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t a i v j get a j if i j for any array state a any value v and any tuples i j for which the operations are defined the first axiom means that each element behaves like a variable the second axiom means that elements with distinct indices behave as disjoint variables so that storing a value in one element does not affect the value of any other element these axioms do not place any constraints on the set of valid index tuples i therefore this abstract model can be used for triangular matrices and other oddly shaped arrays implementations edit in order to effectively implement variables of such types as array structures with indexing done by pointer arithmetic many languages restrict the indices to integer data types 6 7 or other types that can be interpreted as integers such as bytes and enumerated types and require that all elements have the same data type and storage size most of those languages also restrict each index to a finite interval of integers that remains fixed throughout the lifetime of the array variable in some compiled languages in fact the index ranges may have to be known at compile time on the other hand some programming languages provide more liberal array types that allow indexing by arbitrary values such as floating point numbers strings objects references etc such index values cannot be restricted to an interval much less a fixed interval so these languages usually allow arbitrary new elements to be created at any time this choice precludes the implementation of array types as array data structures that is those languages use array like syntax to implement a more general associative array semantics and must therefore be implemented by a hash table or some other search data structure language support edit this section needs more citations please help improve this section by adding citations to reliable sources unsourced material may be challenged and removed january 2026 learn how and when to remove this message type edit languages have differing ways of defining an array type for example in c an array is actually a block of contiguous memory which is essentially treated as a pointer 8 in such cases array declarations can decay to pointers int a 10 array a of 10 ints int p a p points to the first element of a void foo int arr the parameter arr is an int arr is decayed to a pointer to its first element however in other languages such as java an array is an actual type for any type t it has a corresponding array type t which is an object with a length field 9 int a new int 5 declares an array a of 5 ints multi dimensional arrays edit a two dimensional array stored as a one dimensional array of one dimensional arrays rows the number of indices needed to specify an element is called the dimension dimensionality or rank of the array type a there are two common ways to support multi dimensional arrays via pointer chasing edit many languages support only one dimensional arrays in those languages a multi dimensional array is typically represented by an iliffe vector a one dimensional array of references to arrays of one dimension less a two dimensional array in particular would be implemented as a vector of pointers to its rows 10 thus an element in row i and column j of an array a would be accessed by double indexing a i j in typical notation this way of emulating multi dimensional arrays allows the creation of jagged arrays where each row may have a different size or in general where the valid range of each index depends on the values of all preceding indices this representation for multi dimensional arrays is quite prevalent in c and c software however c and c will use a linear indexing formula for multi dimensional arrays that are declared with compile time constant size e g by int a 10 20 or int a m n instead of the traditional int a 11 the c99 standard introduced variable length array types that let define array types with dimensions computed in run time the dynamic 4d array can be constructed using a pointer to 4d array e g int arr t u v w malloc sizeof arr the individual elements are accessed by first de referencing an array pointer followed by indexing e g arr i j k l alternatively n d arrays can be declared as pointers to its first element which is a n 1 dimensional array e g int arr u v w malloc t sizeof arr and accessed using more idiomatic syntax e g arr i j k l via computation edit some languages allow direct computation of element location languages with direct computation of element locations typically enclose the subscript list in a single pair of delimiters e g foo bar baz fortran pl i foo bar baz algol 60 pascal rather than placing each subscript in a pair of delimiters e g foo bar baz the order in which array elements are stored differs among languages e g fortran has row major arrays while pl i has column major arrays indexing notation edit most programming languages that support arrays support the store and select operations and have special syntax for indexing early languages used parentheses e g a i j as in fortran others choose square brackets e g a i j or a i j as in algol 60 and pascal to distinguish from the use of parentheses for function calls index types edit array data types are most often implemented as array structures with the indices restricted to integer or totally ordered values index ranges fixed at array creation time and multilinear element addressing this was the case in most third generation languages and is still the case of most systems programming languages such as ada c and c in some languages however array data types have the semantics of associative arrays with indices of arbitrary type and dynamic element creation this is the case in some scripting languages such as awk and lua and of some array types provided by standard c libraries bounds checking edit main article bounds checking some languages like pascal and modula perform bounds checking on every access raising an exception or aborting the program when any index is out of its valid range compilers may allow these checks to be turned off to trade safety for speed other languages like fortran and c trust the programmer and perform no checks good compilers may also analyze the program to determine the range of possible values that the index may have and this analysis may lead to bounds checking elimination index origin edit further information comparison of programming languages array some languages such as c provide only zero based array types for which the minimum valid value for any index is 0 12 this choice is convenient for array implementation and address computations with a language such as c a pointer to the interior of any array can be defined that will symbolically act as a pseudo array that accommodates negative indices this works only because c does not check an index against bounds when used other languages provide only one based array types where each index starts at 1 this is the traditional convention in mathematics for matrices and mathematical sequences a few languages such as pascal and lua support n based array types whose minimum legal indices are chosen by the programmer the relative merits of each choice have been the subject of heated debate zero based indexing can avoid off by one or fencepost errors 13 highest index edit the relation between numbers appearing in an array declaration and the index of that array s last element also varies by language in many languages such as c one should specify the number of