This chapter examines the properties of the floor, ceiling, and remainder functions, which are involved in 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. The rest of this chapter, although essential to the proofs of subsequent results, may be skipped on first reading by the reader who is interested only in the statements of those results.
Notation: The symbols,
, and
will denote the sets of all rational numbers, integers, and natural numbers (i.e., nonnegative integers), respectively.