To the Student

1
To the Student

• What are we here for?
• To look the insides of computers
• To learn the jargon of computers
• 200-MHz CPU, 8MB RAM, CD-ROM, , etc.
• This course will provide a foundation, but it is
  NOT the end.
• The more you learn, the better prepared you will
  be.
• The most rewarding skill you can develop is the
  ability to learn on your own.

2
The Study of Algorithms(I)

• A set of steps that defines a task is performed
  (an ordered set of unambiguous, executable steps
  that define a terminating activity.)
• Problem to be solving in general.
• Algorithm a solution for solving a problem.
• Program a machine-compatible representation of
  an algorithm.

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The Study of Algorithms(II)

• Software a collection of programs and the
  algorithms.
• Programming Language to transform the conceptual
  algorithm into a set of instructions .
• Hardware a machine to realize or execute
  instructions unambiguously.

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The Euclidean algorithm

• To fine the Greatest Common Divisor of two
  positive integers
• Step 1. Assign two input values M and N (larger
  and smaller respectively).
• Step 2. Divide M by N, and call the remainder R.
• Step 3. If R is not 0, the assign M of N, assign
  N of R, and return to step 2 otherwise, the GCD
  is the value of N.

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The Origins of Computing Machines

• 16??-17??-18??-19??
• Gear-driven calculators
• Information as holes in paper cards
• Vacuum tubes
• Transistors
• Integrated circuits
• Optic
• ...

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The Evolution of Computer Science
Execution of
Limitations of
Algorithms
Analysis of
Communi- cation of
Representation of
Discovery of
The central role of algorithms in computer science
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CHAPTER ONEData Storage

• Storage of Bits
• Main Memory
• Mass Storage
• Representing Information as Bit Patterns
• The Binary System
• Storing Integers
• Storing Fractions
• Data Compression
• Communication Errors

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Storage of Bits

• BIT binary digit
• 0 and 1
• on and off (a switch)
• open and closed (a relay)
• raised and lowered (a flag)
• reflected and scattered (a laser beam)
• any device with two states

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Boolean Operations

• 1815-1864 mathematician George Boole
• truth or falseness manipulation
• P AND Q (both P and Q)
• P OR Q (ether P or Q)
• P XOR Q (ether P or Q but not both)
• NOT P

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Gates and Flip-Flops

• Gate a device that produces the output of
  Boolean operation
• AND gate, OR gate, NOT gate, XOR
• Flip-Flop a circuit that produces an output
  value of 0 or 1 that remains constant until a
  temporary pulse from another circuits causes it
  to change to the other value.

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A Simple Flip-Flop Circuit
Input
Output
Input
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Other Storage Techniques

• Magnetic core a small donut-shaped ring magnetic
  material with two directions of magnetic field.
• Capacitor charged or discharged
• Nonvolatile (core) versus volatile (capacitor)
  storage.
• a refresh circuit to replenish/charge a capacitor
  periodically, so the capacitor-type of storage is
  often called dynamic memory.

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Hexadecimal Notation

• Bit pattern Hexadecimal
• 0000 0
• 0001 1
• 0010 2
• 0011 3
• 0100 4
• 0101 5
• 0110 6
• 0111 7
• 1000 8
• 1001 9
• 1010 A
• 1011 B
• 1100 C
• 1101 D
• 1110 E
• 1111 F

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Main Memory(I)

• Byte a typical storage with size of eight bits
• KB (kilobyte) 210 1,024 1,000 103
• MB (megabyte) 220 1,048,576 1,000,000 106
• GB (gigabyte) 230 1,024MB 1,000,000,000
  109

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Main Memory(II)

• Address to identify individual storage
  cells(bytes) in a machines main memory.
• Each cell has a unique name.
• Addressed as 0, 1, 2, N-1, where N 2n bytes
  in main memory
• each cell can be accessed individually, so often
  referred as random access memory (RAM)
• DRAM, SRAM, SDRAM,

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Main Memory(II)

• MSB
• most significant bit (high-order end) of a cell
• LSB
• least significant bit (low-order end) of a cell

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Mass Storage

• Due to volatility and limited size of a
  computers main memory, most computer systems are
  provided with additional nonvolatile and larger
  size of storage devices such as magnetic
  disks/tapes, CDs, etc.

