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Miraheze>Adám Brudzewsky No edit summary |
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APL unifies certain | APL unifies the notations for certain mathematical constructs, while having a more consistent syntax. Below are some examples comparing traditional mathematical notation (denoted TMN in this article) with the equivalent APL. Note that some of these examples only work in certain APL [[dialect]]s. | ||
== Similarities == | == Similarities == | ||
APL [[Iverson notation|developed from TMN]], and much of APL will be immediately recognised by a mathematician. Interestingly, it seems that look of the typeface and identifiers (names of variables etc.) used in APL makes a big difference in the perception | APL [[Iverson notation|developed from TMN]], and much of APL will be immediately recognised by a mathematician. Interestingly, it seems that the look of the typeface and identifiers (names of variables etc.) used in APL makes a big difference in the reader's perception. A fixed-width typewriter-style font with long identifiers gives the impression of computer source code, while an italic serif typeface and single-letter identifiers gives the impression of scientific formulae. Compare the impression you get when reading the following two expressions which is both valid APL (though it has superfluous parentheses) and traditional mathematics: | ||
<div style="text-align:center"> | <div style="text-align:center"> | ||
< | <syntaxhighlight lang=apl inline>((x + y) × (x - y)) = (f ∘ g)(x, y)</syntaxhighlight> | ||
<math>((x + y) | <math>((x + y) \times (x - y)) = (f \circ g)(x, y)</math> | ||
</div> | </div> | ||
Now note that they are identical, letter for letter | Now note that they are identical, letter for letter. Computer Scientists tend to have great difficulty in approaching APL until they let go of the notion that APL is just another programming language (despite its name!) and instead look at it as an executable mathematical notation. | ||
It is notable that APL traditionally was written with a font wherein letters were always italicised, while everything else wasn't. Here is how it looks in the default font for [[SAX]]: | It is notable that APL traditionally was written with a font wherein letters were always italicised, while everything else wasn't. Here is how it looks in the default font for [[SAX]]: | ||
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! style="width:33%" | Traditional notation !! style="width:33%" | APL !! Article | ! style="width:33%" | Traditional notation !! style="width:33%" | APL !! Article | ||
|- | |- | ||
| <math>\sim q</math> | | <math>\sim q</math> || <syntaxhighlight lang="apl" inline>~ q</syntaxhighlight> || [[Not]] | ||
|- | |- | ||
| <math> | | <math>p \vee q</math> || <syntaxhighlight lang="apl" inline>p ∨ q</syntaxhighlight> || [[Or]] | ||
|- | |- | ||
| <math> | | <math>p \wedge q</math> || <syntaxhighlight lang="apl" inline>p ∧ q</syntaxhighlight> || [[And]] | ||
|- | |- | ||
| <math> | | <math>a \times b</math> || <syntaxhighlight lang="apl" inline>a × b</syntaxhighlight> || [[Multiply]] | ||
|- | |- | ||
| <math> | | <math>a \div b</math> || <syntaxhighlight lang="apl" inline>a ÷ b</syntaxhighlight> || [[Divide]] | ||
|- | |- | ||
| <math>a \in S</math> || < | | <math>a \in S</math> || <syntaxhighlight lang="apl" inline>a ∊ S</syntaxhighlight> || [[Membership]] | ||
