Computer Architecture Reflective Journal Report

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Systems Architecture,
Seventh Edition

Chapter 3

Data Representation

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Systems Architecture, Seventh Edition

Chapter Objectives

• In this chapter, you will learn to:
• Describe numbering systems and their use in data representation
• Compare different data representation methods
• Summarize the CPU data types and explain how nonnumeric data is represented
• Describe common data structures and their uses

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Systems Architecture, Seventh Edition

Data Representation and Processing

• To “process” or manipulate data, it must be converted into a form that the “processor” can understand
• Any data processing system (for example, a human or a computer system) must have the ability to:
• Recognize external data/stimuli and convert them into a suitable internal format
• Store and retrieve data internally
• Transport data among internal and external storage and processing components
• Manipulate the data to produce desired results or actions
• The human brain analogy:
• Our five senses convert touch, taste, smell, hearing, and vision into electrical impulses
• Our brain has dedicated regions for both short-term and long-term memory
• Our brain transmits electrical signals among its various storage and processing regions
• Our brain’s “processing circuitry” is based on electro-chemical reactions among interconnected neurons
• “Processing results” can be stored internally or output as physical actions

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Systems Architecture, Seventh Edition

Automated Data Processing

• The processing circuitry of a modern computer consists of:
• Electrical switches combined into more complex circuits that implement storage and processing functions
• Wiring among the switches and complex circuits
• Switches are two-state devices that are either open (off) or closed (on)
• Each position can represent a binary digit (bit) that is either zero or one
• Thus, any data processed by a modern computer must be converted to a set of binary digits before processing
• Processor outputs are also sets of binary digits which must usually be converted to another form (e.g., text or visual images) for humans to use

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Systems Architecture, Seventh Edition

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FIGURE 3.1 Two electrical inputs on the left flow through processing circuitry that generates their sum on the right

Courtesy of Course Technology/Cengage Learning

Systems Architecture, Seventh Edition

Data Representation and Processing Power

• A computer processor’s ability to manipulate only binary digits is sometimes a shortcoming
• Data that “naturally” converts to binary digits such as accounting data is easily and accurately manipulated
• Data that isn’t easily converted to binary digits such as literary concepts and facial images are difficult to accurately represent and manipulate
• Our brains have greater power than a computer because they can process a wider variety of data types
• Computers have more limited processors in terms of data complexity but they compensate for that with greater accuracy and execution speed

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Systems Architecture, Seventh Edition

Binary Data Representation

• A binary number is a single- or multi-digit number in which each digit is either zero or one
• Each digit of a multi-digit binary numeric value is called a bit (short for binary digit)
• Advantages of using binary numbers for processing:
• Binary numbers can be reliably transported via digital signals (more on this in Chapter 8)
• Two-state processing devices are relatively easy to design and fabricate (more on this in Chapter 4)
• Both advantages are based on existing technology
• Future technology changes may make other data representation and processing schemes a better choice (for example, optical CPUs or quantum computing)

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Systems Architecture, Seventh Edition

Boolean Logic

• Binary digit values correspond directly to the values “true” and “false”, for example
• Zero = “true” and One = “false”, or
• Zero = “false” and One = “true
• The choice is arbitrary though most computers use the first interpretation/mapping
• Boolean logic is a mathematical system of manipulating and combining true and false values
• Computer processing of simple and complex conditions is based on Boolean logic, for example
• IF AccountBalance > WithdrawalAmount THEN …

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Systems Architecture, Seventh Edition

Interpreting Binary Values

• Multi-digit numeric values are interpreted by digit position and weight, for example:
• The value 45610 in base 10 is interpreted as:

(6 × 100) + (5 × 101) + (4 × 102)

• The value 1012 in base 2 (binary) is interpreted as:

(1 × 20) + (0 × 21) + (1 × 22) = 510

• The base or radix of the numbering system is the:
• Number of different digit values (e.g., decimal has 10 digits numbered 0-9)
• The value that is raised to a power based on the position of a digit within a multi-digit value
• The base/radix of the decimal numbering system is 10
• The base/radix of the binary numbering system is 2

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Systems Architecture, Seventh Edition

Interpreting Binary Values - Continued

• Interpretation of multi-digit values with fractional components is a bit more complex, for example:
• The value 45.610 in base 10 is interpreted as:

(6 × 10-1) + (5 × 100) + (4 × 101)

• The value 10.12 in base 2 (binary) is interpreted as:

(1 × 2-1) + (0 × 20) + (1 × 21) = 2.510

• A radix point (the period in the examples above) denotes the boundary between whole and fractional parts of a multi-digit numeric value

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Systems Architecture, Seventh Edition

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TABLE 3.1 Binary and decimal notations for the values 0 through 10

Systems Architecture, Seventh Edition

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FIGURE 3.2 Computing the decimal equivalent of a binary number

Courtesy of Course Technology/Cengage Learning

Systems Architecture, Seventh Edition

Numeric Range

• If we assume that there is no limit on the number of digits in a multi-digit numeric then there is no limit on the size (large or small) of values that can be represented
• If we assume the number of digits is limited then numbering systems with larger bases have greater numeric range, for example:
• A three digit base 10 number can represent 1000 (103) different values
• A three digit base 2 number can represent 8 (23) different values
• Since computers process binary numbers, they require many bits to represent very large or very small values
• Computer circuitry is greatly simplified if all data of the same type uses the same number of bits!

