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{{Built-in|Signum|×}}, '''Sign''', '''Sign of''', or '''Direction''' is a [[monadic]] [[primitive function]] which returns the [[wikipedia:Sign function|sign]] of a real or [[complex]] number. That is, it returns 0 when given an argument of 0, and otherwise returns a number with [[magnitude]] 1 given by [[Divide|dividing]] the argument by its own magnitude.
{{Built-in|Signum|×}}, '''Sign''', '''Sign of''', or '''Direction''' is a [[monadic]] [[scalar function]] which returns the [[wikipedia:Sign function|sign]] of a real or [[complex]] number. That is, it returns 0 when given an argument of 0, and otherwise returns a number with [[magnitude]] 1 given by [[Divide|dividing]] the argument by its own magnitude.


== Examples ==
== Examples ==


The three possible results of Signum on a real argument are <source lang=apl inline>0</source>, <source lang=apl inline>1</source>, and <source lang=apl inline>¯1</source>.
The three possible results of Signum on a real argument are <syntaxhighlight lang=apl inline>0</syntaxhighlight>, <syntaxhighlight lang=apl inline>1</syntaxhighlight>, and <syntaxhighlight lang=apl inline>¯1</syntaxhighlight>.
<source lang=apl>
<syntaxhighlight lang=apl>
       × ¯3 0 5
       × ¯3 0 5
¯1 0 1
¯1 0 1
</source>
</syntaxhighlight>
In dialects with [[complex number]]s, Signum is a somewhat more complicated function, and may return any unit complex number.
In dialects with [[complex number]]s, Signum is a somewhat more complicated function, and may return any unit complex number.
<source lang=apl>
<syntaxhighlight lang=apl>
       × 3j4
       × 3j4
0.6J0.8
0.6J0.8
</source>
</syntaxhighlight>
The result is still equal to the original number [[divide]]d by its [[magnitude]]:
The result is still equal to the original number [[divide]]d by its [[magnitude]]:
<source lang=apl>
<syntaxhighlight lang=apl>
       | 3j4
       | 3j4
5
5
       3j4 ÷ | 3j4
       3j4 ÷ | 3j4
0.6J0.8
0.6J0.8
</source>
</syntaxhighlight>
The [[magnitude]] of the result for a non-zero argument is always 1.
The [[magnitude]] of the result for a non-zero argument is always 1.
<source lang=apl>
<syntaxhighlight lang=apl>
       | × 3j4 ¯2j1 6j¯7
       | × 3j4 ¯2j1 6j¯7
1 1 1
1 1 1
</source>
</syntaxhighlight>


== Zero divided by zero ==
== Zero divided by zero ==


The identity <source lang=apl inline>×z</source> {{←→}} <source lang=apl inline>z÷|z</source> holds only when <source lang=apl inline>z</source> is not zero in most APLs. In "Zero Divided by Zero"<ref>[[Eugene McDonnell|McDonnell, Eugene]]. ["https://www.jsoftware.com/papers/eem/0div0.htm" Zero Divided by Zero]. [[APL76]].</ref>, [[Eugene McDonnell]] gave this identity as a reason to define <source lang=apl inline>0÷0</source> to be equal to <source lang=apl inline>0</source>. In [[J]], which took McDonnell's suggestion, the identity always holds.
The identity <syntaxhighlight lang=apl inline>×z</syntaxhighlight> {{←→}} <syntaxhighlight lang=apl inline>z÷|z</syntaxhighlight> holds only when <syntaxhighlight lang=apl inline>z</syntaxhighlight> is not zero in most APLs. In "Zero Divided by Zero"<ref>[[Eugene McDonnell|McDonnell, Eugene]]. [https://www.jsoftware.com/papers/eem/0div0.htm "Zero Divided by Zero"]. [[APL76]].</ref>, [[Eugene McDonnell]] gave this identity as a reason to define <syntaxhighlight lang=apl inline>0÷0</syntaxhighlight> to be equal to <syntaxhighlight lang=apl inline>0</syntaxhighlight>. In [[J]], which took McDonnell's suggestion, the identity always holds. [[Dyalog APL]] and [[NARS2000]] allow choosing [[division method]] though the default remains 1.
 
== See also ==
* [[Negate]]
* [[Conjugate]]


== External links ==
== External links ==
Line 34: Line 38:
=== Documentation ===
=== Documentation ===


* [http://help.dyalog.com/latest/index.htm#Language/Primitive%20Functions/Direction.htm Dyalog]
* [https://help.dyalog.com/latest/index.htm#Language/Primitive%20Functions/Direction.htm Dyalog]
* [http://microapl.com/apl_help/ch_020_020_050.htm APLX]
* [http://microapl.com/apl_help/ch_020_020_050.htm APLX]
* J [https://www.jsoftware.com/help/dictionary/d110.htm Dictionary], [https://code.jsoftware.com/wiki/Vocabulary/star NuVoc]
* J [https://www.jsoftware.com/help/dictionary/d110.htm Dictionary], [https://code.jsoftware.com/wiki/Vocabulary/star NuVoc]
* [https://mlochbaum.github.io/BQN/doc/arithmetic.html#basic-arithmetic BQN]


== References ==
== References ==
<references />
<references />
{{APL built-ins}}
{{APL built-ins}}[[Category:Primitive functions]][[Category:Scalar monadic functions]]

Latest revision as of 21:32, 10 September 2022

×

Signum (×), Sign, Sign of, or Direction is a monadic scalar function which returns the sign of a real or complex number. That is, it returns 0 when given an argument of 0, and otherwise returns a number with magnitude 1 given by dividing the argument by its own magnitude.

Examples

The three possible results of Signum on a real argument are 0, 1, and ¯1. 

      × ¯3 0 5
¯1 0 1

In dialects with complex numbers, Signum is a somewhat more complicated function, and may return any unit complex number. 

      × 3j4
0.6J0.8

The result is still equal to the original number divided by its magnitude: 

      | 3j4
5
      3j4 ÷ | 3j4
0.6J0.8

The magnitude of the result for a non-zero argument is always 1. 

      | × 3j4 ¯2j1 6j¯7
1 1 1

Zero divided by zero

The identity ×z z÷|z holds only when z is not zero in most APLs. In "Zero Divided by Zero"[1], Eugene McDonnell gave this identity as a reason to define 0÷0 to be equal to 0. In J, which took McDonnell's suggestion, the identity always holds. Dyalog APL and NARS2000 allow choosing division method though the default remains 1.

See also

External links

Documentation

References

  1. ↑ McDonnell, Eugene. "Zero Divided by Zero". APL76.
APL built-ins [edit]
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