|- | |- | ||
| <math>A | | <math>A \cup B</math> || <syntaxhighlight lang="apl" inline>A ∪ B</syntaxhighlight> || [[Union]] | ||
|- | |- | ||
| <math>A | | <math>A \cap B</math> || <syntaxhighlight lang="apl" inline>A ∩ B</syntaxhighlight> || [[Intersection]] | ||
|} | |} | ||
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! style="width:25%" | Traditional notation !! style="width:25%" | <small>Parenthesised<br></small>APL !! style="width:25%" | <small>Normal<br></small>APL !! Article | ! style="width:25%" | Traditional notation !! style="width:25%" | <small>Parenthesised<br></small>APL !! style="width:25%" | <small>Normal<br></small>APL !! Article | ||
|- | |- | ||
| <math>[ | | <math>[a \equiv b]</math> || <syntaxhighlight lang="apl" inline>(a ≡ b)</syntaxhighlight> || <syntaxhighlight lang="apl" inline>a ≡ b</syntaxhighlight> || [[Match]] | ||
|- | |- | ||
| <math>[a=b]</math> | | <math>[a=b]</math> || <syntaxhighlight lang="apl" inline>(a = b)</syntaxhighlight> || <syntaxhighlight lang="apl" inline>a = b</syntaxhighlight> || [[Equal]] | ||
|- | |- | ||
| <math>[ | | <math>[a \ne b]</math> || <syntaxhighlight lang="apl" inline>(a ≠ b)</syntaxhighlight> || <syntaxhighlight lang="apl" inline>a ≠ b</syntaxhighlight> || [[Not Equal]] | ||
|- | |- | ||
| <math>[ | | <math>[a \le b]</math> || <syntaxhighlight lang="apl" inline>(a ≤ b)</syntaxhighlight> || <syntaxhighlight lang="apl" inline>a ≤ b</syntaxhighlight> || [[Less Than or Equal]] | ||
|- | |- | ||
| <math>( | | <math>(f \circ g)(x)</math> || <syntaxhighlight lang="apl" inline>(f ∘ g)(x)</syntaxhighlight> || <syntaxhighlight lang="apl" inline>f ∘ g x</syntaxhighlight> || [[Bind]] | ||
|- | |- | ||
| <math>f(g(x))</math> | | <math>f(g(x))</math> || <syntaxhighlight lang="apl" inline>f(g(x))</syntaxhighlight> || <syntaxhighlight lang="apl" inline>f g x</syntaxhighlight> || | ||
|- | |- | ||
| <math>(f+g)(x)</math> || < | | <math>(f+g)(x)</math> || <syntaxhighlight lang="apl" inline>(f+g)(x)</syntaxhighlight> || <syntaxhighlight lang="apl" inline>(f+g) x</syntaxhighlight> || [[Train]]s | ||
|} | |} | ||
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! style="width:33%" | Traditional notation !! style="width:33%" | APL !! Article | ! style="width:33%" | Traditional notation !! style="width:33%" | APL !! Article | ||
|- | |- | ||
| <math>n \pi</math> || < | | <math>n \pi</math> || <syntaxhighlight lang="apl" inline>○n</syntaxhighlight> || [[Pi Times]] | ||
|- | |- | ||
| <math>x!</math> | | <math>x!</math> || <syntaxhighlight lang="apl" inline>!x</syntaxhighlight> || [[Factorial]] | ||
|- | |- | ||
| <math>|x|</math> | | <math>|x|</math> || <syntaxhighlight lang="apl" inline>|x</syntaxhighlight> || [[Magnitude]] | ||
|- | |- | ||
| <math> | | <math>\lceil x \rceil</math> || <syntaxhighlight lang="apl" inline>⌈x</syntaxhighlight> || [[Ceiling]] | ||
|- | |- | ||
| <math> | | <math>\lfloor x \rfloor</math> || <syntaxhighlight lang="apl" inline>⌊x</syntaxhighlight> || [[Floor]] | ||
|} | |} | ||
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! style="width:33%" | Traditional notation !! style="width:33%" | APL !! Article | ! style="width:33%" | Traditional notation !! style="width:33%" | APL !! Article | ||
|- | |- | ||
| <math>ab</math> || < | | <math>ab</math> || <syntaxhighlight lang="apl" inline>a×b</syntaxhighlight> || [[Multiply]] | ||