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Magnetic Disks(I)

• Organization
• Track a concentric circle of a disk surface
• Sector an arc of a track
• Cylinder a set of tracks
• Types
• Floppy disk 3½-inch diskette with 300 rpm and
  1.44MB
• Hard disk case is sealed at the factory with
  such as 7,200 rpm and 9GB

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Magnetic Disks(II)

• Performance
• Seek time
• Rotation delay or latency time
• Transfer time
• Measurement
• Access time seek time rotation delay
• Transfer rate
• Comparison
• nanoseconds(109)
• milliseconds (103)

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Compact Disks

• CD-ROM
• a single spiraled track with 550630 MB
• CD-WORM
• write-once-and-read-many-times
• DVD
• digital versatile disk
• 10GB
• MO
• etc.

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File Storage and Retrival

• Physical record - a sector of a disk
• Logical record - a natural block of data
• Buffer - main memory to store logical records
• Decreasing degrees of random access
• main memory - individual bytes
• magnetic disk - entire sector of data
• compact disk - entire sector but more time to
  locate spiraling track
• magnetic tape

22
Representing Text

• ASCII American Standard Code for Information
  Interchange
• 7-bit, 128 bit patterns
• ANSI American National Standards Institute
• Unicode
• 16-bit, 65536 bit patterns
• natural language supports
• Chinese, Japanese, etc.
• ISO International Organization for
  Standardization

23
Representing Numeric Values(I)

• Base ten system - decimal system
• Base two system - binary system
• Numeric conversion between number bases
• Base 10 to base 2
• Base 2 to base 10
• Base 10 to another base
• Another base to base 10
• etc.

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Representing Numeric Values(II)

• 2-10 Decoding the binary representation
• 10-2 Encoding the binary representation
• one byte 0255
• two bytes 065535
• two ASCII codes 00 99

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Representing Numeric Values(III)

• Fractional part of numbers 4¾
• Floating-point notation
• Too large of values or too small of fractions
• Errors of overflow or truncation
• etc.

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Representing Images

• Bit map techniques - vector techniques
• Pixel - picture element
• three components per pixel (RGB)
• 3-byte-per-pixel of 1280x1024 pixels
• disadvantage not easy to be rescaled
• Scalable fonts
• TrueType fonts (Microsoft and Apple)
• PostScript (Adobe Systems)

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The Binary System

• Binary addition
• carry
• Fractions in binary
• radix point

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Storage Integers

• Twos complement notation
• The complement of a pattern is by changing all
  the 0s to 1s and all the 1s to 0s
• Sign and magnitude
• MSB is the sign bit of the value
• The problem of overflow
• Excess notation

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CHAPTER TWOData Manipulation

• The Central Processing Unit
• The Stored-Program Concept
• Program Execution
• Arithmetic/Logic Instructions
• Communication with Other Devices
• Other Architectures

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The Central Processing Unit

• An isolated part of the computer
• performs operations such as addition on data is
  NOT directly connected to the storage cells
• Arithmetic/logic unit
• contains the circuitry that performs data
  manipulation
• Control unit
• contains the circuitry for coordinating the
  machines activities

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The Central Processing Unit

• CISC architecture
• Pentium series developed by Intel
• RISC architecture
• PowerPC series developed by Motorola and IBM
• In order to increase performance
• these CPUs are capable of retrieving multiple
  bytes from memory in a single step.
• Capable of fetching and executing more than one
  instruction simultaneously

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Registers

• Temporary storage of information
• general-purpose registers
• holding data immediately applicable to the
  operation of CPU
• memory hierarchy
• registers immediately needed
• cache memory a portion copy of main memory
• main memory needed in the near future
• mass storage
• special-purpose registers
• see section 2.3

33
CPU/Memory Interface

• Bus
• a collection of wires used to connect CPU and
  main memory units
• extract or read or load data from main memory
• place or write or store data into main memory

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Machine Instructions

• Three categories
• data transfer (move) group
• LOAD and STORE instructions
• I/O instructions
• arithmetic/logic group
• logic operations AND, OR, and XOR
• arithmetic operations
• other operations SHIFT and ROTATE...
• control group
• JUMP or BRANCH instructions
• conditional and unconditional jumps

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The Stored-Program Concept

• John von Neumann
• To design a control unit
• To extract/fetch a program from main memory
• To decode the instructions in program
• To execute the instructions
• machine-language
• op-code field
• operand fields

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Program Execution

• Control unit two special-purpose registers
• Program counter
• to hold the address of the next instruction to be
  executed
• Instruction register
• to hold the instruction being executed
• repeat performing a 3-steps algorithm
• fetch, decode, and execute

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Number Systems

• Base 2, 3, , etc.
• Binary arithmetic
• Bit (stands for Binary digit, 0 or 1)
• Byte, Halfword, Word, Doubleword, Short, Long, ,
  etc.

