CSCI13003A

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Lecture-3 (1.2 - 2.9)

• CSCI130-03A
• Instructor Dr. Imad Rahal

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What Computers Do?

• General-purpose computers
• Many tasks using a limited set of operations
• Computing an answer to an equation
• 63 53 33/12 7
• Complicated math functions using simpler ones
• raise to a power 63 6 6 6
• Multiplication and division can even be reduced
  to and -
• 53 (555) or (33333)
• 33/12
• Keep on subtracting the denominator from the
  numerator until denominatorgtnumerator
• 33-1221
• 21-129
• 9lt12 ? quit and return how many s weve done ? 2

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Programs Algorithms

• Assume a computer cant directly multiply
• Multiplication task
• 1 Get 1st number, Number_1
• 2 Get 2nd number, Number_2
• 3 Answer 0
• 4 Add Number_1 to Answer
• 5 Subtract 1 from Number_2
• 6 if Number_2gt0, go to step 4
• 7 Stop

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Programs Algorithms

• What we really need for this example
• Store numbers
• Add numbers (Subtract ?)
• Compare number
• In general, all tasks done by a general-purpose
  computer can be accomplished through the
  following set of operations
• Store data
• numbers (positive, negative or fractions), text,
  pictures, etc
• Compare data (numbers, pictures, sounds, letters)
• Add
• Move data from one storage (memory) location to
  another
• Editing a text document
• Input/output
• Not mentioned in book but important

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Programs Algorithms

• CAVEAT Not all computers are restricted to this
  set
• Most likely not!
• Why only a limited set?
• There are potentially infinite tasks wed like to
  do
• Not possible to hard-wire a specific operation
  for each task
• Like the Arabic numeral system 10 digits (0
  thru 9)
• Develop programs for non hard-wired tasks
• Program A formal representation of a method for
  performing some task
• Written in a code/programming language understood
  by a computer
• Detailed and very well-organized (computers just
  do what they are told)
• Follows an algorithm method for fulfilling the
  task
• Plan to do something VS the actual performance

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Programs Algorithms

• Characteristics of an algorithm
• List of steps that complete a task
• Each step is PRECISELY defined and is suitable
  for the machine used
• Add 5 to variable X
• Make the color brighter
• Increase the speed
• Eat a sandwich!
• Finite number of steps
• The process terminates in a finite amount of time
• No infinite loops

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Programs Algorithms

• Algorithm to compute the average of the first 20
  numbers
• 1 sum 0, value 1, average 0
• 2 add value to sum
• 3 add 1 to value
• 4 if value is small, go to step 2
• 5 set average sum/20
• 6 display average
• 7 stop
• How many steps?
• Finite? Precisely defined?
• Is this an algorithm?
• if value lt 21, go to step 2
• How would you make it get the average of the
  first x numbers where x is specified as input?

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Programs Algorithms

• We can program a task if and only if
• an algorithm exists
• each step is mechanically possible
• ask a computer to eat a sandwich!
• How intelligent are computers?
• They are not!
• They just follow the steps that we define for
  them
• Why use them then?
• Extremely powerful
• Fast every step typically takes way less than a
  second (nanosecond --- 10-9 seconds)
• Flexible enough to solve complicated tasks by
  combining programs that solve simpler tasks
• Powers, multiplications!

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The History of Computing

• A computer originally meant a human being who
  did computations on paper
• Abacus was the first computing device

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The History of Computing

• Pascal (17th), Leibnitz (18th)
• Charles Babbages (19th) machine was the closest
  to todays computers
• Was programmable (not special purpose) using
  punched cards
• The punch card is a medium on which data are
  recorded by punching out holes
• Analogously, data are recorded on a DVD by
  stamping or burning much smaller holes

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An Example of a Punched Card
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The History of Digital Computing

• First generation computers
• First actual computer was developed during WWII
• Crack German code messages
• Calculate trajectories for launching shells
• Used vacuum tubes
• Can switch electricity on/off or amplify a
  current
• Much like a light bulb
• generates a lot of heat and has a tendency to
  burn out
• Slow, big (4-5 inches) and bulky

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The History of Computing

• ENIAC (shown on previous slide)
• weighed over 30 tons, and consumed 200 kilowatts
  of electrical power
• had around 18,000 vacuum tubes that constantly
  burned out, making it very unreliable
• Computers were large, programmed in machine code,
  suffered from mechanical malfunctions, expensive
  and not easy to maintain