elements contained in the array whereas in others such as pascal and visual basic net one should specify the numeric value of the index of the last element this distinction is not present in languages where the indices start at 1 such as lua array algebra edit some programming languages support array programming where operations and functions defined for certain data types are implicitly extended to arrays of elements of those types thus one can write a b to add corresponding elements of two arrays a and b usually these languages provide both the element by element multiplication and the standard matrix product of linear algebra and which of these is represented by the operator varies by language languages providing array programming capabilities have proliferated since the innovations in this area of apl these are core capabilities of domain specific languages such as gauss idl matlab and mathematica they are a core facility in newer languages such as julia and recent versions of fortran these capabilities are also provided via standard extension libraries for other general purpose programming languages such as the widely used numpy library for python string types and arrays edit many languages provide a built in string data type with specialized notation string literals to build values of that type in some languages such as c a string is just an array of characters or is handled in much the same way 14 other languages like pascal may provide vastly different operations for strings and arrays array index range queries edit some programming languages provide operations that return the size number of elements of a vector or more generally range of each index of an array in c and c arrays do not support the size function so programmers often have to declare separate variable to hold the size and pass it to procedures as a separate parameter elements of a newly created array may have undefined values as in c or may be defined to have a specific default value such as 0 or a null pointer as in java in c a std vector object supports the store select and append operations with the performance characteristics discussed above 15 vectors can be queried for their size and can be resized slower operations like inserting an element in the middle are also supported slicing edit an array slicing operation takes a subset of the elements of an array typed entity value or variable and then assembles them as another array typed entity possibly with other indices if array types are implemented as array structures many useful slicing operations such as selecting a sub array swapping indices or reversing the direction of the indices can be performed very efficiently by manipulating the dope vector of the structure the possible slicings depend on the implementation details for example fortran allows slicing off one column of a matrix variable but not a row and treat it as a vector on the other hand other slicing operations are possible when array types are implemented in other ways resizing edit some languages allow dynamic arrays also called resizable growable or extensible array variables whose index ranges may be expanded at any time after creation without changing the values of its current elements for one dimensional arrays this facility may be provided as an operation append a x that increases the size of the array a by one and then sets the value of the last element to x other array types such as pascal strings provide a concatenation operator which can be used together with slicing to achieve that effect and more in some languages assigning a value to an element of an array automatically extends the array if necessary to include that element in other array types a slice can be replaced by an array of different size with subsequent elements being renumbered accordingly as in python s list assignment a 5 5 10 20 30 that inserts three new elements 10 20 and 30 before element a 5 resizable arrays are conceptually similar to lists and the two concepts are synonymous in some languages an extensible array can be implemented as a fixed size array with a counter that records how many elements are actually in use the append operation merely increments the counter until the whole array is used when the append operation may be defined to fail this is an implementation of a dynamic array with a fixed capacity as in the string type of pascal alternatively the append operation may re allocate the underlying array with a larger size and copy the old elements to the new area see also edit array access analysis array database management system bounds checking elimination delimiter separated values index checking parallel array sparse array variable length array notes edit this nomenclature conflicts with the concept of dimension in linear algebra which expresses the shape of a matrix thus an array of numbers with 5 rows and 4 columns hence 20 elements is said to have dimension 2 in computing contexts but represents a matrix that is said to be 4 5 dimensional also the computer science meaning of rank conflicts with the notion of tensor rank which is a generalization of the linear algebra concept of rank of a matrix references edit 1 2 robert w sebesta 2001 concepts of programming languages addison wesley 4th edition 1998 5th edition 2001 isbn 9780201385960 introduction to tensors tensorflow core tensorflow k jensen and niklaus wirth pascal user manual and report springer paperback edition 2007 184 pages isbn 978 3540069508 john mitchell concepts of programming languages cambridge university press lukham suzuki 1979 verification of array record and pointer operations in pascal acm transactions on programming languages and systems 1 2 226 244 deitel harvey m deitel paul j 2005 c for programmers prentice hall professional p 303 isbn 978 0 13 246591 5 retrieved 22 may 2024 friesen jeff 5 march 2014 learn java for android development java 8 and android 5 edition apress p 56 isbn 978 1 4302 6455 2 retrieved 22 may 2024 array declaration cppreference com cppreference retrieved 15 november 2025 arrays the java tutorials docs oracle com oracle corporation retrieved 15 november 2025 van der linden peter 1994 expert c programming deep c secrets englewood cliffs nj sunsoft press isbn 978 0 13 177429 2 brian w kernighan and dennis m ritchie 1988 the c programming language prentice hall p 81 kernighan brian w ritchie dennis m 1988 the c programming language 2nd ed englewood cliffs n j prentice hall p 24 isbn 978 0 13 110370 2 edsger w dijkstra why numbering should start at zero null terminated byte strings cppreference com cppreference com retrieved 15 november 2025 std vector cppreference com cppreference com retrieved 15 november 2025 external links edit wikibooks has a book on the topic of data structures arrays look up array in wiktionary the free dictionary nist s dictionary of algorithms and data structures array v t e data types uninterpreted bit byte trit tryte word bit array numeric arbitrary precision or bignum complex decimal fixed point block floating point floating point reduced precision minifloat half precision bfloat16 single precision double precision quadruple precision octuple precision extended precision long double integer signedness interval rational reference address physical virtual pointer text character string null terminated composite algebraic data type generalized array associative array class dependent equality inductive intersection list object metaobject option type 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