The compositional structure of the decimal system consists not only of making groups of 10 from 10 ones and groups of 100 from 10 groups of 10, but also groups of 1,000 from 10 groups of 100, so that 1,000 = 10 × 10 × 10. The grouping structure of the decimal system continues in such a way that all successive groupings are obtained by repeatedly grouping by 10. The geometric counterpart of this grouping structure of the decimal system takes one into 3-D space and then higher dimensional space. Just as 2-D rectangles can be structured as 2-D arrays of squares, so, too, 3-D rectangular prisms (box shapes) can be structured as 3-D arrays of cubes. As in the case of rectangles, the multiplicative structure of a 3-D array of cubes explains why one multiplies the three dimensions of length, width, and height of a box to find its volume. Box shapes can be built as layers of identical cubes, as in Figure 2-12, and each layer can be viewed as groups of rows, so a box built from cubes can be viewed as a group of a group of cubes in the same way that 1,000 is 10 groups of 10 groups of 10.
When one extends the array structure of rectangular prisms to all of 3-D space, one gets essentially the idea of coordinate space, in which the location of each point in space is described by a triple of numbers that indicate its location relative to three coordinate lines.
The properties (or laws) of arithmetic are the fundamental structural properties of addition and multiplication from which all of arithmetic is derived. These properties include the commutative properties of addition