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The History of Computing

• Second generation computers
• The transistor was invented in 1947
• Small (½ inch), fast, reliable and effective, it
  quickly replaced the vacuum tube (mid 1960s in
  computers)
• A vacuum tube had a size of 4 to 5 inches
• Machines got smaller, faster, reliable
  (transistors vs. vacuum tubes), cheaper, more
  energy-efficient with much more memory (more
  transistors could be connected in a small space)
• Development of advanced programming languages
  and applications
• No machine language anymore
• Computers became much more wide spread
• Initially restricted to governments and big
  businesses

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The History of Computing

• Third generation computers
• Late 1960s with the advent of integrated circuits
  (ICs)
• Scientists found a way to reduce the size of
  transistors so they could place hundreds of them
  on small silicon chips
• Rather than using (discrete) transistors
  separately as units, transistors were
  miniaturized and placed on silicon chips
• ICs are often classified by the number of
  transistors they hold
• More ? better
• More complicated operations

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The History of Computing

• Increased the speed and efficiency of computers
  and decreased their sizes
• Enabled whole computers to sit upon a desk top
  instead of requiring a whole room (desktops)
• More reliable, cheaper, easier to maintain with
  more memory
• Operating systems
• We are still in the 3rd generation
• IC chips are getting smaller and faster

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Analog and Digital Signals

• Examples of signals
• Lights on traffic signals (G, R, Y), grade
  (performance) etc
• Signals are the basis of all communications
• Computers
• Phones
• A piece of data moving from place to another
  signal

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Analog and Digital Signals

• Analog signals take on a continuous set of values
• Grades
• Between any two analog signal values points, you
  can always find a 3rd value no matter what!
• Instruments for measurement usually give
  estimates since any value is possible

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Analog and Digital Signals

• Digital signals take on a discrete/finite set of
  values
• Traffic signals
• R, Y, and G
• Letter grades
• Between any two adjacent digital signal values
  points, you cant find a 3rd value
• We can measure the exact value cannot have
  fractions we can store them in a fixed space
  (continuous cant)
• E.g. of students in a CSBSJU can surely be
  represented by less than 6 digits 0 - 100,000
• This is what makes them attractive for use in
  computers and hence the name digital computers

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Numeric Data Representation
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2s Complement Representation

• Used by almost all computers today
• All places hold the value they would in binary
  except for the leftmost place which is negative
• 8 bit integer -128 64 32 16 8 4 2 1
• Range -128,127
• If last bit
• is 0, then positive ? looks just as before
• is 1, then negative ?add values for first 7
  digits and then subtract 128
• 1000 1101 148-128 -115

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2s Complement Representation

• Converting from decimal to 2s complement
• For positive numbers find the binary
  representation
• For negative numbers
• Find the binary representation for its positive
  equivalent
• Flip all bits
• Add a 1
• 43
• 43 0010 1011
• -43
• 43 0010 1011 ? 1101 0100 ? 1101 0101
• 1101 0101 -128641641 -43
• We can use the usual laws of binary addition

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2s Complement Representation

• 125 - 19 106
• 0111 1101 1110 1101 ------------- 0110 1010
  106 !
• What happened to the one that we carried at the
  end?
• Got lost but still we got the right answer!
• We carried into and carried out of the leftmost
  column

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2s Complement Representation

• 12565 190
• 0111 11010100 0001 -------------1111 1110 -
  2 !!!
• We only carried into ? overflow
• Number too big to fit in 8 bits since
  range-128,127
• 12565 190 gt 127

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2s Complement Representation

• -125-65 -190
• 1000 00101011 1111 -------------0100 0011 67
  !!!
• We only carried out of ? underflow
• -190 lt -127
• Solution
• use larger registers (more than 7 bits)
• Very large positive and very small negative we
  might still have a problem ? combine two
  registers (double precision)

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Conversion Practice

• Binary to decimal
• 1100 00102 0001 11002
• 19410 2810
• Decimal to binary
• 24510,10210 310
• 1111 01012, 0110 01102 , 000 000112
• Hexa to binary
• BA1016, CA0D16 12FE16
• 1011 1010 0001 00002 , 1100 1010 0000 11012 ,
  0001 0010 1111 11102
• Binary to hexa
• 1100 00102 0001 11002
• C216 1C16
• Decimal to 2s complement
• 102, -102, 3, -3. 245 -245
• 0110 01102, 1001 10102, 0000 00112, 1111 11012,
  outside range outside
• 2s complement to decimal
• 1100 0010 0001 1100
• -6210 2810

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Real Number Representation

• We deal with a lot of real numbers in our lives
  (class average) and while using computers
• Fractions or numbers with decimal parts
• 3/100.3 or 82.34
• 82.34 810 21 31/10 41/100
• To represent this number in binary, we use a
  radix point instead of a decimal point
• 1101.11 8 4 1 ½ ¼ 13.75
• But how can we represent the radix point on the
  computer? 0? 1?