|- | |- | ||
| <math>e^x</math> || < | | <math>e^x</math> || <syntaxhighlight lang="apl" inline> *x</syntaxhighlight> || [[Exponential]] | ||
|- | |- | ||
| <math>a^b</math> || < | | <math>a^b</math> || <syntaxhighlight lang="apl" inline>a*b</syntaxhighlight> || [[Power]] | ||
|- | |- | ||
| <math>\sqrt[a]{b}</math> || < | | <math>\sqrt[a]{b}</math> || <syntaxhighlight lang="apl" inline>a√b</syntaxhighlight> || [[Root]] | ||
|- | |- | ||
| <math>\tbinom{n}{k}</math> || < | | <math>\tbinom{n}{k}</math> || <syntaxhighlight lang="apl" inline>k!n</syntaxhighlight> || [[Binomial]] | ||
|- | |- | ||
| <math>f^n(x)</math> || < | | <math>f^n(x)</math> || <syntaxhighlight lang="apl" inline>(f⍣n) x</syntaxhighlight> || [[Power operator]] | ||
|- | |- | ||
| <math>\sin^n(x)</math> || < | | <math>\sin^n(x)</math> || <syntaxhighlight lang="apl" inline>(1○x)*n</syntaxhighlight> || [[Circle function]] | ||
|- | |||
| <math>a\cdot b</math> || <syntaxhighlight lang="apl" inline>a+.×b</syntaxhighlight> || [[Inner product]] | |||
|} | |} | ||
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! style="width:33%" | Traditional notation !! style="width:33%" | APL !! Article | ! style="width:33%" | Traditional notation !! style="width:33%" | APL !! Article | ||
|- | |- | ||
| <math>f^n(x)</math> || < | | <math>f^n(x)</math> || <syntaxhighlight lang="apl" inline>(f⍣n) x</syntaxhighlight> || [[Power operator]] | ||
|- | |- | ||
| <math>\sin^n(x)</math> || < | | <math>\sin^n(x)</math> || <syntaxhighlight lang="apl" inline>(1○x)*n</syntaxhighlight> || [[Circle function]] | ||
|- | |- | ||
| <math>a\pm b</math> || < | | <math>a\pm b</math> || <syntaxhighlight lang="apl" inline>a(+,-)b</syntaxhighlight> || [[Train]]s | ||
|- | |- | ||
| <math>a\mp b</math> || < | | <math>a\mp b</math> || <syntaxhighlight lang="apl" inline>a(-,+)b</syntaxhighlight> || [[Train]]s | ||
|- | |- | ||
| <math>\frac{a}{b}</math> || < | | <math>\frac{a}{b}</math> || <syntaxhighlight lang="apl" inline>a÷b</syntaxhighlight> || [[Divide]] | ||
|- | |- | ||
| <math>e^x</math> || < | | <math>e^x</math> || <syntaxhighlight lang="apl" inline> *x</syntaxhighlight> || [[Exponential]] | ||
|- | |- | ||
| <math>a^b</math> || < | | <math>a^b</math> || <syntaxhighlight lang="apl" inline>a*b</syntaxhighlight> || [[Power function]] | ||
|- | |- | ||
| <math>\log_a{b}</math> || < | | <math>\log_a{b}</math> || <syntaxhighlight lang="apl" inline>a⍟b</syntaxhighlight> || [[Logarithm]] | ||
|- | |- | ||
| <math>\sqrt[a]{b}</math> || < | | <math>\sqrt[a]{b}</math> || <syntaxhighlight lang="apl" inline>a√b</syntaxhighlight> || [[Root]] | ||
|- | |- | ||
| <math>\tbinom{n}{k}</math> || < | | <math>\tbinom{n}{k}</math> || <syntaxhighlight lang="apl" inline>k!n</syntaxhighlight> || [[Binomial]] | ||
|- | |- | ||
| <math>\left\{ | | <math>\left\{ | ||
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a, & \text{if } p(x) \\ | a, & \text{if } p(x) \\ | ||
b, & \text{if } q(x) | b, & \text{if } q(x) | ||
\end{array}\right.</math> || < | \end{array}\right.</math> || <syntaxhighlight lang="apl">{p x: a | ||
q x: b}</ | q x: b}</syntaxhighlight> || [[dfn]]s | ||
|- | |- | ||
| <math>\sum_{i=1}^{N} v_i</math> || < | | <math>\sum_{i=1}^{N} v_i</math> || <syntaxhighlight lang="apl" inline>+/v</syntaxhighlight> || [[Reduce]] | ||
|- | |- | ||
