Number Representation

How do we represent signed (positive & negative) numbers?

Option 3: Two's Complement ☜ ✓ ★ ☺

How did anyone come up with this?

• Use subtraction to derive signed (positive & negative) numbers
• Notice that the resulting number is almost the one's complement; call the new system "two's complement"
•  

  
             00000000     0  (assume a borrow is possible)
           - 00000001  -  1   
           ----------  -----
             ????????    
  
  
            100000000     0  (show the 1 that will be borrowed)
           - 00000001  -  1   
           ----------  -----
             11111111    -1  (two's complement)
  
             11111111    -1  (two's complement)
           - 00000001  -  1   
           ----------  -----
             11111110    -2  (two's complement)
  
             11111110    -2  (two's complement)
           - 00000001  -  1   
           ----------  -----
             11111101    -3  (two's complement)
  
             10000001  -127  (two's complement)
           - 00000001  -  1   
           ----------  -----
             10000000  -128  (two's complement)

• Whoa!
  The range of numbers we can represent is (-128) … 127   not symmetric!
  There are 128 negative numbers and 128 non-negative numbers (but one of these is 0).
  In general, if we have n bits, the range is -2^n-1 …  2^n-1 - 1.