Page 60 - Engineering Digital Design

CHAPTER 2

Number Systems, Binary

Arithmetic, and Codes

2.1 INTRODUCTION

Number systems provide the basis for conveying and quantifying information. Weather
data, stocks, pagination of books, weights and measures — these are just a few examples
of the use of numbers that affect our daily lives. For this purpose we find the decimal (or
Arabic) number system to be reliable and easy to use. This system evolved presumably
because early humans were equipped with a crude type of calculator, their 10 fingers. But a
number system that is appropriate for humans may be intractable for use by a machine such
as a computer. Likewise, a number system appropriate for a machine may not be suitable
for human use.
Before concentrating on those number systems that are useful in computers, it will be
helpful to review those characteristics that are desirable in any number system. There are
four important characteristics in all:

• Distinguishability of symbols
• Arithmetic operations capability
• Error control capability
• Tractability and speed
To one degree or another the decimal system of numbers satisfies these characteristics
for hard-copy transfer of information between humans. Roman numerals and binary are
examples of number systems that do not satisfy all four characteristics for human use. On
the other hand, the binary number system is preferable for use in digital computers. The
reason is simply put: current digital electronic machines recognize only two identifiable
states, physically represented by a high voltage level and a low voltage level. These two
physical states are logically interpreted as binary symbols 1 and 0.
A fifth desirable characteristic of a number system to be used in a computer should be
that it have a minimum number of easily identifiable states. The binary number system
satisfies this condition. However, the digital computer must still interface with humankind.
This is done by converting the binary data to a decimal and character-based form that can
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