Chapter 2: How are data represented

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Chapter 2 How are data represented?
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Why we care?

• The accuracy of our results
• The speed of processing
• The range of alphabets available to us
• The size of the files we must store
• The quality of graphics on screen and on paper
• The time it takes for Internet download

3
Why computers work in binary?

• Cheapest and simplest in design and engineering
• Switch on ? 1 off ? 0
• Circuit voltages
• 1.7 volts higher ? 1
• 0.0 volts - 1.3 volts ? 0
• Voltages (1.3 to 1.7) are avoided in design
• Mathematics binary numbers
• Using digits 0 and 1 only.

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Decimal vs. Binary

• Decimal system
• 10 symbols 1, 2, 3,9, 0
• Base 10 (We have 10 fingers)
• Decimal number 2324 reads 2 thousands 3 hundreds
  twenty four.
• Binary system
• 2 symbols 0 and 1
• Base 2
• Binary number 1101 ?

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Decimal vs. Binary
4
2
3
2
.
Decimal System
21000
3100
210
41
Each digit represents
1000
100
10
1
Position values
103
Position values (base)
102
101
100
Value in Decimal
21000310021041 2324D
1
1
1
0
.
Binary System
23
Position values (base)
22
21
20
8
4
2
1
Position values
18
14
02
11
Each digit represents
Value in Decimal
18140211 13D
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Why do computer work in binary?

• Binary digits bits
• 8 bits 1 byte
• 210 bytes 1024 bytes 1 kilobytes 1KB
• 220 bytes 210 KB 1 megabytes 1MB
• 230 bytes 210 MB 1 gigabytes 1GB
• 240 bytes 210 GB 1 terabytes 1TB

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Types of data

• Instructions
• Computer instructions are coded in sequences of
  0s and 1s
• Numbers
• 2324, -34.35, 34567890123.12345
• Characters and symbols
• A, B, C, Z, a, b, c, z,
• 0, 1, 2, 3 9, , -, ), (, , , etc
• Images
• Photos, charts, drawings
• Audio
• Sound, music, etc
• Video
• Video clips and movies

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Representation of Numbers

• Fixed-size-storage approach
• Computers allocate a specified amount of space
  for a number
• Integers
• 1 bit 0 to 1
• 2 bits 00, 01, 10, 11 ? 0 to 3
• 4 bits 0000, 0001, 0010, 1111 ? 0 to 15
• 1 byte 0 to 255
• 2 bytes -32768 to 32767
• 4 bytes -2,147,648 to 2,147,483,647
• Note with 4 bytes for integers, any number
  smaller than -2,147,648 or larger than
  2,147,483,647 would be incorrectly represented.,

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Representation of Numbers
Binary representation of real numbers
1
1
1
0
.
Binary System
1
21
Position values (base)
20
2-1
2-2
2-3
2
1
1/2
1/4
Position values
1/8
12
01
10.5
10.25
Each digit represents
10.125
Value in Decimal
2 ½ ¼ 1/8 2.875D
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Representation of Numbers

• Floating-point numbers for real numbers
• Three parts of representation
• Sign (always 1 bits 0 for and 1 for -)
• Significant digits (e.g., six bits)
• the power of 2 for the leftmost digit (e.g., 3
  bits)
• Example for binary -1111.01
• Sign 1 (negative)
• Significant digits 111101B
• Power of 2 011B
• Example for binary 100.1101B
• Sign 0 (positive)
• Significant digits 110110B
• Note the last digit is lost, which is 1/16 in
  decimal
• Power of 2 010B

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Representation of Numbers

• Single-precision floating-point numbers
• Sign (always 1 bits 0 for and 1 for -)
• Significant digits 23 bits
• exponent 8
• Double-precision floating-point numbers
• Sign (always 1 bits 0 for and 1 for -)
• Significant digits 52 bits
• exponent 11
• What you should know?
• Computers can represent numbers only in limited
  accuracy.
• E.g., when you enter a 20 digit decimal into a
  program that uses single-precision, only about 7
  digits are actually stored, the rest are lost.
• Real examples
• Designing aircraft on p.35
• The Vancouver Stock Exchange Index on pp. 38-39

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Representation of Numbers

• // file public_html/2005f-html/cil102/accuracy.c
• include ltstdio.hgt
• int main()
• int x, y, result // x, y, and result all
  use 32 bits to represent integers (-2,147,648 to
  2,147,483,647)
• char op
• int i
• for (i 0 i lt 100 i)
• printf("please enter an
  expression\n")
• scanf("d c d", x, op, y)
• if (op '')
• result x y
• else if (op '-')
• result x - y
• else
• printf("Invalid
  operator!!")
• break

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Representation of Numbers

• Variable-size-storage approach
• Allow a wide-range of numbers to be stored
  accurately
• Needs significant more time to process
• Fixed-size approach is used more common than
  variable-size approach.

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Representation of characters

• There are no visual letters A, B, C, etc stored
  in computers like we have in mind.
• Letters and symbols are encoded in 8 bits one
  byte - of 0s and 1s.
• Keyboard converts keys A, B, C etc to their
  corresponding codes and
• monitor converts the code into visual letters A,
  B, C etc on screen.
• Two commonly used coding schemes
• ASCII American Standard Code Information
  Interchange
• EBCDIC Extended Binary Coded Decimal Interchange
  Code

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Representation of characters
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Representation of characters

• Foreign characters two approaches
• Use one byte per char
• Ex.,
• ISO-8859-1 for Western (Roman)
• ISO-8859-7 for Greek
• ISO-2022-CN for simplified Chinese
• Webpage using META charset to specify which
  encoding is used.
• Use two bytes per char/symbols
• 16 bits have 65,536 combinations (characters)
• Unicode coding system

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Representation of Images

• A picture is treated as a matrix of dots, called
  pixels.

