Sign Magnitude

• Why worry about 9's and 10's complement?
• Sign magnitude
  • This is the way we normally work
  • -3, +14, ...
• But as we saw last time, we have a rules based system for addition
  • If the signs are the same, keep the sign and add the magnitudes
  • Of the signs are different, subtract the smaller magnitude from the larger then keep the sign of the larger.
  • If we have subtraction, change the sign of the second and turn it into an addition problem.
• Plus there is a problem with -0
  • Consider a 3 bit representation in sign magnitude.
  • The first bit (msb) is a sign bit, 0 => +, 1=> -
    • What is 010?
    • What is 000?
    • What is 110?
    • What is 100?
• There are (at least) two problems with sign magnitude
  • -0
  • The rules based addition.
• We do use sign-magnitude for floating point, which is coming soon.
• But
  • The CPU does most of the work in integer
  • So integer needs to be fast
  • And 2's complement is faster than sign-magnitude for integer.
• On common misconception: you can represent fewer signed numbers than unsigned numbers with the same number of bits.
  • This is not true (well mostly not true)
  • With 4 bits, we have $2^4 = 16$ different patterns
  • This means we can represent 16 different numbers.
  • Unsigned we can represent 0-15 (or 0 to $2^{n-1})$
  • Sign magnitude we can represent -7 to 7 (a total of 15).
• In one's complement there is a -0
  • 1111₂=> 0000₂ hence -0
• We will use two's complement from here on out.
• Let's do some math in 6 bit two's complement.
  • First what are the ranges?
    • $(-2^{6-1} to 2^{6-1} -1$
    • Or -32 to 31
    • 17 + 12
    • 17 - 12
    • -17 + 12
    • -17 - -12