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Systems Architecture, Seventh Edition

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TABLE 3.2 Binary notations for decimal values up to 16 positions

Systems Architecture, Seventh Edition

Beyond Binary

• Numbering systems based on 8 and 16 digit values are sometimes used by computer programmers because they are:
• More compact than binary values
• Easy to convert to/from binary values
• Even though a programmer may write values in something other than binary, the value must be converted to binary for storage or processing.
• Hexadecimal notation:
• Has 16 digit values (0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F)
• Is commonly used to represent memory and storage addresses
• Octal notation:
• Has 8 digit values (0,1,2,3,4,5,6,7)
• Was commonly used with older IBM mainframe systems and several generations of later systems that were backward compatible)

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Systems Architecture, Seventh Edition

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TABLE 3.3 Hexadecimal and decimal values

Systems Architecture, Seventh Edition

CPU Data Types

• The CPUs of most modern computer recognize and process 5 different data types (formats):
• Integer
• Real
• Character
• Boolean
• Memory address
• Each data type:
• Is represented using binary digits
• Is stored in a one or more format(s) that vary in the arrangement and interpretation of the digits
• Formats are generally chosen by a hardware designer or programmer for their:
• Suitability to the problem being solved
• Efficiency (of processing and communication circuitry)

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Systems Architecture, Seventh Edition

Goals of Computer Data Representation

• The ideal data representation method would be:
• Compact – stored values consume little “space”
• Accurate – any value could be represented and processed without error (e.g., no rounding)
• Have infinite range – infinitely large and small values could be represented
• Easy to manipulate – related processing circuitry would be simple, cheap, and fast
• Standardized – data represented within one computer device would could be recognized and processed by any other device
• Unfortunately:
• The first 4 ideals/goals are in conflict (e.g., compactness and range)
• Different trade-offs among the goals are needed for different data types (e.g., integer vs. real values) and different processing purposes (e.g., sorting alphabetic data vs. performing calculations for astronomy applications)

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Systems Architecture, Seventh Edition

Integers

• An integer is a whole number value (no fractional parts)
• 1234 is an integer value
• 12.34 is not an integer value
• Positive and negative values:
• An unsigned integer can hold only a positive value (e.g., 1234 is assumed to be a positive value)
• A signed integer is an integer that can value a positive or negative value (e.g., +1234 or ‒1234)
• How do we represent a sign as a binary digit?
• Two notation formats are well suited to signed integers:
• Excess notation
• Twos complement
• Most computers use twos complement notation

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Systems Architecture, Seventh Edition

Excess Notation

• Can be used to represent signed integers
• Divides a range of ordinary binary numbers in half; uses lower half for negative values and upper half for nonnegative values
• Always uses a fixed number of bits with the leftmost bit representing the sign (1 for nonnegative and 0 for negative values)

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Systems Architecture, Seventh Edition

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Table 3.4 Excess notation

Systems Architecture, Seventh Edition

Twos Complement Notation

• The complement of a bit value is its opposite
• 1 is the complement of 0
• 0 is the complement of 1
• Twos complement notation represents positive values as ordinary binary values, for example:
• +710 = 0111
• Twos complement notation represents negative values as the complement of the positive value plus 1:
• -710 = complement of (0111) + 1)
• = 1000 + 0001
• = 1001
• Note that the leftmost bit is a sign bit (1 if negative)
• Note also that despite its awkwardness for humans, twos complement notation enables very simple addition and subtraction circuitry

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Systems Architecture, Seventh Edition

Numeric Range

• The numeric range of a twos complement value is
• −(2n−1) to (2n−1−1)
• Where n is the number of bits
• For 32 bits, numeric ranges is approximately ±2 billion
• Any attempt to store a larger value (e.g., 20 billion) in a 32-bit space using twos complement notation results in overflow, loss of high-order bits
• As a practical matter for integers, the problem is disappearing as modern CPUs and software shift to 64-bit data storage
• For older processors with 32-bit data storage, double precision data types were sometimes defined which stored a 64-bit value in two adjacent storage locations

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Systems Architecture, Seventh Edition

Two’s Complement Notation

• Nonnegative integer values are represented as ordinary binary values
• Compatible with digital electronic circuitry
• Leftmost bit represents the sign
• Fixed number of bit positions
• Only two logic circuits required to perform addition on single-bit values
• Subtraction can be performed as addition of a negative value

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Systems Architecture, Seventh Edition

Real Numbers

• Real Numbers can have both whole and fractional parts, for example (e.g., 123.456)
• To store a real numeric value we must “store” both a sign and a radix point
• A simplistic way to accomplish that is:

