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Comparison with traditional mathematics: Difference between revisions
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Comparison with traditional mathematics (view source)
Revision as of 22:22, 10 September 2022
, 22:22, 10 September 2022Text replacement - "</source>" to "</syntaxhighlight>"
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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: | 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 lang=apl inline>((x + y) × (x - y)) = (f ∘ g)(x, y)</syntaxhighlight> | ||
<math>((x + y) \times (x - y)) = (f \circ g)(x, y)</math> | <math>((x + y) \times (x - y)) = (f \circ g)(x, y)</math> | ||
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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> || <syntaxhighlight lang="apl" inline>~ q</ | | <math>\sim q</math> || <syntaxhighlight lang="apl" inline>~ q</syntaxhighlight> || [[Not]] | ||
|- | |- | ||
| <math>p \vee q</math> || <syntaxhighlight lang="apl" inline>p ∨ q</ | | <math>p \vee q</math> || <syntaxhighlight lang="apl" inline>p ∨ q</syntaxhighlight> || [[Or]] | ||
|- | |- | ||
| <math>p \wedge q</math> || <syntaxhighlight lang="apl" inline>p ∧ q</ | | <math>p \wedge q</math> || <syntaxhighlight lang="apl" inline>p ∧ q</syntaxhighlight> || [[And]] | ||
|- | |- | ||
| <math>a \times b</math> || <syntaxhighlight lang="apl" inline>a × b</ | | <math>a \times b</math> || <syntaxhighlight lang="apl" inline>a × b</syntaxhighlight> || [[Multiply]] | ||
|- | |- | ||
| <math>a \div b</math> || <syntaxhighlight lang="apl" inline>a ÷ b</ | | <math>a \div b</math> || <syntaxhighlight lang="apl" inline>a ÷ b</syntaxhighlight> || [[Divide]] | ||
|- | |- | ||
| <math>a \in S</math> || <syntaxhighlight lang="apl" inline>a ∊ S</ | | <math>a \in S</math> || <syntaxhighlight lang="apl" inline>a ∊ S</syntaxhighlight> || [[Membership]] | ||
|- | |- | ||
| <math>A \cup B</math> || <syntaxhighlight lang="apl" inline>A ∪ B</ | | <math>A \cup B</math> || <syntaxhighlight lang="apl" inline>A ∪ B</syntaxhighlight> || [[Union]] | ||
|- | |- | ||
| <math>A \cap B</math> || <syntaxhighlight lang="apl" inline>A ∩ B</ | | <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>[a \equiv b]</math> || <syntaxhighlight lang="apl" inline>(a ≡ b)</ | | <math>[a \equiv b]</math> || <syntaxhighlight lang="apl" inline>(a ≡ b)</syntaxhighlight> || <syntaxhighlight lang="apl" inline>a ≡ b</syntaxhighlight> || [[Match]] | ||
|- | |- | ||
| <math>[a=b]</math> || <syntaxhighlight lang="apl" inline>(a = b)</ | | <math>[a=b]</math> || <syntaxhighlight lang="apl" inline>(a = b)</syntaxhighlight> || <syntaxhighlight lang="apl" inline>a = b</syntaxhighlight> || [[Equal]] | ||
|- | |- | ||
| <math>[a \ne b]</math> || <syntaxhighlight lang="apl" inline>(a ≠ b)</ | | <math>[a \ne b]</math> || <syntaxhighlight lang="apl" inline>(a ≠ b)</syntaxhighlight> || <syntaxhighlight lang="apl" inline>a ≠ b</syntaxhighlight> || [[Not Equal]] | ||
|- | |- | ||
| <math>[a \le b]</math> || <syntaxhighlight lang="apl" inline>(a ≤ b)</ | | <math>[a \le b]</math> || <syntaxhighlight lang="apl" inline>(a ≤ b)</syntaxhighlight> || <syntaxhighlight lang="apl" inline>a ≤ b</syntaxhighlight> || [[Less Than or Equal]] | ||
|- | |- | ||
| <math>(f \circ g)(x)</math> || <syntaxhighlight lang="apl" inline>(f ∘ g)(x)</ | | <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> || <syntaxhighlight lang="apl" inline>f(g(x))</ | | <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> || <syntaxhighlight lang="apl" inline>(f+g)(x)</ | | <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> || <syntaxhighlight lang="apl" inline>○n</ | | <math>n \pi</math> || <syntaxhighlight lang="apl" inline>○n</syntaxhighlight> || [[Pi Times]] | ||
