Conjugate: Difference between revisions

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{{Built-in|Conjugate|+}} is a [[monadic]] [[scalar function]] which negates the imaginary component of a [[complex]] argument. Because many APLs do not have a dedicated [[Identity]] function, but also do not support complex numbers, Conjugate is often used to return the argument unchanged. This usage is discouraged in modern APLs in favor of the Identity function (usually <source lang=apl inline>⊢</source>). Conjugate shares the glyph <source lang=apl inline>+</source> with [[Add]].
{{Built-in|Conjugate|+}} is a [[monadic]] [[scalar function]] which negates the imaginary component of a [[complex]] argument. Because many APLs do not have a dedicated [[Identity]] function, but also do not support complex numbers, Conjugate is often used to return the argument unchanged. This usage is discouraged in modern APLs in favor of the Identity function (usually <syntaxhighlight lang=apl inline>⊢</source>). Conjugate shares the glyph <syntaxhighlight lang=apl inline>+</source> with [[Add]].


== Examples ==
== Examples ==


Conjugate leaves real numbers unchanged but negates the imaginary part of complex numbers.
Conjugate leaves real numbers unchanged but negates the imaginary part of complex numbers.
<source lang=apl>
<syntaxhighlight lang=apl>
       + 0 2 ¯1.1 1E300 ¯200
       + 0 2 ¯1.1 1E300 ¯200
0 2 ¯1.1 1E300 ¯200
0 2 ¯1.1 1E300 ¯200
Line 14: Line 14:


In early APLs Conjugate was frequently used to return the argument unchanged, for example to display a [[shy]] result or break up [[stranding]]:
In early APLs Conjugate was frequently used to return the argument unchanged, for example to display a [[shy]] result or break up [[stranding]]:
<source lang=apl>
<syntaxhighlight lang=apl>
       +a←2 3
       +a←2 3
2 3
2 3
Line 20: Line 20:
256 6561
256 6561
</source>
</source>
Sometimes the name "Identity" was even used for <source lang=apl inline>+</source>. Although this usage is becoming rare among new APLers, it may still affect the behavior of Conjugate. For instance, in [[Dyalog APL]], Conjugate will allow a non-numeric argument and return it unchanged even though other monadic scalar functions give a [[DOMAIN ERROR]]:
Sometimes the name "Identity" was even used for <syntaxhighlight lang=apl inline>+</source>. Although this usage is becoming rare among new APLers, it may still affect the behavior of Conjugate. For instance, in [[Dyalog APL]], Conjugate will allow a non-numeric argument and return it unchanged even though other monadic scalar functions give a [[DOMAIN ERROR]]:
<source lang=apl>
<syntaxhighlight lang=apl>
       + ⎕NULL
       + ⎕NULL
[Null]
[Null]
Line 34: Line 34:
Like [[Negate]], Conjugate is its own [[Inverse]].
Like [[Negate]], Conjugate is its own [[Inverse]].


The conjugate of a number is proportional to (that is, a real multiple of) its [[Reciprocal]]. Specifically, since for any complex number <source lang=apl inline>z</source>, <source lang=apl inline>z×+z</source> {{←→}} <source lang=apl inline>(+z)×z</source> is a real number (the square of the [[Magnitude]] of <source lang=apl inline>z</source>), we have <source lang=apl inline>z×((+z)÷z×+z)</source> {{←→}} <source lang=apl inline>1</source>, so by definition <source lang=apl inline>(+z)÷z×+z</source> is <source lang=apl inline>÷z</source>. If <source lang=apl inline>z</source> is a unit complex number (for instance the result of [[Signum]]), then <source lang=apl inline>+z</source> {{←→}} <source lang=apl inline>÷z</source>.
The conjugate of a number is proportional to (that is, a real multiple of) its [[Reciprocal]]. Specifically, since for any complex number <syntaxhighlight lang=apl inline>z</source>, <syntaxhighlight lang=apl inline>z×+z</source> {{←→}} <syntaxhighlight lang=apl inline>(+z)×z</source> is a real number (the square of the [[Magnitude]] of <syntaxhighlight lang=apl inline>z</source>), we have <syntaxhighlight lang=apl inline>z×((+z)÷z×+z)</source> {{←→}} <syntaxhighlight lang=apl inline>1</source>, so by definition <syntaxhighlight lang=apl inline>(+z)÷z×+z</source> is <syntaxhighlight lang=apl inline>÷z</source>. If <syntaxhighlight lang=apl inline>z</source> is a unit complex number (for instance the result of [[Signum]]), then <syntaxhighlight lang=apl inline>+z</source> {{←→}} <syntaxhighlight lang=apl inline>÷z</source>.


