Complex (function)

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⌾

Complex (⌾) is a dyadic scalar function which combines its arguments into a complex number having with the left argument as real part and the right argument as imaginary part. It was added to J as <syntaxhighlight lang=j inline>j.</source> together with initial support for complex numbers, and was adopted in Extended Dyalog APL using the glyph <syntaxhighlight lang=apl inline>⌾</source> because it was seen as a type of numerical composition (<syntaxhighlight lang=apl inline>∘</source>) for the circular (<syntaxhighlight lang=apl inline>○</source>) domain. Complex shares its glyph with Imaginary which is equivalent to applying Complex with a left argument of 0.

History

The question of whether to include this function is one of Roger Hui's 50 exercises in APL language design:[1]

Complex numbers can be constructed as ordered pairs of real numbers, similar to how integers can be constructed as ordered pairs of natural numbers and rational numbers as ordered pairs of integers. For complex numbers, <syntaxhighlight lang=j inline>j.</source> plays the same role as <syntaxhighlight lang=apl inline>-</source> for integers and <syntaxhighlight lang=apl inline>÷</source> for rational numbers.

In a later blog post, Hui quoted Adám Brudzewsky:[2]

(…) I’d probably go for <syntaxhighlight lang=apl inline>⊕</source> because:

  • it is more clearly related to <syntaxhighlight lang=apl inline>○</source> (for which the monadic form is also a simple multiplication)
  • it consists of two overstruck basic APL symbols
  • it looks more harmonic (in my eyes)

Oh, and Euler’s formula turns out beautiful (to my taste) with it: <syntaxhighlight lang=apl>0 = 1+*⊕○1

0 = 1+*○⊕1</source>

Hui continues:

On further reflection, I like the <syntaxhighlight lang=apl inline>⊕</source> symbol for <syntaxhighlight lang=apl inline>j←{⍺←0 ⋄ ⍺+0j1×⍵}</source>. If there are existing interpretation of it this one should predominate, if the function is as important as I think it is. (Peer to <syntaxhighlight lang=apl inline>-</source> and <syntaxhighlight lang=apl inline>÷</source>, etc.)

Moreover, complex numbers are commonly represented by magnitude and phase. So if you have a magnitude and phase, how do you get the number so represented? Why <syntaxhighlight lang=apl inline>k←{⍺←1 ⋄ ⍺×*0j1×⍵}</source> (or <syntaxhighlight lang=apl inline>{⍺←1 ⋄ ⍺×*⊕⍵}</source>) of course. And what symbol would you use? <syntaxhighlight lang=apl inline>⊗</source> (Monadic <syntaxhighlight lang=apl inline>⊗</source>, with a default magnitude of 1, gets you a point on the unit circle.)

Examples

Try it online!<syntaxhighlight lang=apl> 

     ¯2 9 5 ⌾ 3 ¯4 6

¯2J3 9J¯4 5J6

</source>

Works in: Extended Dyalog APL

Documentation

References

  1. ↑ Roger Hui. Some Exercises in APL Language Design. §8 Imaginary/Complex and sample answer. Jsoftware. 2016.
  2. ↑ Roger Hui. ⊕ and ⊗. Dyalog Forums. 2021-06-13.
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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