| <math>\sum_{n=1}^{N} n</math> || < | | <math>\sum_{n=1}^{N} n</math> || <syntaxhighlight lang="apl" inline>+/⍳N</syntaxhighlight> || [[Reduce]], [[Iota]] | ||
|- | |- | ||
| <math>\prod_{n=1}^{N} n</math> || < | | <math>\prod_{n=1}^{N} n</math> || <syntaxhighlight lang="apl" inline>×/⍳N</syntaxhighlight> || [[Reduce]], [[Iota]] | ||
|} | |} | ||
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! style="width:33%" | Traditional notation !! style="width:33%" | APL !! Article | ! style="width:33%" | Traditional notation !! style="width:33%" | APL !! Article | ||
|- | |- | ||
| <math>e^x</math> || < | | <math>e^x</math> || <syntaxhighlight lang="apl" inline> *x</syntaxhighlight> || [[Exponential]] | ||
|- | |- | ||
| <math>\ln{x}</math> || < | | <math>\ln{x}</math> || <syntaxhighlight lang="apl" inline> ⍟b</syntaxhighlight> || [[Natural Logarithm]] | ||
|- | |- | ||
| <math>\log_a{b}</math> || < | | <math>\log_a{b}</math> || <syntaxhighlight lang="apl" inline>a⍟b</syntaxhighlight> || [[Logarithm]] | ||
|- | |- | ||
| <math>a \ | | <math>a \bmod b</math> || <syntaxhighlight lang="apl" inline>b|a</syntaxhighlight> || [[Modulus]] | ||
|- | |- | ||
| <math>\left\{ | | <math>\left\{ | ||
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a, & \text{if } p(x) \\ | a, & \text{if } p(x) \\ | ||
b, & \text{if } q(x) | b, & \text{if } q(x) | ||
\end{array}\right.</math> || < | \end{array}\right.</math> || <syntaxhighlight lang="apl">{p x: a | ||
q x: b}</ | q x: b}</syntaxhighlight> || [[dfn]]s | ||
|- | |- | ||
| <math>\text{let } n=42</math> || < | | <math>\text{let } n=42</math> || <syntaxhighlight lang="apl" inline>n←42</syntaxhighlight> || [[Assignment]] | ||
|} | |} | ||
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! style="width:50%" | Traditional notation !! APL | ! style="width:50%" | Traditional notation !! APL | ||
|- | |- | ||
| <math>\sum_{i=1}^{N} v_i</math> || < | | <math>\sum_{i=1}^{N} v_i</math> || <syntaxhighlight lang="apl" inline>+/ v </syntaxhighlight> | ||
|- | |- | ||
| <math>\sum_{n=1}^{N} n</math> || < | | <math>\sum_{n=1}^{N} n</math> || <syntaxhighlight lang="apl" inline>+/ ⍳N </syntaxhighlight> | ||
|- | |- | ||
| <math>\prod_{n=1}^{N} n</math> || < | | <math>\prod_{n=1}^{N} n</math> || <syntaxhighlight lang="apl" inline>×/ ⍳N </syntaxhighlight> | ||
|- | |- | ||
| <math> | | <math>\forall x, P(x)</math> || <syntaxhighlight lang="apl" inline>∧/P(x)</syntaxhighlight> | ||
|- | |- | ||
| <math> | | <math>\exists x, P(x)</math> || <syntaxhighlight lang="apl" inline>∨/P(x)</syntaxhighlight> | ||
|} | |} | ||
== Practical example == | |||
{{APL syntax}} | The steps to produce the component of a vector <math>\textbf{a}</math> in the direction of another vector <math>\textbf{b}</math> go as follows: | ||
<div style="text-align:center"> | |||
{| class="wikitable c" style="margin: 1em auto 1em auto" | |||
! style="width:50%" | Traditional notation !! APL | |||
|- | |||
| <math>|\textbf{b}|=\sqrt[2]{\textbf{b}\cdot\textbf{b}}</math> || <syntaxhighlight lang=apl>bNorm ← 2 Root b +.× b</syntaxhighlight> | |||
|- | |||
| <math>\hat{\textbf{b}} = \textbf{b}\div|\textbf{b}|</math> || <syntaxhighlight lang=apl>bHat ← b ÷ bNorm</syntaxhighlight> | |||
|- | |||
| <math>\textbf{a}_\textbf{b}=(\textbf{a}\cdot\hat{\textbf{b}})\hat{\textbf{b}}</math> || <syntaxhighlight lang=apl>a_b ← (a +.× bHat) × bHat</syntaxhighlight> | |||
|} | |||
</div> | |||
{{APL syntax}}[[Category:Essays]] |