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
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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
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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]
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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 ==

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