In mathematics and computer programming, index notation is used to specify the elements of an array of numbers. The formalism of how indices are used varies according to the subject. In particular, there are different methods for referring to the elements of a list, a vector, or a matrix, depending on whether one is writing a formal mathematical paper for publication, or when one is writing a computer program. In mathematics It is frequently helpful in mathematics to refer to the elements of an array using subscripts. The subscripts can be integers or variables. The array takes the form of tensors in general, since these can be treated as multi-dimensional arrays. Special (and more familiar) cases are vectors (1d arrays) and matrices (2d arrays). The following is only an introduction to the concept: index notation is used in more detail in mathematics (particularly in the representation and manipulation of tensor operations). See the main article for further details. One-dimensional arrays (vectors) A vector treated as an array of numbers by writing as a row vector or column vector (whichever is used depends on convenience or context): Index notation allows indication of the elements of the array by simply writing ai, where the index i is known to run from 1 to n, because of n-dimensions. For example, given the vector: then some entries are . The notation can be applied to vectors in mathematics and physics. The following vector equation can also be written in terms of the elements of the vector (aka components), that is where the indices take a given range of values. This expression represents a set of equations, one for each index. If the vectors each have n elements, meaning i = 1,2,…n, then the equations are explicitly Hence, index notation serves as an efficient shorthand for representing the general structure to an equation, while applicable to individual components. Two-dimensional arrays More than one index is used to describe arrays of numbers, in two or more dimensions, such as the elements of a matrix, (see also image to right); The entry of a matrix A is written using two indices, say i and j, with or without commas to separate the indices: aij or ai,j, where the first subscript is the row number and the second is the column number. Juxtaposition is also used as notation for multiplication; this may be a source of confusion. For