Defining dimensions for arrays

Before devising a solution for the problem mentioned earlier, let's first understand what a dimension is and what it means in the context of arrays.

However, mathematically speaking, the dimension of an object is the minimum number of coordinates needed to specify any point in it. When talking about arrays, it is easy to visualize up to three-dimensional arrays and see what these coordinates mean (row, column, depth).

Even though we cannot visualize anything beyond the three dimensions, we can mathematically create and work with any number of dimensions we want. In the context of an array, the terms N-dimensional and multidimensional array are used interchangeably and mean the same thing, i.e., a multidimensional array with N dimensions (where N is a positive integer like 1, 2, 3, and so on).

When we move to higher dimensions, the notion of rows, columns, and depth breaks down. However, we can still define an array in mathematical terms. Let us consider an example of an N-dimensional array that has dimensions D1, D2, D3 ... Dn (here Di denotes the size of the ith dimension, so for a 2 x 3 array we would have D1 = 3 and D2 = 2).

To access a data item at index I1, I2, I3 ... In (where Ii is the index along the ith dimension), we use the same higher-to-lower notation in almost all programming languages. We use the method shown in the drawing below to find the base address (the memory address where the array starts, which we will look at in detail in a later lesson) of the data item at the index [In] [In-1] ... [I1].