Encoding Information as Bit Patterns

1
Encoding Information as Bit Patterns
2
The Big Picture

• Two Key principles of todays computers
• Instructions are represented as numbers
• Programs can be stored in memory to be read or
  written just like numbers
• Initially, computers were designed to crunch
  numbers, but, as soon as they were commercially
  viable, they were designed to process text as
  well.
• We will try to understand more about representing
  numbers

3
Objectives

• Look at different techniques of encoding
  information
• Compare various number systems and positional
  notation
• Get familiar with base conversion
• Introduce binary encoding of text, numerical
  data, images and sound

4
What is a Number?

• Number is a theoretical concept in mathematics
• Number System
• A convenient way to represent the number concept
• Well-known system is our Decimal system

5
Representing Numbers

• Decimal positional notation uses 10 digits and
  few symbols (., /) to represent all of the
  values that a number may assume. The digits are
  0-9.
• Other ways of representing numbers
• Octal, Hexadecimal, Binary
• Roman Numeral system does not use positional
  notation

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Base of the number representation

• The number of digits used to represent a number
  is called the base of the number (or base of the
  number representation).
• Decimal notation is base 10
• 0,1,2,3,4,5,6,7,8,9
• Binary notation is base 2
• 0,1
• Octal notation is base 8
• 0,1,2,3,4,5,6,7
• Hexadecimal notation is base 16
• 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F

7
Base 60 number representation

• Babylonians had an advanced positional number
  system with base 60
• Approximate days for earth to go around sun was
  known 365 days 1 year
• Babylonians used their base 60 to define number
  of degrees in a circle (360o 6 x 60o)
• 60 seconds 1 minute
• 60 minutes 1 hour
• 24 hours 1 day

8
Why binary for computers?

• Decimals is natural for humans because we have 10
  fingers or digits, so, why not use decimal for
  computers?
• First commercial computer did offer decimal
  arithmetic but it was inefficient.
• The Problem Computers can only use on, off
  signals, so, decimal digit was simply represented
  by several binary digits which was more
  cumbersome
• Less is more.
• - Robert Browning, Andrea del Sarto, 1855

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Positional Notation

• Positional notation represents numbers as a
  string of digits, e.g. 123
• Positions in the string are numbered from the
  rightmost position, and start with position 0
  (zero).
• 3 is in position 0
• 2 is in position 1
• 1 is in position 2

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Positional Notation

• Positional notation means that the value of a
  digit depends on the position of the digit
• The value of the 5 in 50 is different than
  the value of the 5 in 2005
• Binary, Octal, Decimal, and Hex are all
  positional notation schemes
• Roman Numerals are NOT a positional notation
  scheme
• Value of XIII is determined by adding

11
Positional Notation

• The value of the digit n in position p of a
  string representing a number in base b is
• n x b p
• The value of the number is calculated by adding
  the value of each digit in the number
• ? (n x b p )
• sum over the terms (n x b p )

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Digits are the same

• What do we mean?
• The value of 0-9 is the same in each base where
  those digits appear
• Not all bases use all the 10 decimal digits
• Octal (base 8) uses 0-8
• Hex (base 16) uses extra 6 digits with the
  following decimal values
• A 10
• B 11
• C 12
• D 13
• E 14
• F 15

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Converting TO Decimal

• Use the formula ? (n x b p ) to calculate the
  decimal value of the string 123 in
• Octal 1238 is
• 1 x 82 2 x 81 3 x 80 8310
• Hex 12316 is
• 1 x 162 2 x 161 3 x 160 29110

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Converting FROM Decimal

• Converting from decimal into a different base
  requires long division
• Divide the decimal number by the base
• Remainder is 1st digit of the new number
• Divide the quotient by the base
• Remainder is 2nd digit of the new number
• Repeat until quotient is zero and remainder is
  less than the base.

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Algorithm for Decimal to Binary Conversion

• Divide the value by the base (2 for binary) and
  record the remainder
• As long as the quotient obtained is not zero,
  continue to divide the newest quotient by 2 and
  record the remainder
• Now that quotient zero has been obtained, binary
  representation is just the string of remainders
  listed right to left in the order in which they
  were recorded

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Example in text, Page 39
Figure 1.16Applying the algorithm in Figure 1.15
to obtain the binary representation of thirteen
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Lets try it!

• 35 10 converted to binary
• Let R remainder, Q Quotient
• 35/2 gt Q 17, R 1
• 17/2 gt Q 8, R 1
• 8/2 gt Q 4, R 0
• 4/2 gt Q 2, R 0
• 2/2 gt Q 1, R 0
• 1 / 2 gt Q 0, R 1
• Answer 1000112

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Binary to Decimal Conversion
Figure 1.14Decoding the binary representation
100101
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Lets Try it!

• Convert 1000112 to decimal
• 1 x 25 0 x 24 0 x 23 0 x 22 1 x 21 1 x
  20
• 32 0 0 0 2 1
• 3510

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Hexadecimal Notation
Figure 1.6The hexadecimal coding system

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Hexadecimal Notation

• Only strings of 0s and 1s could get too long
  and hard to process (stream of bits)
• Bit patterns in machines tend to have lengths in
  multiples of four
• Four bits per symbol
• What is 10110010?
• What is 101110101101?

