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...751:2001]]), and complex numbers are available in many APLs. Usually these numbers are written with a syntax such as <syntaxhighlight lang=apl inline>5j2</syn ...e usually written with a "J" joining the real and imaginary parts. Complex numbers are added component-wise.

...ng=j inline>j.</syntaxhighlight> together with initial support for complex numbers, and was adopted in [[Extended Dyalog APL]] using the [[glyph]] <syntaxhigh ...gers and <syntaxhighlight lang=apl inline>÷</syntaxhighlight> for rational numbers. </blockquote>

...ot have a dedicated [[Identity]] function, but also do not support complex numbers, Conjugate is often used to return the argument unchanged. This usage is di ...te leaves real numbers unchanged but negates the imaginary part of complex numbers.

...uals the original number [[times]] (or [[Divide|divided]] by, for non-zero numbers) its [[Signum|sign]]. For complex numbers, the magnitude is defined as the Euclidean distance from the number 0 on th

...mple scalar). However, mixed arrays with depth one can be formed by mixing numbers and characters. While such an array is [[Simple array|simple]], it is neith

...ion for the built-in function [[Floor]] to accept [[complex number|complex numbers]]. It was originally designed by [[Eugene McDonnell]]<ref>McDonnell, Eugene ...uses of [[Floor]]: calculating the quotient and [[remainder]] between two numbers, and calculating the [[GCD]] via the [[wikipedia:Euclidean algorithm|Euclid

...]. Traditionally, APL provides only one user-visible numeric type, so that numbers round identically regardless of how they are represented, but newer APLs ma ...erformance reasons but converts them automatically on overflow to give all numbers the semantics of 64-bit [[wikipedia:IBM_hexadecimal_floating_point|HFP]] fl

...gh it differs from the native implementation in ordering characters before numbers instead of the opposite.<ref>[[Dfns workspace]]. [http://dfns.dyalog.com/n_ ...be compatible with multiplication in the sense that the product of any two numbers greater than zero is greater than zero.<ref name=bfh/>

...ex (function)|Complex]]. In other APL implementations that support complex numbers, [[Circular]] with the left argument of <syntaxhighlight lang=apl inline>¯

...m''', is a [[dyadic]] [[scalar function]] that [[wikipedia:Addition|adds]] numbers together. As a basic arithmetic operation, Add is subject to the language's ...also be considered an element-wise operation, since each of these types of numbers forms a vector space over the reals. Addition of scalars is always performe

...teger arguments, the GCD is the largest positive number which divides both numbers. If one of the arguments is zero, the GCD function returns the other number ...edia:Gaussian integer|Gaussian integers]], when X and/or Y are [[complex]] numbers).

...common multiple is the smallest positive number which is divisible by both numbers. If one of the arguments is zero, the LCM function returns zero. ...edia:Gaussian integer|Gaussian integers]], when X and/or Y are [[complex]] numbers).

...tion''' and is a part of token formation rather than execution. Only plain numbers are stranded. ...tibility]] with APL and does not allow stranding by juxtaposition even for numbers. Instead, the "ligature" character <code>‿</code> is used for a more expl

Negating a vector of positive numbers. Note that the result array is displayed with a [[high minus]] on each elem ...sign bit because they distribute numbers exponentially. But only positive numbers can be represented as the result of an exponent, so an additional sign bit

...her be parenthesized or cannot be stranded at all. It also allows negative numbers to be used as left [[argument]]s to functions, or as [[operand]]s. * [https://help.dyalog.com/latest/index.htm#Language/Introduction/Variables/Numbers.htm Dyalog]

Ceiling rounds up the given numbers to the nearest integers. The extension for [[complex number|complex numbers]] is derived from [[complex Floor]] via this property.