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Numbers as a Physical Representation

• Rome I, II, III, IV, V, VI, , etc.
• Base 2 101102
• Base 3 2103 1103
• Base 8 1076543218
• Base 10 987654321010
• Base 16 fedcba987654321016

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Counting in Different Base

• R Bk
• 6248 682 281 480

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CHAPTER FOURPresenting Integer Data

• Nearly, every high-level computing language
  provides a method for storing, manipulation, and
  calculation of signed integer and real numbers.
• Thus, we need to consider methods of representing
  and manipulating there numbers within the
  zeros-and-ones constraint of the computer.

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Unsigned Binary

• An 8-bit storage location can store any unsigned
  integer of value between 0 and 255.
• 16-bit storage location
• 0-65535 (216-1)
• 32-bit storage location
• 0-4,294,967,295 (232-1)

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Storage of a 32-bit Data Word
Memory location
1 byte
31
24
23
16
15
8
7
0
Most significant bits (31-24) bits (23-16) bits
(15-8) least significant bits (7-0)
M
32-bit data word
M1
M2
M3
Next data item
M4
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Binary-Coded Decimal (BCD)

• Each decimal digit is individually converted to
  binary with 4 bits per digit.
• 6810 0110 10002
• Disadvantage BCD arithmetic is nearly always
  much slower.
• Advantages
• easy decimal rounding and precision
• easy translation to character forms.

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BCD Multiplication (Example)

• 76 0111 0110x
  7 0111 42
  101010 7 x 6
  42 49 110001 7 x 7
  49 (1)32 0100
  0010 convert to BCD 4
  0100 1001 convert to BCD 532
  0100 1101 0010
  (13) convert to BCD
  (carry) 0001 0011
  
  0101 0011 0010 532 BCD

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Representations for Signed Integers

• Negative numbers in binary form
• sign-and-magnitude
• 1s complement
• 2s complement
• the most common method

49
Sign-and-Magnitude(I)

• Sign most significant bit (leftmost)
• 0 plus sign, 1 minus sign
• 37 0100101, - 37 1100101
• 1 0000 0000 0000 0001
• - 1 1000 0000 0000 0001
• - 0 1000 0000 0000 0000
• - 32767 1111 1111 1111 1111

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Sign-and-Magnitude(II)

• A signed 16-bit BCD - 7999 value 7999
• Disadvantages
• sign is confusing2 - 4 - 2 but 12 - 4 8
• - 0 is confusing

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Nines Decimal Representation

• Negative each digit - the largest numeral in
  the radix.
• Advantages no carry, converted independently
  like a mirror
• Base 10 radix 9s complement

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9s Complement

• - 467 999 - 467 532 (3-digit number)
• - 467 9999 - 467 9532 (4-digit number)
• 9990 9999 - 9990 9 (represents - 9)
• normal rules of arithmetic
• - (- value) value
• comp base - value
• base - ( base - value) value
• modular rotation
• reach 999, next 000, and then rotated

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9ComplementCounting Process
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Wraparound and Modulus Crossing
(1099 End-Around Carry)
300
500 799 999
0 99 499
- 499 - 200 - 0
0 100 499
300
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Summary of 9s Complement(I)

• Positive numbers are stated with 0, 1, 2, 3, or 4
  (the same of sign-and-magnitude).
• Negative numbers are started with 5, 6, 7, 8, or
  9 in complementary form.
• Negative 0 exists.

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Summary of 9s Complement(II)

• To change sign is to subtract each digit from 9
  (the largest digit of base).
• To add two numbers is to simply add and
  end-around-carry if carry exists.
• To subtract two numbers is to change sign of the
  subtrahend and then add.

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1complement

• Overflow 6465 - 12901000000 01000001
  10000001

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10s Complement

• Only a single ZERO
• Add two numbers adding digits and ignoring the
  carry.
• Subtract two numbers inverting the subtrahend
  and adding.
• Negative Modulus - Value
• 1000 - 247 753 (value of -247)
• 10s comp 9s comp 1

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2s Complement

• A signed 8-bit with range - 128 value 127
  
• A signed 16-bit with range - 32768 value
  32767

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Other Bases

• Even-numbered base can be split the same way
• Old-numbered base is split unevenly
• A 4-digit hexadecimal with 16s Complement
• positive numbers start with 0-7
• negative numbers start with 8-F

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CHAPTER FIVEFloating Point Numbers

• Real, floating point, numbers can expressed
  numbers with a decimal fraction a limited, fixed
  number of digits of precision with a power.
• A potential loss of precision better precision
  in terms of larger storage requirements and
  slower calculations.

