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{{Built-in|Imaginary|⌾}} is a [[monadic]] [[scalar function]] which multiplies the [[argument]] by the imaginary unit <source lang=apl inline>0J1</source>. This can be seen as a mapping from a real number to a pure imaginary number (a [[complex number]] with the real part of 0). It was added to [[J]] as <source lang=j inline>j.</source> together with initial support for complex numbers, and was adopted in [[Extended Dyalog APL]] using the [[glyph]] <source lang=apl inline>⌾</source> as a monadic counterpart of [[Complex (function)|Complex]]. In other APL implementations that support complex numbers, [[Circular]] with the left argument of <source lang=apl inline>¯11</source> has the same functionality as Imaginary.
{{Built-in|Imaginary|⌾}} is a [[monadic]] [[scalar function]] which multiplies the [[argument]] by the [[wikipedia:imaginary unit|imaginary unit]] <syntaxhighlight lang=apl inline>0J1</syntaxhighlight>, usually denoted <math>i</math> or <math>j</math> in traditional mathematics. This can be seen as a mapping from a real number to a pure imaginary number (a [[complex number]] with the real part of 0). It was added to [[J]] as <syntaxhighlight lang=j inline>j.</syntaxhighlight> together with initial support for complex numbers, and was adopted in [[Extended Dyalog APL]] using the [[glyph]] <syntaxhighlight lang=apl inline>⌾</syntaxhighlight> as a monadic counterpart of [[Complex (function)|Complex]]. In other APL implementations that support complex numbers, [[Circular]] with the left argument of <syntaxhighlight lang=apl inline>¯11</syntaxhighlight> has the same functionality as Imaginary.


== Examples ==
== Examples ==


[https://tio.run/##SyzI0U2pTMzJT9dNrShJzUtJTfn//1Hf1EdtEx717FMwVji03kTB7P9/AA Try it online!]<source lang=apl>
[https://tio.run/##SyzI0U2pTMzJT9dNrShJzUtJTfn//1Hf1EdtEx717FMwVji03kTB7P9/AA Try it online!]<syntaxhighlight lang=apl>
       ⌾ 3 ¯4 6
       ⌾ 3 ¯4 6
0J3 0J¯4 0J6
0J3 0J¯4 0J6
</source>{{Works in|[[Extended Dyalog APL]]}}
</syntaxhighlight>{{Works in|[[Extended Dyalog APL]]}}


Imaginary is equivalent to [[Complex (function)|Complex]] with the default left argument of 0.
Imaginary is equivalent to [[Complex (function)|Complex]] with the default left argument of 0.


[https://tio.run/##SyzI0U2pTMzJT9dNrShJzUtJTfn//1Hf1EdtEwwUHvXsUzBWOLTeRMHs/38A Try it online!]<source lang=apl>
[https://tio.run/##SyzI0U2pTMzJT9dNrShJzUtJTfn//1Hf1EdtEwwUHvXsUzBWOLTeRMHs/38A Try it online!]<syntaxhighlight lang=apl>
       0 ⌾ 3 ¯4 6
       0 ⌾ 3 ¯4 6
0J3 0J¯4 0J6
0J3 0J¯4 0J6
</source>{{Works in|[[Extended Dyalog APL]]}}
</syntaxhighlight>{{Works in|[[Extended Dyalog APL]]}}


== See also ==
* [[Negate]]
* [[Reciprocal]]
== External links ==
=== Documentation ===
=== Documentation ===


* J [https://www.jsoftware.com/help/dictionary/djdot.htm Dictionary], [https://code.jsoftware.com/wiki/Vocabulary/jdot NuVoc]
* J [https://www.jsoftware.com/help/dictionary/djdot.htm Dictionary], [https://code.jsoftware.com/wiki/Vocabulary/jdot NuVoc]
{{APL built-ins}}[[Category:Primitive functions]][[Category:Scalar monadic functions]]
{{APL built-ins}}[[Category:Primitive functions]][[Category:Scalar monadic functions]]

Latest revision as of 22:05, 10 September 2022

⌾

Imaginary (⌾) is a monadic scalar function which multiplies the argument by the imaginary unit 0J1, usually denoted or in traditional mathematics. This can be seen as a mapping from a real number to a pure imaginary number (a complex number with the real part of 0). It was added to J as j. together with initial support for complex numbers, and was adopted in Extended Dyalog APL using the glyph ⌾ as a monadic counterpart of Complex. In other APL implementations that support complex numbers, Circular with the left argument of ¯11 has the same functionality as Imaginary.

Examples

Try it online!

      ⌾ 3 ¯4 6
0J3 0J¯4 0J6
Works in: Extended Dyalog APL

Imaginary is equivalent to Complex with the default left argument of 0.

Try it online!

      0 ⌾ 3 ¯4 6
0J3 0J¯4 0J6
Works in: Extended 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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