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Representation of Images

• The pixels are so small and close together we
  cannot really see them as separate dots.
• Resolution dots per inch (dpi)
• 72 dpi for Web images
• 600 or 1200 dpi for professional printers or home
  photo printers

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Representation of Images

• The color of each pixel is represented using
  bits.
• Black/White one bit per pixel
• 1-white and 0-black
• Gray scale one byte per pixel
• 256 different degrees of gray (00000000 to
  11111111)
• 00000000 black, 01111111 intermediate gray,
  11111111 white
• Color three bytes per pixel
• Red, green, blue color
• One byte for the intensity of each of the three
  color
• 256 possible red, 256 green, 256 blue
• Pure red 11111111 for red byte, 00000000 for
  green and blue
• White 11111111 for all three bytes
• Black 00000000 for all three bytes

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Representation of Images

• Image storage -- size
• Gray scale one byte per pixel
• E.g., A 3 X 5 picture with 300 dpi resolution
• 3 300 900 pixels per column
• 5 300 1500 pixels per row
• 900 1500 1,350,000 pixels/picture
• Needed storage 1,350,000 bytes/picture
  1MB/picture
• Color three bytes per pixel
• E.g., A 3 X 5 picture with 300 dpi resolution
• 3 300 900 pixels per column
• 5 300 1500 pixels per row
• 900 1500 1,350,000 pixels/picture
• Needed storage 3 (bytes per pixel)
  1,350,000
• 4,050,000
  bytes/picture
• 4MB/picture ---
  TOO BIG

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Representation of Images

• Image compression
• Color table
• Most pictures contain a small of different
  colors
• Use a table to define colors that are actually
  used in the picture
• Each pixel has an index to the color table.
• Each image contains a color table and table
  indices
• Example
• For a picture with 100 different colors, the
  color table would contain 100 entries, three
  bytes each entry for each color. One byte can be
  used as index to the table for each pixel.

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Representation of Images

• Drawing commands
• Draw picture using basic commands
• Just as artists draws using a pencil or a brush
  and other basic movements
• Example,
• A house is drawn by sketching various elements
  (doors, windows, walls), adding color to them,
  and moving to the desired position.

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Representation of Images

• Data averaging or sampling
• Condense the size by selecting a smaller
  collection of information to store.
• Many different ways of sampling and data
  averaging
• An example choose to store only every other
  pixel in an image (sampling) reducing the size
  to half. To display the full picture, the
  computer need to fill in the missing data with,
  for example, the average of neighboring pixels
  (data averaging)
• The resulting picture cannot be as sharp as the
  original
• Lossy data compression

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What are .gif, .ps, .jpg, .bmp formats?

• Commonly used image file formats -1
• Bitmap (.bmp)
• Pixel-by-pixel storage of all color information
  for each pixel.
• Lossless representation
• Files are huge.
• Graphics Interchange Format (.gif)
• Use one or more color tables the color table
  technique
• Each table contains 256 colors.
• Suitable for pictures with a small (lt256) of
  different colors (e.g., organization charts)
• Not suitable for pictures with shading (e.g.,
  photos)

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What are .gif, .ps, .jpg, .bmp formats?

• Commonly used image file formats - 2
• PostScript (.ps)
• Employ the drawing commands technique
• moveto draws a line from current position to a
  new one and arc draws an arc given its center,
  radius, etc
• General shapes can be used in multiple places
• Fonts can be reused.
• Useful when the picture can be rendered as a
  drawing or its contains many of the same elements
  (e.g., text of the same fonts)
• Joint Photographic Experts Group (JPEG) (.jpg)
• use the data averaging and sampling on 88 pixel
  blocks
• User determines the level of details and clarity
• High-quality image 88 blocks maintain their
  contents
• Low-quality image info in 88 blocks is
  discarded ? smaller files

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Comparison b/w jpg, gif, and ps

• Pictures in the textbook
• http//www.cs.grinnell.edu/walker/fluency-book/fi
  gures/chapter2/fig-2-overview.html
• Comparison of .jpg and .gif
• http//www.siriusweb.com/tutorials/gifvsjpg/
• More on .jpg and .gif
• http//www.wfu.edu/matthews/misc/jpg_vs_gif/JpgVs
  Gif.html

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Summary chapter 2

• Computers work in binary
• Integers may be constrained in size
• Real numbers may have limited accuracy
• Computations may produce roundoff errors,
  affecting accuracy
• Characters and languages are encoded in binary
• Pictures are displayed pixel by pixel
• Color table, draw commands, and data averaging
  and sampling compression techniques
• .bmp, jpg, .gif, .ps formats

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Terminology

• Binary vs. decimal
• Position value
• The base of a system
• Bit/byte/KB/MB/GB/TB
• Integer binary s
• Real in binary
• Floating point numbers
• Representational error
• Roundoff errors

• ASCII/EBCDIC/Unicode
• Pixels
• Dots per inch (dpi)
• Bitmap
• Color table
• Data averaging
• Data sampling
• Data compression
• .jpg, .bmp, .gif, .ps