This chapter examines the properties of the floor, ceiling, and modulus functions, which are central to our formulation of the RTL primitives and floating-point rounding modes. The properties of these functions that are derived here will be required in later chapters. Thus, their definitions (Definitions 1.1.1, 1.1.2, and 1.2.1) are prerequisite to understanding the definitions appearing in later chapters. We also define and investigate the properties of the function chop, which is related to the floor and is important in the analysis of the algorithms discussed in Part V.
Notation: The symbols , , , and will denote the sets of all real numbers, rational numbers, integers, and natural numbers (i.e., nonnegative integers), respectively.