FIGURE 3.3 A 32-bit storage format for real numbers using a fixed radix point

Courtesy of Course Technology/Cengage Learning

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Systems Architecture, Seventh Edition

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Floating Point Notation

• Similar to scientific notation, except that 2 is the base

value = mantissa x 2exponent

• Many CPU-specific implementations of floating-point notation are possible
• IEEE standard 754 defines formats for floating-point data

Systems Architecture, Seventh Edition

Floating Point Notation

• The previous slide depicts fixed point notation in which the location of the radix point never varies
• Pros – simple data format that’s easy for humans to understand
• Cons – limited numeric range – extremely large or small values can’t be represented unless a large number of bits are used
• Floating point notation is similar to scientific notation (e.g., 13,526,473 × 10−16) except that the radix is 2 instead of 10
• As in scientific notation, floating point notation uses both a:
• Mantissa (e.g., 13,526,473 in the above example)
• Exponent (e.g., -16 in the above example)
• The Institute of Electrical and Electronics Engineers (IEEE) has defined floating point notations with several different lengths and all CPU manufacturers now follow those standards

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Systems Architecture, Seventh Edition

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FIGURE 3.4 Conversion of scientific notation to decimal notation

Courtesy of Course Technology/Cengage Learning

Systems Architecture, Seventh Edition

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FIGURE 3.5 IEEE binary32 floating-point format

Courtesy of Course Technology/Cengage Learning

Systems Architecture, Seventh Edition

Processing Floating Point Numbers

• Floating point notation is considerably more complex than twos complement notation
• The greater format complexity yields greater complexity in processing circuitry
• Computational circuits for real numbers generally have at least 2X more components than for integers
• More components and wiring yields larger and slower circuits
• Floating point notation is also subject to truncation errors
• For example, the value 0.1010 has a non-terminating mantissa in floating point notation
• Thus, adding 0.1010 + 0.1010 yields a result slightly smaller than 0.2010
• Such errors can accumulate and compound in long computational sequences (e.g., some of the formulas used in MGMT 326)
• Floating point notation is also subject to overflow and underflow errors

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Systems Architecture, Seventh Edition

Character Data

• Characters in written human languages have no obvious binary numeric equivalents
• But we can represent characters with integer values via a coding table and store the characters using twos complement or as ordinary binary values (usually the former so that the number of different data formats is minimized)
• A coding table is simply a set of correspondences between characters and integer values (for example, see next slide)

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Systems Architecture, Seventh Edition

ASCII

• The American Standard Code for Information Interchange (ASCII) is a coding table for English
• Represents uppercase and lowercase letters and the digits 0-9
• Represents some punctuation marks
• Represents a small set of device control characters (e.g., line feed)
• ASCII is a 7-bit coding format

Burd, Systems Architecture, seventh edition, Table 3-5, Copyright © 2015 Course Technology

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Systems Architecture, Seventh Edition

ASCII Limitations

• Insufficient range
• Uses 7-bit code, providing 128 table entries (33 for device control)
• 95 printable characters can be represented
• English-based
• Latin-1
• Lower 128 entries ASCII-7 characters
• Upper 128 entries multinational characters

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Systems Architecture, Seventh Edition

Unicode

• Unicode is an advanced character coding table developed by an international consortium
• Unicode incorporates ASCII as a subset (thus, it’s at least partly backward-compatible)
• The goal of Unicode is to include the written forms of all human languages (including the dead ones)
• Unicode has advanced through a number of version toward that goal but is still under development
• Unicode currently defines over 100,000 characters and uses a 32-bit code

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Systems Architecture, Seventh Edition

Boolean Data

• When a CPU processes a programming language statement such as:
• IF AccountBalance < WithdrawalAmount GOTO 1000
• It must execute one instruction to perform the comparison and another instruction to perform the conditional branch
• The “result” of the comparison is a data value that must be stored in a register by the first instruction and used as input to the second instruction
• Thus, CPUs need a Boolean data type that can store the values “true” and “false”
• Most CPUs adopt a simplistic approach:
• A multi-digit binary value with all zeros is “false”
• Any other binary value is “true”

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Systems Architecture, Seventh Edition

Memory Address

• CPUs constantly read and write to/from memory so efficient processing of memory addresses is a critical performance issue
• Memory is treated a sequence of storage locations numbered start at zero
• Each storage location is usually assumed to hold 8 bits (one byte)
• Thus, a memory address is an unsigned integer, usually stored as a binary value with no sign bit
• Addressable memory is limited by the number of bits used to store this unsigned integer
• 32 bits – addressable memory is 4 gigabytes (232 bytes)
• 64 bits – 264 bytes
• Older Intel CPUs used a more complex segmented memory address format

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Systems Architecture, Seventh Edition

Summary

• Data can be represented in many ways
• Data types are used as building blocks to create more complex data structures
  (e.g., arrays, records)
• Data representation is key to understanding hardware and software technology

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