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Real Number Representation

• We resort to the floating-point representation
• We represent the number without the radix
  point!!!
• Based on a popular scientific notation that has
  three parts
• Sign
• Mantissa (one digit before the decimal point, and
  two or more after), and
• Exponent (written with an E following by a sign
  and an integer which represents what power to
  raise 10 to)
• 6.124E5
• 6.124 105 612,400
• Sign? Mantissa? Exponent?
• Scientific to decimal
• -9.14E-3 -9.14 10-3 -9.14/1000 - 0.00914

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Real Number Representation

• Decimal to scientific
• Divide (or multiply) by 10 until you have only
  one digit (usually non-zero) before the decimal
  point
• Add (or subtract) one to the exponent every time
  you divide (or multiply)
• 123.8 ?
• 1.238E2
• 0.2348 ?
• 2.348E-1

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Real Number Representation

• Floating-point numbers is very similar
• But use only 0s and 1s
• For an eight bit number
• 0 000 0000
• Sign, exponent, mantissa (not standard lab 2)
• Sign 0 for positive and 1 for negative
• The exponent has three digits 0,7, but can be
  positive or negative
• shift by 4 ? -4,3
• 000?-4, 001?-3, 010?-2, , 111?3
• i.e. subtract 4 from the real value
• Use base two now (i.e. 2 raised to the power of
  the exponent value)

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Real Number Representation

• The mantissa has one (non-zero) place before the
  decimal point and a number of places after it
• in binary our digits are 0 and 1 and so we always
  have 1.xxxx
• No need to represent the one (add one to the
  result)
• 11100010 ? 1 110 0010
• Negative
• 1106, -4 ? 2, exponent22
• Mantissa 0010 01/2 01/4 11/8 01/16
  1/8
• without the initial 1 which weve omitted ? 11/8
• Or 1 decimal equivalent/16 1 2/16
• - (22)(11/8) -4 1/2

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Real Number Representation

• Floating point to decimal conversion
• 1 break bit pattern into sign (1 bit), exponent
  (3 bits), mantissa (4 bits)
• 2 sign is if 1 and otherwise
• 3 exponent decimal equivalent -4
• 4 mantissa 1 decimal equivalent/16
• 5 number sign (mantissa2exponent)

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Real Number Representation

• What about from decimal to floating point?
• Format the number into the form 1.xxxx2exponent
• Multiply (or divide) by 2 until we have a 1
  before the decimal point
• Subtract (or add) 1 from (to) the exponent for
  every such multiplication (or division)
• Add 4 to result (in floating-to-decimal
  conversion we subtracted 4)
• Convert exponent to binary
• Subtract 1 from the resulting mantissa
• Multiply mantissa by 16
• Round to the nearest integer
• Convert mantissa to binary

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Real Number Representation

• -8.48
• Sign is negative ? 1
• 8.48 ? 4.24 ? 2.12? 1.06 ? 3 divisions
• exponent34 7 or 1112
• mantissa mantissa -1 ? .06
• multiply mantissa by 16 (write it in terms of
  16ths) ? 0.96 1 0r 0001
• 1 111 0001

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Real Number Representation

• Decimal to floating-point conversion
• Sign bit 1 if negative and 0 otherwise
• Exponent 0, mantissa absolute(number)
• While mantissa lt 1 do
• mantissa mantissa 2
• exponent exponent -1
• While mantissa gt 2 do
• mantissa mantissa / 2
• exponent exponent 1
• Exponent exponent 4
• Change to binary
• Mantissa (mantissa -1)16
• Round off to nearest integer and change to binary
• Assemble number sign exponent mantissa

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Real Number Representation

• 0.319
• Positive ? sign bit 02
• mantissa 0.319 2 0.638 ? exponent -1
• mantissa 0.638 2 1.276 ? exponent -2
• exponent -2 4 2 0102
• mantissa (mantissa -1)16 4.416 4 01002
• 0 010 0100
• - 0.319
• Negative ? sign bit 12
• exponent 0102
• mantissa 01002
• 1 010 0100
• 0
• Sign bit 02
• mantissa? Not going to change to 1.xxxx no matter
  what!

