CSCI E2 Section 21109

1
CSCI E-2 Section 2/11/09

• Binary representation
• Character encoding
• Exponential growth

2
Binary Representation

• Computers just understand 0 and 1
• We are used to looking at numbers in base 10
• Can easily convert between decimal and binary by
  dividing by 2 repeatedly

3
Convert 43 to binary

• 43/2 21 remainder 1
• 21/2 10 remainder 1
• 10/2 5 remainder 0
• 5/2 2 remainder 1
• 2/2 1 remainder 0
• 1/2 0 remainder 1
• 43 101011 in binary (read from bottom up)

4
Binary representation

• How do we check our result?
• Start with the binary 101011
• 125 024 123 022 121 120
• 32 0 8 0 2 1 43
• We can do a similar process if we wanted to
  convert to base 5 or base 17 or whatever

5
Character Encoding

• Machines store everything in binary, so how do we
  represent things like text?
• We have to agree on a code that maps characters
  to binary
• One common code ASCII
• Standard ASCII 7 bits
• Extended ASCII 8 bits

6
Some ASCII examples
7
Limitations of ASCII

• How many characters can we represent in 7- and
  8-bit ASCII?
• What about 16-bit Unicode?
• For some character sets, that isnt even enough
• What does doubling the number of bits per
  character do to the number of characters that can
  be represented?

8
Character Sets

• Why are character sets such a big deal in the
  real world?
• Data comes in all sorts of formats and alphabets
  ASCII, Greek, Cyrillic, EBCDIC, Shift_JIS,
• What happens if my computer tries to interpret
  your message using the wrong character set?

9
Exponential Growth

• y kx
• Every time you increase x by 1, y gets multiplied
  by k
• Adding 1 character to the number of characters on
  a license plate allows the state to issue 36
  times more license plates as before

10
Paper Folding

• Start with a sheet of paper .1mm thick and 10
  inches square
• Fold it in half once, and its .2mm thick and
  5x10 inches
• Fold it in half again, and its .4mm thick and
  either 5x10 inches or 2.5x10 inches
• What happens if we keep folding it?

11
Paper Folding, continued

• If we fold it n times, its thickness is 0.1mm
  2n
• If n50, then its 0.1mm250, or about 113
  million km
• Earth to sun is about 150 million km

12
Paper Folding, continued

• If you alternate directions of the fold to keep
  it square after even numbered folds, then the
  resulting paper is 10in 2(n/2)
• If n50, then the paper is about 7nm square
• Thickness of a carbon atom is .22nm, so this is
  the thickness of 35 carbon atoms

13
Solving when x is in the exponent

• I started with paper .1mm thick and folded until
  it was 16.68 miles thick
• How many times did I fold it?
• 16.68 miles 1.61 km/mile 1000000 mm/km
  26,854,800 mm
• .12x 26,854,800
• x lg(26854800 mm/0.1mm) 28 times

14
Mythbusters

• The Mythbusters (on the Discovery channel) showed
  that in normal situations, 7 folds is the limit
• Specially-shaped and super-thin paper can maybe
  get 8 folds
• Ridiculously huge paper in a hangar with a
  steamroller and forklift got 11 folds