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Decimal to Hex Conversion

• 2310 converted to hexadecimal
• 23 / 16 gives Q 1, R 7
• 1 / 16 gives Q 0, R 1
• So 2310 1716

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Exercise Base Conversions

• Purpose Practice representing numbers in
  different bases
• Who Groups of 2-3
• Task Perform the conversions shown on the next
  slide
• Presentation Spokesperson for each group will
  report their results when asked, and be able to
  explain how the results were derived
• Time 10 minutes

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Exercise Working with Bases

• Find the value of ?
• 2510 ?2
• 1101012 ?10
• 1310 ?8
• 4316 ?10
• 9610 ?16

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Answers Working with Bases

• Find the value of ?
• 2510 110012
• 1101012 5310
• 1310 158
• 4316 6710
• 9610 6016

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Conversion Trick

• Binary, Octal, and Hex conversions can be done
  very easily by using a binary grouping
  technique
• Binary numbers use 1 bit groups
• Octal numbers use 3 bit groups
• Hex numbers use 4 bit groups

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Conversion Trick
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Addition

• Remember your addition algorithm?
• Remember sum and carry?
• To add numbers
• Add the digits in a given position
• Divide the result by the base
• The remainder is the new digit for that position
• The quotient is the carry to the next position

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Binary Addition
Figure 1.17The binary addition facts
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Addition

• Binary Addition
• 1 1 10
• 10 10 100
• Octal Addition
• 7 7 16
• 15 5 22
• 12 6 20
• Hex Addition
• A F 19
• 9 B 14
• 18 9 21

• To add numbers
• Add the digits in a given position
• Divide the result by the base
• The remainder is the new digit for that position
• The quotient is the carry to the next position
• A 10
• B 11
• C 12
• D 13
• E 14
• F 15

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Identifying Bases

• Purpose Think about number bases
• Who Groups of 2-3
• Task Identify the base used in each of the
  following operations
• Presentation Spokespeople will provide answers
  when requested
• Time 5 minutes

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Archaeological Find!

• You are part of an archaeological team that has
  found a valuable stone tablet somewhere on the
  islands that were once part of the great
  Atlantis. This stone tablet is the first clue in
  several thousand years that gives an insight into
  the greatness of this ancient culture. Skeletal
  remains found years ago shared a common feature
  that was classified as an anomaly and never
  studied. This tablet contains, to your utter
  surprise and dismay, the following calculations.

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Solve the Mystery!

• It was surprising that this civilization,
  well-known for its greatness, appeared to not
  have mastered simple arithmetic. However, renewed
  study of the skeletal remains along with this
  tablet, proved the Atlanteans were indeed superb
  mathematicians.
• Can you guess the common feature about the
  skeletons that proved this tablet correct?

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Combinations

• Combinatorics is a branch of mathematics that
  studies combinations and complexity
• The number of possible combinations resulting
  from multiple inputs is the product of the number
  of possible input values
• How many possible outfits does a wardrobe of 2
  suits, 3 ties, and 4 pairs of shoes results in ?

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Result 24 outfits!
4 shoes
3 ties
4 shoes
4 shoes
2 suits
4 shoes
4 shoes
3 ties
4 shoes
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Combinations

• Example
• In arranging your schedule for next term, you
  must select one humanities course, one social
  science course, and one math course. If there are
  10 humanities courses, 7 social science courses,
  and 3 math courses, from how many different
  possible schedules could you select?
• (Assume each course can be taken independently
  meaning, no conditional pre-requisites and so on).

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Combinations and Computer Science

• Deals with the number of possible outcomes from a
  set of inputs
• How many different numbers can be represented by
  8 bits?
• How many computers can be connected to the
  Internet using 32-bit addresses?
• How many bits are required to represent true and
  false in a computer?

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Exponents

• Remember these rules for working with exponents?
• XN XM XNM
• (XN)M XNM

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Power of 2

• 24 28 212
• 4 22
• 8 4 x 2 22 x 21 23
• 16 4 x 4 22 x 22 24
• 1612 819 (24)12 ((23)19
• 248 257 24857 2105
• How many bits to represent 2n?
• 1 n zeroes n1 bits

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Binary Encoding

• Binary Encoding is the act of converting
  non-numeric data into binary numbers so that the
  data may be represented by a computer

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Representing Text

• Character codes have been invented to represent
  alphanumeric data on a computer
• ASCII (American Standard Code for Information
  Interchange) uses 8 bits to represent characters
• http//www.asciitable.com
• Online reference for ASCII encoding
• EBCDIC (Extended Binary Coded Decimal Interchange
  Code) is an alternative 8 bit code
• Unicode is a set of 8-,16-,32-bit codes. Unicode
  is designed to work in an internationalized
  environment.
• http//www.unicode.org/

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Images

• Bitmap formats
• Each pixel is encoded
• Bitmap, GIF, JPEG
• Could get large and cumbersome
• Vector formats
• Instructions for drawing the picture are encoded
• Postscript, TrueType fonts
• Makes scaling easier

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Representing Images Bitmap Formats

• Monochrome 1 bit per pixel
• 16 color 4 bits per pixel
• 256 color 8 bits per pixel
• 24-bit color 24 bits per pixel
• JPEG
• Encodes every pixels brightness
• averages color over 4 pixels
• records changes not actual values
• compresses the image
• GIF
• Uses 8 bits per pixel maximum and selective color
  palettes to change the appearance of an image.
  May reduce the file size by detecting the number
  of colors and reducing the number of bits per
  pixel.
• Compresses the image.

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Bitmap Formats
BMP Format 177Kb
GIF Format 31Kb
JPEG Format 10Kb