Chapter overview

This chapter describes some of the basic ideas in digital arithmetic that are necessary to understand the polynomial and linear system optimizations presented in the later chapters. The chapter is divided into five sections.Section 5.2 discusses elementary properties of number systems, focusing on binary number representations including signed digit representations. It provides a high-level overview of both fixed and floating point representations. Section 5.3 gives background material on two-operand addition including ripple carry, carry propagate, and pipelined addition architectures. Then Section 5.4 provides an overview of multiple-operand addition. It gives details on sequential and parallel carry propagate multiple-operand architectures, as well as redundant digit summations using carry save adders and other higher-order counters and compressors. The section ends with a discussion on distributed arithmetic architecture for multiple-operand summations that are typically found in dot product operations. Section 5.5 summarizes the chapter.

Basic number representation

Number representation plays an important role in the design of algorithms for digital arithmetic. This section provides a brief and concise discussion on number representations that serves as a basis for understanding the topics presented in this book. There are a number of textbooks on digital arithmetic that deal with these topics in great detail [1–3].

Properties of number systems

Before we consider specific number representations, we should discuss fundamental properties that can be used to characterize a number system.