Two's Complement

Objectives

1. Understand the basics of two's complement
2. Perform computations in two's complement
3. Understand the advantages of two's complement over sign magnitude

Notes

• This is based in part on Carpinelli
• Two's complement is a system to represent signed integers in a fixed number of bits.
• Let's look at some addition in four bits.
  • 15 + 1 = 1111₂ + 0001₂ = 10000₂
  • 14 + 2 = 1110₂ + 0010₂ = 10000₂
  • 13 + 3 = 1101₂ + 0011₂ = 10000₂
  • ...
  • 9 + 7 = 1001₂ + 0111₂ = 10000₂
  • There is one more, but we will ignore that for now.
• Somehow the bit patterns for 1 and 15 "complement" each other
  • Or there is a sum of 0000₂ and a carry of 1.
• What would happen if we let the bit pattern for 15 be a -1?
  • We saw that 15 + 1 = 10000₂
  • What is 15 + 2 0010₂ + 1111₂ = 10001₂ 1, 1
  • What is 15 + 3 0011₂ + 1111₂ = 10010₂ 1, 2
  • Check up to 7, but not beyond.
  • Do a few more, note the carry in and carry out for the MSB.
• What if we let the bit pattern for 14, 1110₂ be a -2
  • Check a few, 2 through 7.
  • Check the carry in and carry out to the MSB
  • Check 1 (1 + -2) = 0001₂ + 1110₂ = 011112
  • Is there a problem here?
  • Check the carry in and carry out to the MSB
  • Try -1 + -2 (1111₂ + 1110₂)
  • This is 13, but following the pattern what would 13 be in our representation?
  • Check the carry in and carry out to the MSB
• We have just "discovered" two's complement.
  • In a n bit system, if the MSB is 0, the number is positive.
    • 000...0₂ is 0
    • 000...01₂ is 1
  • If the MSB is 1, the number is negative.
    • But what number?
    • The clue comes from where we started
    • 1+15 or 0001 and 1111
    • 2+14 or 0010 and 1110
    • 3+13 or 0011 and 1101
    • It is a little hard to see but these bit patterns are one off
    • 1 = 0001 -> 1110 +1 = 1111
    • 2 = 0010 -> 1101 +1 = 1110
    • 3 = 0011 -> 1100 +1 = 1101
• Representation an integer in two's complement
  • To represent a n-1 bit positive number, just represent it in binary.
  • To represent a n-1 bit negative number,
    1. Represent the magnitude in binary
    2. "flip" the bits
    3. Add 1.
• Let's try a few numbers in 5 bits.
  • 6 is 00110₂ so -6 => 11001₂+1 = 11010₂
  • What is -6 + 10, -6 + 2?
  • 14 is 01110₂ so -14 => 10001₂+1 = 10010₂
  • Add -14 + 15, -14 + 7
• To convert from two's complement to sign magnitude
  • If the MSB is a 0 stop
  • Otherwise flip the bits and add 1, represent the sign as -
• There is one strange number, in four bits 1000₂
  • 1000₂=> 0111+1 -> 1000₂
  • But try adding 1, 2, 3 .. 7
  • This is -2^n-1 where n is the number of bits.
• Benefits of two's complement
  • Subtraction is addition
  • Overflow detection, if c_in ≠ c_out of the MSB.
  • One representation for 0
  • Simple sign test, simple conversion to/from sign magnitude
• In n bits, -2^n-1 to 2^n-1-1 can be represented.