Data Representation in Computer Systems

• Bits, Bytes
• Words
• Positional numbering systems
  • base or radix
  • 41325.23
• Decimal to binary conversion
  • Continue to divide by two (integer part)
  • 49
  • Continue to multiply by two (fractional part)
  • .625
  • .1
• Hex and Octal
  • Conversion to/from hex/octal and binary
• Signed Integer Representation
  • Sign magnitude
    • first bit is a sign, 0 is positive, 1 is negative
    • Usually use a fixed number of bits
    • 100101 is -5, 000101 is 5
    • The sign bit is the most significant bit
    • Not we can represent 1 number with 1 bit
    • 4 numbers with 2 bits
    • 8 numbers with 3 bits
    • 2ⁿ numbers with n bits
    • If we use 1 bit for a sign, we decrease the size of the numbers we can represent by half.
    • Consider 4 bit representations 0000 - 1111
      • Either 0 to 7
      • Or -3 to 3
      • Also note, we have a representation for 0 and -0, we lost a number here
    • How do you do + on sign magnitude numbers?
      • If the sign is the same, add, keep the sign
      • If the sign is different, subtract the smaller from the larger and keep the sign of the larger.
      • This is what you do when you add by hand.
      • Note, in 4 bits, 3+1 0111+0001 = 1000 (-0)
      • This condition is called overflow
      • The result of a computation does not fit in the number of bits in the representation.
    • One's compliment
      • Flip the bits, don't add one.
      • Worry about the carry
      • Doesn't work so well
      • Forget the section
    • Two's Compliment
      • Represent a negative number by flipping the bits and adding 1.
      • Usually in a fixed number of bits
      • Represent -12 in 6 bits two's compliment
      • Represent 12 in 6 bits two's compliment
      • Add the two.
      • In general, to do math it works
      • If the carry in to the last bit doesn't match the carry out from the last bit we have overflow.
      • The biggest problem is the range of numbers is asymmetrical
      • -2^n-1 to 2^n-1-1
    • Multiplication
      • Use long multiplication
      • Store the partial sum
      • Can produce numbers that are two times too big.

        	           101 X 110
        
        		   101  (5)
        		   110  (6)
        		   ---
        		   000
        		  101
        		 101
        		 -----
        		 11110 (16+8+4+2=30)
        
        		 or 
        		 101
        		   0  = 000
        
        		 101
        		   1  - 1010 + 000 = 1010
        
        		 101
        		   1 - 10100+1010 = 11110
        	       

      • A series of shifts and adds.
    • Division
      • Multiple methods, but none as clean as multiplication
      • This takes even longer.