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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 < | {{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>⊢</syntaxhighlight>). Conjugate shares the glyph <syntaxhighlight lang=apl inline>+</syntaxhighlight> 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. | ||
< | <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 | ||
+ 2J3 ¯4J¯3 ¯12 1.5J1.5 | + 2J3 ¯4J¯3 ¯12 1.5J1.5 | ||
2J¯3 ¯4J3 ¯12 1.5J¯1.5 | 2J¯3 ¯4J3 ¯12 1.5J¯1.5 | ||
</ | </syntaxhighlight> | ||
== As identity function == | == 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]]: | 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 | +a←2 3 | ||
2 3 | 2 3 | ||
*∘2⍣3 +a | *∘2⍣3 +a | ||
256 6561 | 256 6561 | ||
</ | </syntaxhighlight> | ||
Sometimes the name "Identity" was even used for < | Sometimes the name "Identity" was even used for <syntaxhighlight lang=apl inline>+</syntaxhighlight>. 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] | [Null] | ||
| Line 28: | Line 28: | ||
-⎕NULL | -⎕NULL | ||
∧ | ∧ | ||
</ | </syntaxhighlight> | ||
== Properties == | == Properties == | ||
| 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 < | 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</syntaxhighlight>, <syntaxhighlight lang=apl inline>z×+z</syntaxhighlight> {{←→}} <syntaxhighlight lang=apl inline>(+z)×z</syntaxhighlight> is a real number (the square of the [[Magnitude]] of <syntaxhighlight lang=apl inline>z</syntaxhighlight>), we have <syntaxhighlight lang=apl inline>z×((+z)÷z×+z)</syntaxhighlight> {{←→}} <syntaxhighlight lang=apl inline>1</syntaxhighlight>, so by definition <syntaxhighlight lang=apl inline>(+z)÷z×+z</syntaxhighlight> is <syntaxhighlight lang=apl inline>÷z</syntaxhighlight>. If <syntaxhighlight lang=apl inline>z</syntaxhighlight> is a unit complex number (for instance the result of [[Signum]]), then <syntaxhighlight lang=apl inline>+z</syntaxhighlight> {{←→}} <syntaxhighlight lang=apl inline>÷z</syntaxhighlight>. | ||
== 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 < | 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</syntaxhighlight> {{←→}} <syntaxhighlight lang=apl inline>(+z)×z</syntaxhighlight> 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 == | ||
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. | 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 == | |||
* [[Circular]] | |||
* [[Complex (function)|Complex]] | |||
* [[Imaginary]] | |||
== External links == | == External links == | ||
| Line 48: | Line 53: | ||
=== Mathematics === | === Mathematics === | ||
* [[wikipedia: | * [[wikipedia:Complex conjugation|Complex conjugation]] | ||
=== Documentation === | === Documentation === | ||
* [ | * [https://help.dyalog.com/latest/index.htm#Language/Primitive%20Functions/Conjugate.htm Dyalog] | ||
* [http://microapl.com/apl_help/ch_020_020_010.htm APLX] | * [http://microapl.com/apl_help/ch_020_020_010.htm APLX] | ||
| Line 58: | Line 63: | ||
* [https://www.jsoftware.com/help/dictionary/d100.htm J Dictionary], [https://code.jsoftware.com/wiki/Vocabulary/plus J NuVoc] | * [https://www.jsoftware.com/help/dictionary/d100.htm J Dictionary], [https://code.jsoftware.com/wiki/Vocabulary/plus J NuVoc] | ||
{{APL built-ins}}[[Category:Primitive functions]] | * [https://mlochbaum.github.io/BQN/doc/arithmetic.html#basic-arithmetic BQN] | ||
{{APL built-ins}}[[Category:Primitive functions]][[Category:Scalar monadic functions]] | |||