Difference between revisions of "Comparison with traditional mathematics"

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APL unifies certain notation for some mathematical constructs, and has a more consistent syntax. Below are some examples comparing traditional mathematical notation (denoted TMN in this article) with equivalent APL. Note that some of these examples only work in certain APL dialects.


APL 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 of the reader. 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:

((x + y) × (x - y)) = (f g)(x, y)

Failed to parse (syntax error): {\displaystyle ((x + y) × (x - y)) = (f ∘ g)(x, y)}

Now note that they are identical, letter for letter. In fact, even the MediaWiki markup code (the language used to define APL Wiki pages) is identical for the two lines identical, except one is tagged as "source" while the other is tagged as "math". 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:

SAX font sample


For many basic operations, APL is identical to TMN:

Traditional notation APL Article
~ q Not
Failed to parse (syntax error): {\displaystyle p∨q} p q Or
Failed to parse (syntax error): {\displaystyle p∧q} p q And
Failed to parse (syntax error): {\displaystyle a×b} a × b Multiply
Failed to parse (syntax error): {\displaystyle a÷b} a ÷ b Divide
a S Membership
Failed to parse (syntax error): {\displaystyle A ∪ B} A B Union
Failed to parse (syntax error): {\displaystyle A ∩ B} A B Intersection

Redundant brackets and parentheses

Often, the only difference is that APL allows omitting brackets or parenthesis. This is because the Iverson bracket is implied and functions have long right scope.

Traditional notation Parenthesised
Failed to parse (syntax error): {\displaystyle [a≡b]} (a b) a b Match
(a = b) a = b Equal
Failed to parse (syntax error): {\displaystyle [a≠b]} (a b) a b Not Equal
Failed to parse (syntax error): {\displaystyle [a≤b]} (a b) a b Less Than or Equal
Failed to parse (syntax error): {\displaystyle (f∘g)(x)} (f g)(x) f g x Bind
f(g(x)) f g x
(f+g)(x) (f+g) x Trains


While based on TMN, APL goes a long way to harmonise and generalise its notation.


In APL, all functions that only take a single argument are placed to the immediate left of their argument. TMN uses prefix, omnifix, and suffix notations.

Traditional notation APL Article
n Pi Times
!x Factorial
|x Magnitude
Failed to parse (syntax error): {\displaystyle ⌈x⌉} x Ceiling
Failed to parse (syntax error): {\displaystyle ⌊x⌋} x Floor


APL writes out all functions explicitly while TMN often implies functions by argument positioning without giving the function an explicit symbol.

Traditional notation APL Article
a×b Multiply
*x Exponential
a*b Power
ab Root
k!n Binomial
(fa) x Power operator
(1x)*n Circle function


APL is strictly linear in the sense that it is written as single lines of uniformly sized and positioned characters, as opposed to TMN which often uses font size and/or position to give meaning.

Traditional notation APL Article
(fa) x Power operator
(1x)*n Circle function
a(+,-)b Trains
a(-,+)b Trains
a÷b Divide
*x Exponential
a*b Power function
ab Logarithm
ab Root
k!n Binomial
{p x: a 
 q x: b}
+/v Reduce
+/N Reduce, Iota
×/N Reduce, Iota

Named functions

APL has no reserved words, and user defined names can have multiple characters. TMN uses multi-character names for many functions and uses many ad-hoc control words.

Traditional notation APL Article
*x Exponential
b Natural Logarithm
ab Logarithm
b|a Modulus
{p x: a 
 q x: b}
n42 Assignment

Other differences

APL harmonises and generalises all folding and dispenses with TMN's counting variables. Every one of the following calculations is a reduction.

Traditional notation APL
+/ v
+/ N
×/ N
Failed to parse (syntax error): {\displaystyle ∀ x, P(x)} /P(x)
Failed to parse (syntax error): {\displaystyle ∃ x, P(x)} /P(x)

APL syntax [edit]
General Comparison with traditional mathematicsPrecedenceTacit programming
Array Numeric literalStringStrand notationObject literalArray notation
Function ArgumentFunction valenceDerived functionDerived operatorNiladic functionMonadic functionDyadic functionAmbivalent functionTradfnDfnFunction train
Operator OperandOperator valenceTradopDopDerived operator
Assignment MultipleIndexedSelectiveModified
Other Function axisBranchQuad nameSystem commandUser commandKeywordDot notationFunction-operator overloading