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Real Number Representation

• No representation for 0
• 0 000 0000 1.0 2-4 (we assumed there is a 1
  before the mantissa)
• Assume this to be zero!
• Another issue, is the method exact?
• rounding? truncation (close numbers give same
  floating point values)
• Mantissa 4.416 or 4.0123 or 4.400 are all 4
  01002
• Problem is also due to using only 8 bits (4-bit
  mantissa)
• Floating-point numbers require 32 or even 64 bits
  usually
• But still we will have to round off and truncate
• That happens regularly with us even when not
  using computers
• PI 22/7
• 2/3 or 1/3
• We approximate a lot but we should know it !
• Higher precision applications use much larger
  size registers

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Practice

• Scientific to decimal
• -1.23E-10 9.01E3
• -0.000000000123 9010
• Decimal to scientific
• 123.23 0.001911
• 1.23E2 1.911E-3
• Floating point to decimal
• 1 101 1010 -21x1.625 -3.25
• 0 001 1110 2-3x1.875 0.109375
• 1 000 1111 -2-4x2 0.125
• Decimal to floating point
• 3.23 0 101 1010
• 0.2911 0 010 0011

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Non-numeric Data Representation
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Non-numeric Data Representation

• Text data
• The most common type of data people use on
  computers
• How can we transform it to binary --- not
  intuitive as numbers!
• Words can be divided into characters
• Each character can then be encoded by some binary
  code
• Every language has its set of letters ? we will
  limit ourselves to the Latin alphabet
• Numbers were (relatively) easy to map to binary
• Decimal to binary change
• What about letters and other symbols?

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Non-numeric Data Representation

• Many transformations exist and all are arbitrary
• Two popular
• EBCDIC (Extended Binary Coded Decimal Interchange
  Code) by IBM
• ASCII (American Standard Code for Information
  Interchange) by American National Standards
  Institute (ANSI)
• Most widely used (table in book page 22)
• Every letter/symbol is represented by 7 bits
• How many letters/symbols do we have in total?
• A-Z (26) , a-z (26) , 0-9 (10), symbols (33),
  control characters (33)
• If using 1 byte/character ? we have one extra bit
• Extended ASCII-8 (more mathematical and graphical
  symbols or phonetic marks) --- 256

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Non-numeric Data Representation

• Given an integer 00110100
• 2s complement ? 52
• Floating point ? 0.625
• ASCII ? character 4 (check ASCII table in book)
• There must be some code blocks preceding such
  values informing the computer of the type
• Sometimes called meta-data
• What would you use if (2s, floating pt. or
  ASCII)
• Do calculations?
• Phone numbers?

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Non-numeric Data Representation

• Picture/Image Data
• Divide the screen into a grid of cells each
  referred to as a pixel
• 512156 image ? grid has 512 columns and 256 rows
• Pixel values and sizes depend on the type of the
  image
• For black white images, we can use 1 bit for
  every pixel such that 1 ? black and 0 ? white
• For grayscale images, we use 1 byte where 255 ?
  black and 0 ? white and anything in between is
  gray (higher/lower values are closer to
  black/white)
• For color images, we need three values per pixel
  (depends on the color scheme used)
• Red/Green/Blue
• We need 1 byte per value ? 3 bytes per pixel
• For an 512x256 image
• 512x256x3 384K bytes

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Non-numeric Data Representation

• Image movies are built from a number of images
  (or frames) that are displayed in a certain
  sequence at high speeds
• 30 frames per second
• 2-hr movie needs (assume previous image used)
• 384K 30 60 120 83 GB (billion) bytes!
• People use compression to reduce large movie
  sizes
• Usually the change between two consecutive images
  is small ? store only difference between frames
  (Temporal compression)
• Large areas with the same color can be stored by
  saving the boundary pixels only (everything
  within the boundaries has the same value)
  (Spatial compression)

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Non-numeric Data Representation

• Sound/Audio Data
• An object produces sound when it vibrates in
  matter (e.g. air)
• A vibrating object sends a wave of pressure
  fluctuations through the atmosphere
• We hear different sounds from different vibrating
  objects because of variations in the sound wave
  frequency
• Higher wave frequency means air pressure
  fluctuation switches back and forth more quickly
  during a period of time
• We hear this as a higher pitch (intensity)
• When there are fewer fluctuations in a period of
  time, the pitch is lower
• The level of air pressure in each fluctuation,
  the wave's amplitude or height, determines how
  loud the sound is

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Non-numeric Data Representation

• Numbers used to represent the amplitude of sound
  wave
• Analog is continuous and we need digital
• Digitize the sound signal
• Measure the voltage of the signal at certain
  intervals (40,000 per sec)
• Reconstruct wave
• Compression can also be used for audio files
• MP3 reduces size to 1/10th
• ? faster transfer over the Internet

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Non-numeric Data Representation

• Digital image and audio have a lot of advantages
  over non-digital ones
• Can easily be modified by changing the bit
  pattern
• Image enhancement, noise/distortion removal, etc
  
• Superimpose one sound on another or image on
  another results in newer ones

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Non-numeric Data Representation

• It makes you wonder if images can be trusted

• adapted from a course offered at BU

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Non-numeric Data Representation