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{{Built-in|Add|+}}, '''Plus''', '''Addition''', or '''Sum''', is a [[dyadic]] [[scalar function]] that [[wikipedia:Addition|adds]] numbers together. As a basic arithmetic operation, Add is subject to the language's [[number]] specification. Add shares the glyph < | {{Built-in|Add|+}}, '''Plus''', '''Addition''', or '''Sum''', is a [[dyadic]] [[scalar function]] that [[wikipedia:Addition|adds]] numbers together. As a basic arithmetic operation, Add is subject to the language's [[number]] specification. Add shares the glyph <syntaxhighlight lang=apl inline>+</syntaxhighlight> with the [[monadic]] function [[Conjugate]], and is closely related to [[Subtract]] (<syntaxhighlight lang=apl inline>-</syntaxhighlight>). | ||
== Examples == | == Examples == | ||
< | <syntaxhighlight lang=apl> | ||
1 2 3 + 2 3 4 | 1 2 3 + 2 3 4 | ||
3 5 7 | 3 5 7 | ||
0.1 + 3 4 5 | 0.1 + 3 4 5 | ||
3.1 4.1 5.1 | 3.1 4.1 5.1 | ||
</ | </syntaxhighlight> | ||
== Properties == | == Properties == | ||
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Addition is [[wikipedia:Commutative property|commutative]] in almost all number systems. It is [[wikipedia:Associative property|associative]] when performed with no rounding, but is unlikely to be associative when rounded to an inexact precision (see [[#Floating point error|Floating point error]]). | Addition is [[wikipedia:Commutative property|commutative]] in almost all number systems. It is [[wikipedia:Associative property|associative]] when performed with no rounding, but is unlikely to be associative when rounded to an inexact precision (see [[#Floating point error|Floating point error]]). | ||
The [[identity element]] for Add is < | The [[identity element]] for Add is <syntaxhighlight lang=apl inline>0</syntaxhighlight>. The [[inverse]] of the function <syntaxhighlight lang=apl inline>n∘+</syntaxhighlight> or the equivalent function <syntaxhighlight lang=apl inline>+∘n</syntaxhighlight> is <syntaxhighlight lang=apl inline>-∘n</syntaxhighlight>. Using [[Commute]] (<syntaxhighlight lang=apl inline>⍨</syntaxhighlight>), we can write <syntaxhighlight lang=apl inline>+⍣¯1</syntaxhighlight> {{←→}} <syntaxhighlight lang=apl inline>-⍨</syntaxhighlight>. | ||
Since adding a number to itself is equivalent to doubling that number, we can express the ''double'' function as < | Since adding a number to itself is equivalent to doubling that number, we can express the ''double'' function as <syntaxhighlight lang=apl inline>+⍨</syntaxhighlight>. | ||
=== Reduction === | === Reduction === | ||
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When using a limited-precision format such as 8-byte floats, the results of addition are rounded in order to fit into the result type. This rounding can cause the results of adding several numbers to be different depending on the order in which they are added, that is, addition fails to be associative: | When using a limited-precision format such as 8-byte floats, the results of addition are rounded in order to fit into the result type. This rounding can cause the results of adding several numbers to be different depending on the order in which they are added, that is, addition fails to be associative: | ||
< | <syntaxhighlight lang=apl> | ||
1e20 + ¯1e20 + 1 | 1e20 + ¯1e20 + 1 | ||
0 | 0 | ||
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(1e20 + ¯1e20) + 1 ⍝ Not equivalent | (1e20 + ¯1e20) + 1 ⍝ Not equivalent | ||
1 | 1 | ||
</ | </syntaxhighlight> | ||
Addition of floating-point numbers may also be subject to [[overflow]], resulting in a [[DOMAIN ERROR]] or an [[infinite]] result. | Addition of floating-point numbers may also be subject to [[overflow]], resulting in a [[DOMAIN ERROR]] or an [[infinite]] result. | ||
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* [http://microapl.com/apl_help/ch_020_020_020.htm APLX] | * [http://microapl.com/apl_help/ch_020_020_020.htm APLX] | ||
* [https://www.jsoftware.com/help/dictionary/d100.htm J Dictionary], [https://code.jsoftware.com/wiki/Vocabulary/plus#dyadic J NuVoc] | * [https://www.jsoftware.com/help/dictionary/d100.htm J Dictionary], [https://code.jsoftware.com/wiki/Vocabulary/plus#dyadic J NuVoc] | ||
* [https://mlochbaum.github.io/BQN/doc/arithmetic.html#basic-arithmetic BQN] | |||
{{APL built-ins}}[[Category:Primitive functions]][[Category:Scalar dyadic functions]] | {{APL built-ins}}[[Category:Primitive functions]][[Category:Scalar dyadic functions]] | ||