|- | |- | ||
| <math>x!</math> || <syntaxhighlight lang="apl" inline>!x</ | | <math>x!</math> || <syntaxhighlight lang="apl" inline>!x</syntaxhighlight> || [[Factorial]] | ||
|- | |- | ||
| <math>|x|</math> || <syntaxhighlight lang="apl" inline>|x</ | | <math>|x|</math> || <syntaxhighlight lang="apl" inline>|x</syntaxhighlight> || [[Magnitude]] | ||
|- | |- | ||
| <math>\lceil x \rceil</math> || <syntaxhighlight lang="apl" inline>⌈x</ | | <math>\lceil x \rceil</math> || <syntaxhighlight lang="apl" inline>⌈x</syntaxhighlight> || [[Ceiling]] | ||
|- | |- | ||
| <math>\lfloor x \rfloor</math> || <syntaxhighlight lang="apl" inline>⌊x</ | | <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> || <syntaxhighlight lang="apl" inline>a×b</ | | <math>ab</math> || <syntaxhighlight lang="apl" inline>a×b</syntaxhighlight> || [[Multiply]] | ||
|- | |- | ||
| <math>e^x</math> || <syntaxhighlight lang="apl" inline> *x</ | | <math>e^x</math> || <syntaxhighlight lang="apl" inline> *x</syntaxhighlight> || [[Exponential]] | ||
|- | |- | ||
| <math>a^b</math> || <syntaxhighlight lang="apl" inline>a*b</ | | <math>a^b</math> || <syntaxhighlight lang="apl" inline>a*b</syntaxhighlight> || [[Power]] | ||
|- | |- | ||
| <math>\sqrt[a]{b}</math> || <syntaxhighlight lang="apl" inline>a√b</ | | <math>\sqrt[a]{b}</math> || <syntaxhighlight lang="apl" inline>a√b</syntaxhighlight> || [[Root]] | ||
|- | |- | ||
| <math>\tbinom{n}{k}</math> || <syntaxhighlight lang="apl" inline>k!n</ | | <math>\tbinom{n}{k}</math> || <syntaxhighlight lang="apl" inline>k!n</syntaxhighlight> || [[Binomial]] | ||
|- | |- | ||
| <math>f^n(x)</math> || <syntaxhighlight lang="apl" inline>(f⍣n) x</ | | <math>f^n(x)</math> || <syntaxhighlight lang="apl" inline>(f⍣n) x</syntaxhighlight> || [[Power operator]] | ||
|- | |- | ||
| <math>\sin^n(x)</math> || <syntaxhighlight lang="apl" inline>(1○x)*n</ | | <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</ | | <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> || <syntaxhighlight lang="apl" inline>(f⍣n) x</ | | <math>f^n(x)</math> || <syntaxhighlight lang="apl" inline>(f⍣n) x</syntaxhighlight> || [[Power operator]] | ||
|- | |- | ||
| <math>\sin^n(x)</math> || <syntaxhighlight lang="apl" inline>(1○x)*n</ | | <math>\sin^n(x)</math> || <syntaxhighlight lang="apl" inline>(1○x)*n</syntaxhighlight> || [[Circle function]] | ||
|- | |- | ||
| <math>a\pm b</math> || <syntaxhighlight lang="apl" inline>a(+,-)b</ | | <math>a\pm b</math> || <syntaxhighlight lang="apl" inline>a(+,-)b</syntaxhighlight> || [[Train]]s | ||
|- | |- | ||
| <math>a\mp b</math> || <syntaxhighlight lang="apl" inline>a(-,+)b</ | | <math>a\mp b</math> || <syntaxhighlight lang="apl" inline>a(-,+)b</syntaxhighlight> || [[Train]]s | ||
|- | |- | ||
| <math>\frac{a}{b}</math> || <syntaxhighlight lang="apl" inline>a÷b</ | | <math>\frac{a}{b}</math> || <syntaxhighlight lang="apl" inline>a÷b</syntaxhighlight> || [[Divide]] | ||
|- | |- | ||
| <math>e^x</math> || <syntaxhighlight lang="apl" inline> *x</ | | <math>e^x</math> || <syntaxhighlight lang="apl" inline> *x</syntaxhighlight> || [[Exponential]] | ||
|- | |- | ||
| <math>a^b</math> || <syntaxhighlight lang="apl" inline>a*b</ | | <math>a^b</math> || <syntaxhighlight lang="apl" inline>a*b</syntaxhighlight> || [[Power function]] | ||
|- | |- | ||
| <math>\log_a{b}</math> || <syntaxhighlight lang="apl" inline>a⍟b</ | | <math>\log_a{b}</math> || <syntaxhighlight lang="apl" inline>a⍟b</syntaxhighlight> || [[Logarithm]] | ||
|- | |- | ||
