Arithmetic operations among shapes using shape numbers

In this work, a notation is given called the Discrete Geometry of Shapes, which describes the forms or shapes of flat regions limited by simply connected curves. A procedure is given that deduces from every region a unique number (its shape number) independent of translation and rotation, and optionally, of size and origin.All the integer numbers contain all the universe of discrete shapes (of course with different precision). In this universe there are shapes such as straight lines, circumferences, ellipses, parabolas, trigonometric functions, graphics of time, absorption waves, etc.The Discrete Geometry of Shapes is one-dimensional. It does not use the definition of equation and function to define shapes in a rectangular co-ordinate plane. With this notation it is possible to generate shapes with any characteristics by generating numerical sequences; also it is possible to do arithmetic operations among shapes. For example, the addition of a square and a circle, the average of a triangle and a circle, the square root of a pentagon, the numerical relations between given shapes, etc.Section V of this work describes the third dimension in the Discrete Geometry of Shapes for surfaces and volumes by means of a vector of shape numbers. It is possible to add surfaces, to divide volumes, to obtain the square root of a volume, etc.The main objective of this notation is the simplification of some mathematical and geometrical processes in this analysis of shapes and surfaces.