Lots of Bits

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Lots of Bits

• Representations and Moores Law

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How many bit strings of length n?

• Adding one bit doubles the number

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Text

• 8 bits per character
• A 01000001
• ( 00101000
• How many combinations of 8 bits?
• 2 2 2 2 2 2 2 2 28 256

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Hexadecimal Digits
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ASCIIAmerican Standard Code for Information
InterchangeCharacter represented by Hex xy, e.g.
4B is K
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ASCII Underneath

• Emails
• Web pages

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http//farm4.static.flickr.com/3021/2494096946_2bf
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What if you need more than 256 characters?

• Unicode
• 32 bits per character

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How many Unicode characters?32 bits each, so
there are in all

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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
OVERFLOW!
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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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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 bytes
• 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
• 2008 2 billion 231
• 25 doublings in 43 years 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 by a factor of 225 is about 30
  millionfold
• If human speed had increased that much over the
  past 43 years, we would now be traveling faster
  than the speed of light