Reciprocal

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Reciprocal (÷) is a monadic scalar function which gives the multiplicative inverse of a real or complex number. Reciprocal shares the glyph <syntaxhighlight lang=apl inline>÷</source> with the dyadic arithmetic function Divide.

Examples

<syntaxhighlight lang=apl> 

     ÷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 </source>

Properties

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

<syntaxhighlight lang=apl> 

     ÷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 </source>

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.

<syntaxhighlight lang=apl> 

     (×∘÷ ≡ +∘×)1 2 3 ¯2 0.5 1J2

1 

     (|∘÷ ≡ ÷∘|)1 2 3 ¯2 0.5 1J2

1

</source>

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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