== Hypercomplex numbers ==
== Hypercomplex numbers ==


The conjugate of a hypercomplex number (a quaternion or octonion) negates ''all'' imaginary components, that is, every component but the real part. Somewhat surprisingly, this maintains the property that <source lang=apl inline>z×+z</source> {{←→}} <source lang=apl inline>(+z)×z</source> is a real number. Therefore the conjugate can be used to define the [[reciprocal]] of a complex number using only real division (dividing a hypercomplex number by a real number divides each component by that numer).
The conjugate of a hypercomplex number (a quaternion or octonion) negates ''all'' imaginary components, that is, every component but the real part. Somewhat surprisingly, this maintains the property that <syntaxhighlight lang=apl inline>z×+z</source> {{←→}} <syntaxhighlight lang=apl inline>(+z)×z</source> is a real number. Therefore the conjugate can be used to define the [[reciprocal]] of a complex number using only real division (dividing a hypercomplex number by a real number divides each component by that numer).


== Numerical precision ==
== Numerical precision ==

Revision as of 21:56, 10 September 2022

+

Conjugate (+) is a monadic scalar function which negates the imaginary component of a complex argument. Because many APLs do not have a dedicated Identity function, but also do not support complex numbers, Conjugate is often used to return the argument unchanged. This usage is discouraged in modern APLs in favor of the Identity function (usually <syntaxhighlight lang=apl inline>⊢</source>). Conjugate shares the glyph <syntaxhighlight lang=apl inline>+</source> with Add.

Examples

Conjugate leaves real numbers unchanged but negates the imaginary part of complex numbers. <syntaxhighlight lang=apl> 

     + 0 2 ¯1.1 1E300 ¯200

0 2 ¯1.1 1E300 ¯200 

     + 2J3 ¯4J¯3 ¯12 1.5J1.5

2J¯3 ¯4J3 ¯12 1.5J¯1.5 </source>

As identity function

In early APLs Conjugate was frequently used to return the argument unchanged, for example to display a shy result or break up stranding: <syntaxhighlight lang=apl> 

     +a←2 3

2 3 

     *∘2⍣3 +a

256 6561 </source> Sometimes the name "Identity" was even used for <syntaxhighlight lang=apl inline>+</source>. Although this usage is becoming rare among new APLers, it may still affect the behavior of Conjugate. For instance, in Dyalog APL, Conjugate will allow a non-numeric argument and return it unchanged even though other monadic scalar functions give a DOMAIN ERROR: <syntaxhighlight lang=apl> 

     + ⎕NULL

[Null] 

     - ⎕NULL

DOMAIN ERROR 

     -⎕NULL
     ∧

</source>

Properties

Like Negate, Conjugate is its own Inverse.

The conjugate of a number is proportional to (that is, a real multiple of) its Reciprocal. Specifically, since for any complex number <syntaxhighlight lang=apl inline>z</source>, <syntaxhighlight lang=apl inline>z×+z</source> <syntaxhighlight lang=apl inline>(+z)×z</source> is a real number (the square of the Magnitude of <syntaxhighlight lang=apl inline>z</source>), we have <syntaxhighlight lang=apl inline>z×((+z)÷z×+z)</source> <syntaxhighlight lang=apl inline>1</source>, so by definition <syntaxhighlight lang=apl inline>(+z)÷z×+z</source> is <syntaxhighlight lang=apl inline>÷z</source>. If <syntaxhighlight lang=apl inline>z</source> is a unit complex number (for instance the result of Signum), then <syntaxhighlight lang=apl inline>+z</source> <syntaxhighlight lang=apl inline>÷z</source>.

Hypercomplex numbers

The conjugate of a hypercomplex number (a quaternion or octonion) negates all imaginary components, that is, every component but the real part. Somewhat surprisingly, this maintains the property that <syntaxhighlight lang=apl inline>z×+z</source> <syntaxhighlight lang=apl inline>(+z)×z</source> is a real number. Therefore the conjugate can be used to define the reciprocal of a complex number using only real division (dividing a hypercomplex number by a real number divides each component by that numer).

Numerical precision

Like Negate, Conjugate almost never suffers issues with numerical precision. This is because negating a component of a complex number is exact in all common number systems.

See also

External links

Mathematics

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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