Digital Circuit Analysis and Design with Simulink Modeling and Introduction to CPLDs and FPGAs, Second Edition

Chapter 1: Common Number Systems and Conversions

This chapter is an introduction to the decimal, binary, octal, and hexadecimal numbers, their representation, and conversion from one base to another. The conversion procedures are illustrated with several examples. Throughout this text, a left justified horizontal bar will denote the beginning of an example, and a right justified horizontal bar will denote the end of the example. These bars will not be shown whenever an example begins at the top of a page or at the bottom of a page. Also, when one example follows immediately after a previous example, the right justified bar will be omitted.

1.1 Decimal, Binary, Octal, and Hexadecimal Systems

The familiar decimal number system has base or radix 10. It referred to as base 10 because it uses ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. These digits are referred to as the coefficients of the decimal system. Thus, in the decimal system the coefficients are multiplied by the appropriate powers of 10 to form a number. For example, the decimal number 58, 392.46 is interpreted as:


In general, any number may be represented by a series of coefficients as:


In the decimal system, the A k coefficients are the ten coefficients (zero through nine), and the subscript value k denotes the power of ten by which the coefficient must be multiplied. Thus, the last expression above can also be written as


Digital computers use the binary (base 2) system which has only two coefficients,...