Reciprocal: Difference between revisions
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{{Built-in|Reciprocal|÷}} is a [[monadic]] [[scalar function]] which gives the [[wikipedia:Multiplicative inverse|multiplicative inverse]] of a real or [[complex]] number. Reciprocal shares the [[glyph]] < | {{Built-in|Reciprocal|÷}} is a [[monadic]] [[scalar function]] which gives the [[wikipedia:Multiplicative inverse|multiplicative inverse]] of a real or [[complex]] number. Reciprocal shares the [[glyph]] <syntaxhighlight lang=apl inline>÷</source> with the dyadic arithmetic function [[Divide]]. | ||
== Examples == | == Examples == | ||
< | <syntaxhighlight lang=apl> | ||
÷1 2 3 4 5 | ÷1 2 3 4 5 | ||
1 0.5 0.3333333333 0.25 0.2 | 1 0.5 0.3333333333 0.25 0.2 | ||
| Line 21: | Line 21: | ||
== Properties == | == Properties == | ||
The reciprocal of any real or complex number is equal to 1 [[divide]]d by that number. Therefore the monadic < | The reciprocal of any real or complex number is equal to 1 [[divide]]d by that number. Therefore the monadic <syntaxhighlight lang=apl inline>÷</source> can be seen as dyadic <syntaxhighlight lang=apl inline>÷</source> with default left argument of 1. This applies even to the reciprocal of 0; <syntaxhighlight lang=apl inline>÷0</source> and <syntaxhighlight lang=apl inline>1÷0</source> show identical behavior for both <syntaxhighlight lang=apl inline>⎕DIV←0</source> (raising [[DOMAIN ERROR]]) and <syntaxhighlight lang=apl inline>⎕DIV←1</source> (returning 0). | ||
< | <syntaxhighlight lang=apl> | ||
÷1 2 3 4 5 | ÷1 2 3 4 5 | ||
1 0.5 0.3333333333 0.25 0.2 | 1 0.5 0.3333333333 0.25 0.2 | ||
| Line 33: | Line 33: | ||
For any non-zero real or complex numbers, the [[signum]] of reciprocal is equal to the [[conjugate]] of signum, and the [[magnitude]] of reciprocal is equal to the reciprocal of magnitude. | For any non-zero real or complex numbers, the [[signum]] of reciprocal is equal to the [[conjugate]] of signum, and the [[magnitude]] of reciprocal is equal to the reciprocal of magnitude. | ||
< | <syntaxhighlight lang=apl> | ||
(×∘÷ ≡ +∘×)1 2 3 ¯2 0.5 1J2 | (×∘÷ ≡ +∘×)1 2 3 ¯2 0.5 1J2 | ||
1 | 1 | ||
Revision as of 21:44, 10 September 2022
÷
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Reciprocal (÷) is a monadic scalar function which gives the multiplicative inverse of a real or complex number. Reciprocal shares the glyph <syntaxhighlight lang=apl inline>÷</source> with the dyadic arithmetic function Divide.
Examples
<syntaxhighlight lang=apl>
÷1 2 3 4 5
1 0.5 0.3333333333 0.25 0.2
÷¯2 0.5 1J2
¯0.5 2 0.2J¯0.4
÷0
DOMAIN ERROR: Divide by zero
÷0
∧
⎕DIV←1 ⍝ this sets division by 0 to always return 0
÷0
0 </source>
Properties
The reciprocal of any real or complex number is equal to 1 divided by that number. Therefore the monadic <syntaxhighlight lang=apl inline>÷</source> can be seen as dyadic <syntaxhighlight lang=apl inline>÷</source> with default left argument of 1. This applies even to the reciprocal of 0; <syntaxhighlight lang=apl inline>÷0</source> and <syntaxhighlight lang=apl inline>1÷0</source> show identical behavior for both <syntaxhighlight lang=apl inline>⎕DIV←0</source> (raising DOMAIN ERROR) and <syntaxhighlight lang=apl inline>⎕DIV←1</source> (returning 0).
<syntaxhighlight lang=apl>
÷1 2 3 4 5
1 0.5 0.3333333333 0.25 0.2
1÷1 2 3 4 5
1 0.5 0.3333333333 0.25 0.2 </source>
For any non-zero real or complex numbers, the signum of reciprocal is equal to the conjugate of signum, and the magnitude of reciprocal is equal to the reciprocal of magnitude.
<syntaxhighlight lang=apl>
(×∘÷ ≡ +∘×)1 2 3 ¯2 0.5 1J2
1
(|∘÷ ≡ ÷∘|)1 2 3 ¯2 0.5 1J2
1
</source>