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Exponential Notation

• - 0.0000003579 - 0.3579 10-6

Sign of mantissa
Sign of exponent
- 0.35790 10-6
Location ofdecimal(radix) point
mantissa
base
exponent
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Floating Point Format

• SEEMMMMM
• S sign of mantissa
• 0 for positive, 5 for negiative
• E exponent
• excess-50 (next page)
• M mantissa
• stored using sing-and-magnitude format
• Implied decimal point
• assumed at the beginning of the five-digit
  mantissa
• .MMMMM10EE

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Excess-N notation

• N is the midvalue of the exponent
• Two-digit exponent with excess-50value 00 to 99
  stands for -50 to 49and the magnitude range
  of0.00001 10-50 lt N lt 0.99999 1049

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Overflow and Underflow

• Overflow magnitude too large
• Underflow magnitude too small

0
-10-55
0.999991049
10-55
-0.999991049
Overflowregion
Overflowregion
Underflowregion
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Normalization

• To maximize the precision, numbers are stored
  with no leading zeros.
• Normalization process is to eliminate leading
  zeros by adjusting (shifting) the exponent
  .MMMM 10EE
• 246.8035 0.2468035 103 05324680
• 1255 10-3 0.1255 101 05112550
• -0.00000075 -0.75 10-6 54475000

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Convert Decimal toFloating Point(I)

• Procedure ConvertToFloat (DecimalIn Real,
  FloatOut Integer)
• Var Mantissa Real
• Exponent, Sign, IntegerMantissa Integer
• Procedure StandardizeNumber
• Begin
• Mantissa Abs(Mantissa)
• ( Adjust the decimal to fall between 0.1 and
  1.0 )
• While Mantissa gt 1.00 Do
• Begin
• Mantissa Mantissa / 10.0
• Exponent Exponent 1
• End
• While Mantissa lt 0.1 Do
• Begin
• Mantissa Mantissa 10.0
• Exponent Exponent - 1
• End
• IntegerMantissa Round(10000.0 Mantissa)
• End

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Convert Decimal toFloating Point(II)

• Procedure ConvertToFloat (DecimalIn Real,
  FloatOut Integer)
• Var Mantissa Real
• Exponent, Sign, IntegerMantissa Integer
• Procedure StandardizeNumber
• ...
• Begin ( main procedure )
• If DecimalIn 0.0 Then FloatOut 0
• Else Begin
• If DecimalIn gt 0.0 Then Sign 0
• Else Sign 50000000
• Exponent 50
• StandardizeNumber
• FloatOut Sign Explonent 100000
  IntegerMantissa
• End
• End

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Addition and Subtraction

• 05199520 04967850 ?
• alignment 05199520 0510067850
• add mantissa () means carry 099520 00067850
  (1)0019850
• normalize result 05210019(850)
• round result 05210020

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Multiplication and Division

• 05220000 x 04712500 ?
• add explonents 52 47 - 50 49
• multiply mantissas0.20000 x 0.12500
  0.025000000
• normalize result 04825000

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Floating Point in the Computer

• 32 bits approximately 10-38 to 1038
• 8-bit (256 levels) exponent excess-128
• 23-bit mantissa stretched to 24-bit
• an implicit or hidden leading bit 1 is not
  stored

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IEEE Standard 754(I)

• IEEE Computer Society
• N (-1)S2E-127(1.M)
• single-precision (32-bit, E8-bit, M23-bit)
• double-precision (64-bit, E11-bit, M52-bit)

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IEEE Standard 754(II)

• 1.MMMMMM
• an implied most significant bit
• excess-127 (excess-1023) base 2

Exponent Mantissa Value 0 0 0 0 ¹ 0 2-126
0.M 1254 any 2E-127 1.M 255 0 255 ¹
0 special condition
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IEEE Standard 754(III)

• special condition (not a number)
• dividing zero by zero
• square root of a negative number

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ConversionBetween Base 10 and Base 2(I)

• 25375 1100011000111112 1.10001100011111
  214
• IEEE 754
• 1-bit sign 0
• 8-bit exponent 14127 141 100011012
• 23-bit mantissa 100011000111110000000002

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ConversionBetween Base 10 and Base 2(II)

• -25.375 -11001.0112 -1.1001011 24
• IEEE 754
• 1-bit sign 1
• 8-bit exponent 4127 131 100000112
• 23-bit mantissa 100101100000000000000002

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Packed Decimal

• A number converted from decimal to binary to
  decimal may not be the same
• each decimal digit is stored in BCD form
• sign is stored in the low-order bits of the last
  byte
• 1111 unsigned
• 1100 positive
• 1101 - negative