...mutation by selecting <syntaxhighlight lang=apl inline>k</syntaxhighlight> numbers from the first <syntaxhighlight lang=apl inline>n</syntaxhighlight> [[Index numbers←'23456789TJQKA'

Minimum finds the smaller of two numbers: Together with [[Maximum]], it can clamp an array of numbers to a range (closed interval), here from 0 to 1:

...n]] <math>\Gamma(n)</math>, so that it is defined for real and [[complex]] numbers. Because <math>\Gamma(n)</math> equals <math>(n-1)!</math>, <syntaxhighligh The Gamma function diverges at 0 or negative numbers, so <syntaxhighlight lang=apl inline>!Y</syntaxhighlight> is undefined at n

...is a [[monadic]] [[scalar function]] which returns random or pseudo-random numbers. The name ''Roll'' comes from the analogy of rolling dice. Each [[scalar]] ...to predict algorithm) or taken from the operating system. They way random numbers are generated is controlled by the [[random link]] <syntaxhighlight lang=ap

...[[wikipedia:Remainder|remainder]] of [[divide|division]] between two real numbers. It takes the divisor as the left [[argument]], and the dividend as the rig For negative arguments, one may decide to return non-negative numbers in all cases or follow the sign of the dividend or the divisor. For complex

The rule for tolerant comparison is that numbers <syntaxhighlight lang=apl inline>a</syntaxhighlight> and <syntaxhighlight l ...evaluated intolerantly. This means that the allowed difference between the numbers increases as they become larger in magnitude (a relative tolerance). Compar

...as) Normed Division Algebra] datatypes: Complex, Quaternion, and Octonion numbers, along with several Multi-Precision datatypes, and signed Infinities: ...http://wiki.nars2000.org/index.php?title=Rational_and_VFP_Numbers Rational Numbers]

...Ler for hire, can be a difficult task. This is both because of the limited numbers involved, but also because there is no obvious place for potential employee

...ell, Eugene. [https://www.jsoftware.com/papers/satn40.htm SATN-40: Complex Numbers].</ref>. His suggestion that zero divided by zero should be zero rather tha * SATN-40: [https://www.jsoftware.com/papers/satn40.htm Complex Numbers]. 1981-06-20.

:''This page is about array rank, a number associated with every array. For numbers associated with a function specifying its argument rank, see [[function ran

In [[K]], characters are converted to numbers to participate in arithmetic. This system differs from affine character ari

It is subject to [[comparison tolerance]]: floating-point numbers which are very close together are considered equal, even though they are sl x ← 1 + (⎕CT×0.6)ׯ2 ¯1 0 1 2 ⍝ Numbers close to 1

...s Group to arrays, where an array corresponds to a dictionary with natural numbers for its keys.<ref>[[Marshall Lochbaum]]. [https://mlochbaum.github.io/BQN/c

...|nested array theory]] the elements can only be arrays, and characters and numbers are held to be contained in [[simple scalar]]s, arrays which by convention ...Ls such as APL2 allow any array to be used as an element, including scalar numbers or characters (written with quotes) as well as larger arrays. In order to i

...nclude all arrays of a single type, as well as arrays which mix types like numbers and characters. Arrays which would not be representable in flat array theor

GNU APL supports both 64-bit floats and 64-bit integers, as well as complex numbers whose components are floating-point. The integer type is used when a value

...such as [[Less Than]] is always composed of Booleans. Because Booleans are numbers, arithmetic functions can be used on them directly, for example by [[sum]]m The use of numbers for Booleans also allows functions to be extended from a Boolean domain to

...can store any array, boxes may be used to define arrays which contain both numbers and characters, as well as arrays with a nested structure.

...nships between them. On some implementations with [[complex number|complex numbers]], Circular also includes functions to decompose and assemble a complex num ...with <syntaxhighlight lang=apl inline>5○¯6○Y</syntaxhighlight> for complex numbers.

On APL implementations that support [[complex]] numbers, one can demonstrate [[wikipedia:Euler's identity|Euler's identity]] (with

Floor rounds down the given numbers to the nearest integers. ...nnell]] designed the domain extension of Floor to [[complex number|complex numbers]].<ref>McDonnell, Eugene. [https://www.jsoftware.com/papers/eem/complexfloo

...r numbers before the first interval boundary, index 0 is returned, and for numbers after the last, index 4 is returned.

