5 1J2
1

      (|∘÷ ≡ ÷∘|)1 2 3 ¯2 0.5 1J2
1
Works in: Dyalog APL

See also

External links

Documentation

APL built-ins [edit]
Primitives (Timeline) Functions
Scalar
Monadic Conjugate ∙ Negate ∙ Signum ∙ Reciprocal ∙ Magnitude ∙ Exponential ∙ Natural Logarithm ∙ Floor ∙ Ceiling ∙ Factorial ∙ Not ∙ Pi Times ∙ Roll ∙ Type ∙ Imaginary ∙ Square Root ∙ Round
Dyadic Add ∙ Subtract ∙ Times ∙ Divide ∙ Residue ∙ Power ∙ Logarithm ∙ Minimum ∙ Maximum ∙ Binomial ∙ Comparison functions ∙ Boolean functions (And, Or, Nand, Nor) ∙ GCD ∙ LCM ∙ Circular ∙ Complex ∙ Root
Non-Scalar
Structural Shape ∙ Reshape ∙ Tally ∙ Depth ∙ Ravel ∙ Enlist ∙ Table ∙ Catenate ∙ Reverse ∙ Rotate ∙ Transpose ∙ Raze ∙ Mix ∙ Split ∙ Enclose ∙ Nest ∙ Cut (K) ∙ Pair ∙ Link ∙ Partitioned Enclose ∙ Partition
Selection First ∙ Pick ∙ Take ∙ Drop ∙ Unique ∙ Identity ∙ Stop ∙ Select ∙ Replicate ∙ Expand ∙ Set functions (Intersection ∙ Union ∙ Without) ∙ Bracket indexing ∙ Index ∙ Cartesian Product ∙ Sort
Selector Index generator ∙ Grade ∙ Index Of ∙ Interval Index ∙ Indices ∙ Deal ∙ Prefix and suffix vectors
Computational Match ∙ Not Match ∙ Membership ∙ Find ∙ Nub Sieve ∙ Encode ∙ Decode ∙ Matrix Inverse ∙ Matrix Divide ∙ Format ∙ Execute ∙ Materialise ∙ Range
Operators Monadic Each ∙ Commute ∙ Constant ∙ Replicate ∙ Expand ∙ Reduce ∙ Windowed Reduce ∙ Scan ∙ Outer Product ∙ Key ∙ I-Beam ∙ Spawn ∙ Function axis ∙ Identity (Null, Ident)
Dyadic 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)
Quad names Index origin ∙ Comparison tolerance ∙ Migration level ∙ Atomic vector
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