ABSTRACT
This chapter investigates the extension of number into new directions and new dimensions. These new directions include positive, negative, and even imaginary directions. The idea is generalized in linear algebra, where vectors represent numbers in any number of dimensions. Endowed with both magnitude and direction, numbers become integrated with geometry, and geometric actions, such as reflections and rotations, undergird their construction. The chapter demonstrates how matrix multiplication represents such mental actions and how it transforms units and area. Connections are made to determinant and change of basis. The chapter closes with a discussion of two principal operations that appear in every domain of number: addition and multiplication. This discussion elucidates why those are the two principal operations in terms of their roles in preserving or transforming units.