| <math>\sqrt[a]{b}</math> || <syntaxhighlight lang="apl" inline>a√b</ | | <math>\sqrt[a]{b}</math> || <syntaxhighlight lang="apl" inline>a√b</syntaxhighlight> || [[Root]] | ||
|- | |- | ||
| <math>\tbinom{n}{k}</math> || <syntaxhighlight lang="apl" inline>k!n</ | | <math>\tbinom{n}{k}</math> || <syntaxhighlight lang="apl" inline>k!n</syntaxhighlight> || [[Binomial]] | ||
|- | |- | ||
| <math>\left\{ | | <math>\left\{ | ||
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b, & \text{if } q(x) | b, & \text{if } q(x) | ||
\end{array}\right.</math> || <syntaxhighlight lang="apl">{p x: a | \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> || <syntaxhighlight lang="apl" inline>+/v</ | | <math>\sum_{i=1}^{N} v_i</math> || <syntaxhighlight lang="apl" inline>+/v</syntaxhighlight> || [[Reduce]] | ||
|- | |- | ||
| <math>\sum_{n=1}^{N} n</math> || <syntaxhighlight lang="apl" inline>+/⍳N</ | | <math>\sum_{n=1}^{N} n</math> || <syntaxhighlight lang="apl" inline>+/⍳N</syntaxhighlight> || [[Reduce]], [[Iota]] | ||
|- | |- | ||
| <math>\prod_{n=1}^{N} n</math> || <syntaxhighlight lang="apl" inline>×/⍳N</ | | <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> || <syntaxhighlight lang="apl" inline> *x</ | | <math>e^x</math> || <syntaxhighlight lang="apl" inline> *x</syntaxhighlight> || [[Exponential]] | ||
|- | |- | ||
| <math>\ln{x}</math> || <syntaxhighlight lang="apl" inline> ⍟b</ | | <math>\ln{x}</math> || <syntaxhighlight lang="apl" inline> ⍟b</syntaxhighlight> || [[Natural Logarithm]] | ||
|- | |- | ||
| <math>\log_a{b}</math> || <syntaxhighlight lang="apl" inline>a⍟b</ | | <math>\log_a{b}</math> || <syntaxhighlight lang="apl" inline>a⍟b</syntaxhighlight> || [[Logarithm]] | ||
|- | |- | ||
| <math>a \bmod b</math> || <syntaxhighlight lang="apl" inline>b|a</ | | <math>a \bmod b</math> || <syntaxhighlight lang="apl" inline>b|a</syntaxhighlight> || [[Modulus]] | ||
|- | |- | ||
| <math>\left\{ | | <math>\left\{ | ||
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b, & \text{if } q(x) | b, & \text{if } q(x) | ||
\end{array}\right.</math> || <syntaxhighlight lang="apl">{p x: a | \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> || <syntaxhighlight lang="apl" inline>n←42</ | | <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> || <syntaxhighlight lang="apl" inline>+/ v </ | | <math>\sum_{i=1}^{N} v_i</math> || <syntaxhighlight lang="apl" inline>+/ v </syntaxhighlight> | ||
|- | |- | ||
| <math>\sum_{n=1}^{N} n</math> || <syntaxhighlight lang="apl" inline>+/ ⍳N </ | | <math>\sum_{n=1}^{N} n</math> || <syntaxhighlight lang="apl" inline>+/ ⍳N </syntaxhighlight> | ||
|- | |- | ||
| <math>\prod_{n=1}^{N} n</math> || <syntaxhighlight lang="apl" inline>×/ ⍳N </ | | <math>\prod_{n=1}^{N} n</math> || <syntaxhighlight lang="apl" inline>×/ ⍳N </syntaxhighlight> | ||
|- | |- | ||
| <math>\forall x, P(x)</math> || <syntaxhighlight lang="apl" inline>∧/P(x)</ | | <math>\forall x, P(x)</math> || <syntaxhighlight lang="apl" inline>∧/P(x)</syntaxhighlight> | ||
|- | |- | ||
| <math>\exists x, P(x)</math> || <syntaxhighlight lang="apl" inline>∨/P(x)</ | | <math>\exists x, P(x)</math> || <syntaxhighlight lang="apl" inline>∨/P(x)</syntaxhighlight> | ||
|} | |} | ||
== Practical example == | == Practical example == | ||
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! style="width:50%" | Traditional notation !! APL | ! 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</ | | <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</ | | <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</ | | <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> | </div> | ||
{{APL syntax}}[[Category:Essays]] | {{APL syntax}}[[Category:Essays]] | ||