Hello World! An Invitation to Computer Science

Lecture Slides: Binary Numbers

Thursday, February 14, 2008

DON'T PANIC!

Key ideas from last time

• sequential search requires linear-time (O(n)
• binary search requires logarithmic-time (O(lg(n))
    logarithms are nothing to be afraid of
• binary search effective on any list, no matter how large
    but list needs to be ordered
• sorting takes more much work than searching
• some problems are intractable
    solution exists, but takes too long

What are computing agents made of?

computing agents should be simple and substrate neutral:
  it does not matter what it is made out of

thus, standardized binary representations:
  we want simplest possible way of representing information

Bits

• smallest unit of information is a bit
    something that can be in one of two "states"
    on/off, yes/no, true/false, 1/0, etc.
• "bit" : contraction of "Binary digIT"
• bit has no inherent meaning
    impose it by interpreting information in context
    computer does not "know" what 1 or 0 is
      just acts if switch is in one position or other

External v. internal representations

• first, how to represent numbers on a computer
    shortly, how to represent just about anything else
• goal: human users to be able to input data and
    to understand data that is output
• use hardware like keyboards and computer displays
    translate from external rep. (e.g., key labeled "5")
      to internal rep. (sequence of bits)
    and back again

Positional number systems

• why do we use base 10? quirk of evolution!
• positional systems: digits, columns, 0 as place holder
• same idea for base 10 will work for any base
• other interesting bases: 12, 60, 2, 16
• need to have base distinct symbols from 0 up to base-1

Numbers as lists of digits

  3705 =   3000    +   700                +    5 
       = 3 x 1000  +  7 x 100  +  0 x 10  +  5 x 1
       = 3 x 103   +  7 x 102  +  0 x 101  +  5 x 100

power as column number, starting at 0, reading right to left

number as list of digits, backwards
  3705 as L₁, L₂, L₃, L₄: 5, 0, 7, 3

Binary numbers

  base ten   base two           base ten   base two
  -------------------------------------------------
       0        0                  8         1000
       1        1                  9         1001
       2       10                 10         1010
       3       11                 11         1011
       4      100                 12         1100
       5      101                 13         1101 
       6      110                 14         1110
       7      111                 15         1111

From binary to decimal

• break into columns
• figure out to which power of 2 column corresponds
• add up powers of 2 where there 1 appears

From binary to decimal

  input list of bits 
  d ← 0
  i ← 1
  while i ≤ n
    d ← d * 2
    d ← d + Bi
    i ← i + 1
  output d

From decimal to binary

• 0 goes to 0
• find largest power of 2 ≤ #
    write down 1 for that power of 2 column
• subtract and check if next power of 2 ≤ what is left
• repeat, moving to right

From decimal to binary

  input base-10 number d
  if d is 0
     B is list with just one item on it: 0
  else 
     create a list B with nothing on it
     x ← d
     while x > 0
         let q, r be quotient and remainder of x / 2
         add r onto front of list B
         x ← q
  output list B