2’s Complement Another system that lets us represent negative numbers MSB is STILL the sign bit, but there is no negative zero Negative numbers count backwards and wrap around Calculating 2’s complement (Pos  Neg) Flip the bits ( 01 and 10 ) Add 1

Example ( Pos  Neg ) 110  -110 00012  11112 Flip bits: 1110 110  -110 00012  11112 Flip bits: 1110 Add 1: 1110 +0001 1111

Another Example ( Pos  Neg ) 2510  -2510 000110012  11100111 2 Flip bits: 11100110 Add 1: 11100110 +00000001 11100111

Your Turn Assuming an 8-bit restriction, what is -2110 in 2’s complement? Flip bits Add 1 Answer: 111010112

Your Turn Assuming an 8-bit restriction, what is -3010 in 2’s complement? Flip bits Add 1 Answer: 111000102

Example ( Neg Pos ) -410  410 11002  01002 Flip bits: 0011 -410  410 11002  01002 Flip bits: 0011 Add 1: 0011 +0001 0100

Another Example ( Neg  Pos ) -2910  2910 111000112  000111012 Flip bits: 00011100 Add 1: 00011100 +00000001 00011101

Your Turn Assuming 2’s complement, what is the decimal value of 111110012? Flip bits Add 1 Answer: -710

Your Turn Assuming 2’s complement, what is the decimal value of 111010102? Flip bits Add 1 Answer: 22

2’s Complement Chart Binary Decimal 0111 7 0110 6 0101 5 0100 4 0011 3 0010 2 0001 1 0000 1111 -1 1110 -2 1101 -3 1100 -4 1011 -5 1010 -6 1001 -7 1000 -8 Binary Decimal 00000111 7 00000110 6 00000101 5 00000100 4 00000011 3 00000010 2 00000001 1 00000000 11111111 -1 11111110 -2 11111101 -3 11111100 -4 11111011 -5 11111010 -6 11111001 -7 11111000 -8

SHORTCUT! Find the 1 on the farthest right Flip all the bits to the left of the 1 (DO NOT FLIP THE 1) Example: 4210  -4210 001010102  110101102

Awesomeness of 2’s Complement No more negative zero Lower minimum value: -(2N-1) So what’s the big deal? Everything is addition No need for special hardware to do subtraction

2’s Complement Addition Just like normal positive binary addition You MUST restrict the number of bits IGNORE any overflow bits maintain bit-size restriction

Positive Addition Example 1210 + 410 = 1610 Assuming 2’s complement 000010102 1210 +000000102 + 410 000011002 1610

Negative Addition Example -1210 + -410 = -1610 111101002 -1210 +111111002 + -410 111100002 -1610 NOTE: We ignored the last overflow bit on the left!

Your Turn Show the binary addition of -14 + -3 = -17

Subtraction Example 1610 – 410 = 1610 + -410 = 1210 000100002 +111111002 000011002 NOTE: We ignored the last overflow bit on the left!

Your Turn Show the binary subtraction of 23 – 10 = 13

Overflow / Underflow Problem Addition and bit-size restriction allow for possible overflow / underflow Overflow – when the addition of two binary numbers yields a result that is greater than the maximum possible value Underflow – when the addition/subtraction of two binary numbers yields a result that is less than the minimum possible value

Overflow Example Assume 4-bit restriction and 2’s complement Maximum possible value: 24-1 – 1 = 7 610 + 310 = 910 01102 610 +00112 +310 10012 -710  not good!

Underflow Example Assume 4-bit restriction and 2’s complement Minimum possible value: -(24-1) = -8 -510 + -510 = -1010 10112 -510 +10112 +-510 01102 610  not good!