and must therefore be implemented by a hash table or some other search data structure. Language support Multi-dimensional arrays The number of indices needed to specify an element is called the dimension, dimensionality, or rank of the array type. (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.) 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. 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. Indexing notation 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 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