The term “mathematics” is difficult to define, in part because it includes so many concepts. Even so, the primary goal of this topic is to organize mathematical concepts and geometrical relationships for the convenience of spatial data users. The approach is to start with simple, well-defined ideas and add understandable pieces as needed to develop tools for handling spatial data more efficiently. Mathematics has been concisely defined as the study of quantities and relationships through the use of numbers and symbols. The terms “quantities” and “relationships” may be somewhat abstract, but their meaning should become clearer with use. “Numbers” includes the set of all real values from negative infinity to positive infinity, and “symbols” includes letters of the alphabet (English, Greek, or otherwise) used to represent numerical values. Symbols also include other markings to indicate mathematical operations such as addition, subtraction, multiplication, division, and square roots. As illustrated by the definition and the two following examples, the goals of presenting simple well-defined concepts and keeping a focus on practical applications for spatial data will not be easy to achieve. Understandably, some readers will be distracted by temptations to pursue peripheral interests. Although that is acceptable and encouraged, space and print limitations do not permit joint excursions. The reader is always welcome back.
• With respect to numbers, few people (certainly not the author) can comprehend the vastness of infinity, yet reference is made to negative infinity and positive infinity with the implication that they might somehow be the same size (but in opposite directions on the real number line). The point is not whether that implication is true, but to note instead that there are just as many numbers between zero and one (0 and 1) as there are numbers greater than one. That statement is proved by taking the reciprocal of any number greater than 1.
• Symbols for mathematical operations such as +, -, x, +, and V are simple and are used the world over. Symbols also include letters that are used to represent certain numerical values. Perhaps the most common mathematical symbol is the Greek letter pi (π), used to represent the ratio of the circumference of a circle divided by its diameter. The definition is simple, and that ratio finds many applications when working with spatial data. However, mathematicians (in what could be called esoteric pursuits) have spent years chasing an increasing number of digits for pi (Beckman 1971). Access to millions of digits for π is now as simple as typing “pi” into a World Wide Web search engine.