...ion, using exact rationals when possible and high-precision floating-point numbers otherwise.

| numeric types = [[wikipedia:Common_Lisp#Scalar_types|CL numbers]] including [[complex]] and [[rational]]

=== Negative numbers === ...side is a [[singleton]]). [[APL2]] defined a different extension: negative numbers do not correspond to any element of the right argument, but still indicate

* [[Dyalog '09]] 15: Complex Numbers Implementation Stories ([https://www.dyalog.com/uploads/conference/dyalog09 * Dyalog '11 D05: [https://dyalog.tv/Dyalog11/?v=CkaQQYcxpfM Rational Numbers] ([https://www.dyalog.com/uploads/conference/dyalog11/presentations/D05_rat

For any non-zero real or complex numbers, the [[signum]] of reciprocal is equal to the [[conjugate]] of signum, and

Maximum finds the larger of two numbers:

...t]], that is, a number whose square is the argument. As there are two such numbers for any non-zero argument (negating one possible result gives another), the

...d true cases, it's placed first, and returns a scalar or vector of natural numbers. Each number indicates one of the following groups of statements (separated

...ated as a different type from the same value without the dressing. Complex numbers, matrices, polar coordinate tuples, sets and vectors are implemented in Lan

...ang=apl inline>○</syntaxhighlight>) to left arguments designed for complex numbers. ...lang=apl inline>⎕BIG</syntaxhighlight>, while [[Kap]] uses exact rational numbers by default. [[NARS2000]] supports numerous numeric types, and [[April]] pro

The result of a comparison function on real numbers is determined by their ordering, which falls into one of three possible cas

The choice of which numbers are indices is subjective and must be made by the language designer. In som

...allowed, they indicate that the index of the number is repeated. Negative numbers are never allowed.

...s of APL. Only a single shift state was available (which would switch from numbers and uppercase letters to APL glyphs), so for example <syntaxhighlight lang=

* [https://doi.org/10.1145/97808.97821 "ACORN: APL to C on real numbers"] at [[APL90]]. With [[Bob Bernecky|Robert Bernecky]], Stephen B. Jaffe, an

...integers or booleans, and may [[Array model#Numeric type coercion|coerce]] numbers to a common type to ensure that an array is homogeneous in memory. ...panded]], adding 1-byte and 2-byte integers as well as 16-byte [[complex]] numbers containing the other types (however, Dyalog also allows the user access to

SHARP originally supported only real numbers using double (8-byte) precision. Numbers were stored in one of three types: ...jsoftware.com/papers/satn40.htm SATN-40] describes the addition of complex numbers to SHARP APL.

...ay be considered a representation of sets rather than multisets of natural numbers.

...igin]]-agnostic alternative to [[Index Generator]] for creating a range of numbers. Range is currently implemented in [[NARS2000]] (using the digraph <syntaxh

...McDonnell]]. SATN-40: [https://www.jsoftware.com/papers/satn40.htm Complex Numbers]. 1981-06-20.</ref> and [[Donald McIntyre]].<ref>[[Donald McIntyre|D. B. Mc

* The notion that APL only has two data types: numbers and characters

...xpressions involving exactly the digits 2 0 1 6 in that order to equal the numbers 0 to 100

...ible to make it work on arrays of a higher rank. Applying Rank with larger numbers will produce a function whose rank is higher, but the function's results wi

...on of [[strand notation]] allowing the elements to be any arrays, not just numbers.

...ric data. In the prototype, all characters are converted to spaces and all numbers to zeros. Thus the prototype retains type and structure information but not

G×⍵[1]×⍵[2]÷⍵[3]*2 ⍝ the argument is a vector of numbers

...on by [[Minimum]] or [[Maximum]] gives the minimum or maximum over several numbers. Same goes for [[And]], [[Or]], [[GCD]], [[LCM]], and XOR ([[Not Equal]] on

...erwise counting the number of true values would require converting them to numbers or using conditional logic. For a live cell the sum is 1 more than the numb ..., recent versions of Dyalog APL use [[vector instructions]], enabling many numbers to be added at once.

...ent]]s. In normal cases, the result is identical to the argument, with all numbers replaced by zeros and all characters replaced by spaces, and the substituti

...de.com/problems/add-two-numbers/ a LeetCode problem] which asks to add two numbers given as reversed digit lists:

...is an APL program to calculate the average (arithmetic mean) of a list of numbers, written as a [[dfn]]: ...refers to the argument of the function, a list (or 1-dimensional array) of numbers. The <syntaxhighlight lang=apl inline>≢</syntaxhighlight> denotes the [[t

We can convert decimal numbers to fractions. For example, we can convert <math>2.625</math> to the imprope +.×⍨2 3 5 7 11 13 17 ⍝ numbers 2 through 17 without those appearing in their multiplication table are prim

Since we will be using binary numbers to index arrays, it is more convenient for us to have arrays start at 0.

