Comairs Glitch, Moores Law, and Morse Code

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Comairs Glitch, Moores Law, and Morse Code
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From last time

• What is this?

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Binary counting1110, or 110 and carry 1
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Positive and Negative Numbers

• Signed and unsigned numbers
• Unsigned 28256 bit patterns represent 0 255
• Signed 28 bit patterns represent -128 127
• Leftmost bit sign bit 0 gt 0 or pos, 1 gt neg
• Largest 8-bit positive number 01111111 127
• 0 00000000
• Most negative negative number 10000000 -128

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Negative numbers
-1 11111111so addition works the same for
positive and negative numbers
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Biggest Numbers

• Biggest positive number 01111111 (like 999999
  on a car odometer)
• Most negative negative number 10000000

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Biggest Positive Number 1 Most Negative
Negative Number
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The Comair Christmas Glitch

• 16 bits for monthly count of crew changes
• Biggest positive 16-bit number 32,767
• December was a bad month, lots of snowstorms,
  lots of flights rescheduled
• As Christmas approached the count went from
  32,767 to -32,768 by adding 1

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How many Bytes?

• 1 byte 8 bits 2 hex digits 1 character
• 210 1024 bytes 1 kilobyte 1KB
• 220 1,048,576 bytes 1 megabyte 1MB
• 230 bytes 1 gigabyte 1GB a billion
• 240 bytes 1 terabyte 1TB a trillion
• 250 bytes 1 petabyte 1PB a quadrillion
• 260 bytes 1 exabyte 1EB a quintillion
• 270 zetta
• 280 yotta

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K

• All this terminology based on the accident that
• Which is 1K?
• There are new standard names
• 1 kibibyte 1000 bytes
• vs. 1 kilobyte 1024 bytes
• But almost no one uses kibi-, mebi-, etc.

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Moores Law (1965)

• The number of transistors on a silicon chip
  doubles every 18 or 12, or 24 months
• 1965 64 26
• 2006 1 billion
• Since 1965 there has been one doubling every 20
  months

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Example of linear increase
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Example of exponential increase

• Now for the y axis use instead lg(y) the
  exponent e such that 2ey

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Same plot, using lg(y) instead of y
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One of the Greatest Engineering Achievements

• An increase of a factor of 224 is about 16
  millionfold
• Increase in disk capacity and processor speed
  have also been exponential
• If human speed had increased that much since
  Moores paper, we would now be traveling faster
  than the speed of light

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The Incomprehensibly Fast Rate of Exponential
Growth

• A 27-decimal-digit counter is enough to have
  counted in nanoseconds since the origin of the
  universe
• 17 letters are enough to name all the stars in
  the universe

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The 40-bit Key Key

• "If you were to tell a cryptographer that this
  system uses 40-bit keys, you'd immediately
  conclude that the system is weak and that you'd
  be able to break it," said Ari Juels, a scientist
  with the research arm of RSA Security
• 240 about a trillion

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Probabilities

• Fair coin P(heads) 1/2
• Fair die P(rolling 3) 1/6
• Fair card deck P(hearts) 1/4
• P(ace) 1/13

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Probabilities of Independent Events Multiply

• P(heads and then heads) 1/2 1/2 1/4
• P(3 and then 4) 1/6 1/6 1/36
• P(ace and ace) 1/131/13 1/169 .0059 but
  only if the first card drawn is replaced and the
  deck is completely reshuffled, otherwise the
  events are not independent
• P(ace and ace without reshuffling)
• 1/13 3/51 .0045

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Unlikely Events

• How likely are 100 heads in a row?
• (1/2)100 10-32 .000000000000000000000000000000
  01

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How Small is 2-100 10-32?

• Age of universe 1018 sec 1027 nanoseconds
• (1 nanosecond 1 ns 1 billionth of a
  second 10-9 sec)
• If you flip a coin 100 times every billionth of
  a second, you will get 100 heads in a row about
  once every hundred thousand lifetimes of the
  universe
• 1032 105 1027
• This is never for all practical purposes

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Morses telegraph1844 1848
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Morse Code (1838)
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Morse Code (1838)
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How Long are Morse Codes on Average?

• Not the average of the lengths of the letters
  (2443)/26 82/26 3.2
• We want the average a to be such that in a
  typical real sequence of say 1,000,000 letters,
  the number of dots and dashes should be about
  a1,000,000
• The weighted average
• (freq of A)(length of code for A)
• (freq of B)(length of code for B)
• .082 .014 .034 .043 2.4

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Data vs. Information

• Message sequence
• yea, nay, yea, yea, nay, nay
• In ASCII, 38 24 bits of data per message
• But if the only possible answers are yea and
  nay, there is only 1 bit of information per
  message
• Entropy is a measure of the information content
  of a message, as opposed to its size
• Entropy 1 bit/message
• 1, 0, 1, 1, 0, 0 same information content but
  24 times more efficient

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Squeezing out the Air

• Suppose you want to ship pillows in boxes and are
  charged by the size of the box
• To use as few boxes as possible, squeeze out all
  the air, pack into boxes, fluff them up at the
  other end
• Lossless data compression
• Entropy lower limit of compressibility