There are 10 kinds of people in the world—those who understand binary and those who don’t.
—Anonymous
CHAPTER
2
Data Representation in Computer Systems
2.1
INTRODUCTION
T
he organization of any computer depends considerably on how it represents numbers, characters, and control
information. The converse is also true: Standards and conventions established over the years have determined certain
aspects of computer organization. This chapter describes the various ways in which computers can store and
manipulate numbers and characters. The ideas presented in the following sections form the basis for understanding
the organization and function of all types of digital systems.
The most basic unit of information in a digital computer is called a
bit
, which is a contraction of
binary digit
. In
the concrete sense, a bit is nothing more than a state of “on” or “off” (or “high” and “low”) within a computer circuit.
In 1964, the designers of the IBM System/360 mainframe computer established a convention of using groups of 8
bits as the basic unit of addressable computer storage. They called this collection of 8 bits a
byte
.
Computer
words
consist of two or more adjacent bytes that are sometimes addressed and almost always are
manipulated collectively. The
word size
represents the data size that is handled most efficiently by a particular
architecture. Words can be 16 bits, 32 bits, 64 bits, or any other size that makes sense in the context of a computer’s
organization (including sizes that are not multiples of eight). An 8-bit byte can be divided into two 4-bit halves called
nibbles
(or
nybbles
). Because each bit of a byte has a value within a positional numbering system, the nibble
containing the least-valued binary digit is called the low-order nibble, and the other half the high-order nibble.
2.2 POSITIONAL NUMBERING SYSTEMS
At some point during the middle of the sixteenth century, Europe embraced the decimal (or base 10) numbering
system that the Arabs and Hindus had been using for nearly a millennium. Today, we take for granted that the
number 243 means two hundreds, plus four tens, plus three units. Notwithstanding the fact that zero means
“nothing,” virtually everyone knows that there is a substantial difference between having 1 of something and having
10 of something.
The general idea behind positional numbering systems is that a numeric value is represented through increasing
powers of a
radix
(or base). This is often referred to as a
weighted numbering system
because each position is
weighted by a power of the radix.
The set of valid numerals for a positional numbering system is equal in size to the radix of that system. For
example, there are 10 digits in the decimal system, 0 through 9, and 3 digits for the ternary (base 3) system, 0, 1, and
2. The largest valid number in a radix system is one smaller than the radix, so 8 is not a valid numeral in any radix
system smaller than 9. To distinguish among numbers in different radices, we use the radix as a subscript, such as in
33
10

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