The last step is to round these numbers down. Traditional mathematics writes ''floor'' as <math>\lfloor x \rfloor</ ...zero [[axes]]) paired up with other individual numbers. As pairing up the numbers in <syntaxhighlight lang=apl inline>3 1 4</syntaxhighlight> with those in <

* [[APL90]]: [https://doi.org/10.1145/97808.97821 ACORN: APL to C on real numbers] ([https://www.snakeisland.com/acorn.htm download]). With [[Charles Brenner

...t, APL being a language well-suited to the manipulation of large arrays of numbers. As technology was moving from the mainframe to distributed systems, there

* [[Dyalog '09]] 19: [https://dyalog.tv/Dyalog09/?v=8ic0ctto1HU Complex Numbers]

...ich was used internally for the primitive) was was wrong on floating point numbers, due to some false assumptions about [[Index of]],<ref>[[Roger Hui|Hui, Rog

...ge.com/rooms/52405/conversation/lesson-43-complex-numbers-in-depth Complex numbers in depth] || Jun 2, 2020

...at it permits [[mixed array]]s, so while in J an array containing a mix of numbers, characters, and boxes cannot be mixed, in Dyalog any array at all can be m

...>0</syntaxhighlight> indicating numbers. Indeed, it is the "vector without numbers", equivalent to <syntaxhighlight lang=apl inline>0~0</syntaxhighlight>. ...ng=apl inline>⍒</syntaxhighlight> pointing down is on the lower of the two numbers 3 and 4. This can be remembered as the pound going down due to [[wikipedia:

| <code>∠</code> || Atangent || Take the arctangent of two numbers || atan | <code>⇡</code> || [[Index Generator|Range]] || Make an array of all natural numbers less than a number

S←,⍎{⍞←⍵ ⋄ (≢⍵)↓⍞}'Please, enter 11 numbers: '

...the operator. See [[Rank]] for the number associated with every array. For numbers associated with a function specifying its argument rank, see [[function ran

...bility in the [[Power]] function was broken: fractional powers of negative numbers, such as <syntaxhighlight lang=apl inline>¯8 * ÷3</syntaxhighlight>, coul Support was added for 128-bit [[decimal floating point]] numbers (from [[wikipedia:IEEE 754|IEEE 754]]-2008), using the [[wikipedia:densely

...s the generalization of [[strand notation]] to any arrays rather than just numbers. APL2 was released in 1984, and Brown continued to work at IBM on design as

...syntax colouring, TCPSocket object, OLE client/server, automatic file tie numbers ...character arrays are always stored using one of the above types. For both numbers and characters, an array may be represented using any type that can contain

Using similar techniques, we can define a function that generates prime numbers by using an outer product of [[Residue]].

...g [[Del]]s (<syntaxhighlight lang=apl inline>∇</syntaxhighlight>) and line numbers, after having such a definition has been made.<ref>[[Dyalog Ltd.]] Language

| 8 || [https://problems.tryapl.org/psets/2020.html?goto=P8_Zigzag Zigzag Numbers] || [https://chat.stackexchange.com/transcript/52405?m=64081060#64081060 Ju

|rowspan=2| 2. || Ascending cardinal numbers (ranking, shareable) ||style="text-align: right;"|<syntaxhighlight lang=apl |rowspan=2| 19. || Ascending cardinal numbers (ranking, all different) ||style="text-align: right;"|<syntaxhighlight lang

* '''Let the Numbers Do the Talking—with MAGIC'''. David Keith.

...problems faster. (The complete code for calculating the mean of a list of numbers requires 3 characters.) Ironically, experience shows APLs are easier for no

* '''15: Complex Numbers Implementation Stories''', [[Roger Hui]] ([https://www.dyalog.com/uploads/c * '''19: Complex Numbers''', [[John Scholes]] ([https://dyalog.tv/Dyalog09/?v=8ic0ctto1HU video (65

It is good programming style to avoid using numbers in code. Instead of talking about 1001, for example, we should use a meanin

* '''Hospital administration of large numbers of multi-trauma patients'''. Mitchell Snyder, Bruno Lunenfeld, and Daniel L * '''ACORN: APL to C on real numbers'''. [[Bob Bernecky|Robert Bernecky]], [[Charles Brenner]], Stephen B. Jaffe