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This chapter delves into key concepts of data representation, focusing on digital versus analog systems, binary foundations, and the significance of bits in modern computing. It provides an overview of number systems—binary, decimal, hexadecimal, and octal—explaining their structure and conversion methods. Readers will learn the hierarchy of numbers defined by their base, how electronic circuits thrive on binary logic, and the methods for converting between different numeral systems. Prepare to enhance your understanding of the language of computers and the essential terminology in data representation.
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Data Representation – Chapter 3 Section 3-1
Terminology • “Digital” • Discrete, well defined values/steps • Opposite of analog • Analogy: digital is to analog as int is to double • “Binary” • A system consisting of two states • on/off, true/false, yes/no, high/low, 0/1 • Basis for modern computers
Terminology • “Bit” • Binary-digit • Smallest unit of storage in modern computers
Data Representation • 1000001 – what does this “mean”? • one million, one • sixteen million, seven hundred seventy seven thousand, two hundred, seventeen • two hundred sixty two thousand, one hundred forty five • sixty five • “A” • AJMP assembly language instruction
Data Representation • 1000001 • Decimal number • Hexadecimal number • Octal number • Binary number • ASCII character • 8051 machine instruction
Number Systems • A number system is defined by its base or radix • The number of unique digits used in the system • Digits range in value from 0 to radix-1 • Larger values are created by stringing together digits • Resultant value is defined by • d: digit, b: base, i: position with 0 being the first position to the left of the “base point”, increasing to the left, decreasing to the right
Number Systems • Binary is convenient/efficient for use in a computer… • Electronic circuits can be easily designed to deal with two distinct levels • e.g. TTL 0-volts and 5-volts • … but extremely inconvenient for human consumption • Humans were designed to work with ten distinct levels • e.g. fingers • We’ll concentrate on decimal, hexadecimal, octal, and binary
Conversion • Base b to decimal, b = 2 (binary) • Decimal to base b • Integer divide value by b • Output remainder • Repeat on quotient • Until quotient is zero
Conversion • Binary to octal • Separate binary number into groups of 3 binary digits padding the left with 0’s if necessary • Convert groups to decimal digits 10012 -> 001001 -> 118
Conversion • Binary to hexadecimal • Separate binary number into groups of 4 binary digits padding the left with 0’s if necessary • Convert groups to decimal digits 10102 -> 1010 -> 1016(?) • Not exactly • 1010—1510 -> A16—F16 • i.e. 0123456789ABCDEF
Conversion • Octal to hexadecimal and hexadecimal to octal • Convert to binary then to the target radix using previous methods
Conversion • Octal and hexadecimal are useful when working closely with the architecture • Designing circuits • Designing device interfaces • Writing assembly language programs • In such situations one is generally concerned with bit patterns rather than the decimal value • The conversions can be done in your head
Homework – chapter 3 • 3-1, 3-2, 3-3, 3-4, 3-5, 3-6, 3-7, 3-8 • Due Thursday • Will discuss in class
