The compositional structure of the decimal system is more complex than just making groups of 10 from 10 ones, since every 10 groups of 10 are composed into a unit of 100. A geometric version of this group’s idea occurs when shapes are put together to form a new, composite shape, and composite shapes are then put together to make another composite shape—a composite of the composite shapes.
An especially important case of geometric structuring as composites of composites occurs when analyzing rectangles and their areas. When considering the area of a rectangle, one views the rectangle as composed of identical square tiles that cover the rectangle without gaps or overlaps. Each square tile has area one square unit. The area of the rectangle (in square units) is the number of squares that cover the rectangle. Although these squares can be counted one by one, to develop and understand the length × width formula for the area of a rectangle, the squares must be seen as grouped, either into rows or into columns (see Figure 2-6). Each row has the same number of squares in it, and the number of rows in the rectangle is equal to the number of squares in a column (likewise, each column has the same number of squares in it, and the number of columns is the number of squares in a row). Because of this grouping structure, the area of the rectangle is # rows × # in each row or length × width (square units). Similarly, the decimal system has a multiplicative structure because 100 is formed (by definition) by making 10 groups of 10, and so 100 = 10 × 10.
The idea of structuring rectangles as arrays of squares can be extended to structuring an entire infinite plane (in the imagination) as an infinite array of squares. This idea of a plane structured by an infinite array is essentially the idea of the Cartesian coordinate plane, in which each point in the plane is described by a pair of numbers that indicate its location relative to two coordinate lines (axes) (see Figure 2-9).