Computer Engineering page 1 Integer arithmetic Depends what you mean by “integer”. Assume at 3-bit string. –Then we define: zero = 000 one = 001 Use zero, one and binary addition: Zero 000 One Zero + one = one. Makes sense!

Computer Engineering page 2 Add one repeatedly, use up all possible patterns: Zero Called the; Unsigned Integer System. No negative integers!

Computer Engineering page 3 Two additions:

Computer Engineering page 4 Two additions: Yes! 5 = But 001 represents one. is = 1???

Computer Engineering page 5 Addition of unsigned integers Error detected by presence of “carry”

Computer Engineering page 6 How do we subtract unsigned integers? We need the concept of the; “ Two’s complement”

Computer Engineering page 7 One’s complement Take any string; Invert every bit; 0 1 This is One’s complement. "NOT”.

Computer Engineering page 8 Two’s complement Given a string; One's complement; then add one. This is called; two’s complement

Computer Engineering page 9 To subtract unsigned A- B Perform: A + 2’s compl (B) = A + Not (B) + 1

Computer Engineering page 10 Example: Carry! =2; Good! No Carry! =6; BAD!

Computer Engineering page 11 Subtraction of unsigned integers Error detected by absence of carry! –Warning: Some machines invert the carry bit on subtraction –So that "carry" => Error for both add and sub

Computer Engineering page 12 Conclusion For unsigned arithmetic we are interested in carry Pay attention! I never used the word "overflow" that's something completely different. Also notice: –3-bit operands gave 3-bit results. –Don't be tempted to write that 4'th bit down!

Computer Engineering page 13 How about negative numbers? How should we represent -1 ? How would we compute 0 - 1? 0 + 2's compl (1) We choose this as our "-1" 1 = =

Computer Engineering page 14 Repeatedly add -1: Zero Less than zero No! High order bit called "sign bit"

Computer Engineering page 15 Signed 3-bit integers Not symmetrical around zero!!!

Computer Engineering page 16 Sign bit The high order bit in a number Also called "N"-bit Value is negative when this bit is "1"

Computer Engineering page 17 Let's try A + B (-1) *000 Both results is OK But: Left case: no carry Right case: carry Conclusion: For signed addition carry is worthless Same conclusion for signed subtraction * carry

Computer Engineering page 18 Some additions A (-3) (-4) (-2)

Computer Engineering page 19 Some additions B (-3) (-4) C C (-2)

Computer Engineering page 20 Some additions C OK BAD (-3) (-4) C C OK BAD (-2) OK OK

Computer Engineering page 21 Some additions D OK BAD (-3) (-4) C C OK BAD (-2) OK OK

Computer Engineering page 22 Error during signed addition: R = A + B A, B same sign and R opposite sign called overflow Notice: Mathematically, signed addition is the same as unsigned addition The same is true for signed subtraction and unsigned subtraction A - B –> A + (-B) –> A + 2's compl (B)

Computer Engineering page 23 Some subtractions A (-1) (-1)

Computer Engineering page 24 Some subtractions B (-1) C (-1) C C C 0 1 1

Computer Engineering page 25 Some subtractions C (-1) C OK BAD (-1) C OK OK C C OK BAD

Computer Engineering page 26 Some subtractions D (-1) C OK BAD (-1) C OK OK C C OK BAD

Computer Engineering page 27 Error during signed subtraction: R = A - B A, B different sign and B, R same sign called overflow

Computer Engineering page 28 Arithmetic- logic unit (ALU) C = carry V = overflow N = sign bit of R Z = 1 if R = 0 32 A B C Operation Condition codes C, V, N, Z

Computer Engineering page 29 Compare two unsigned numbers? is A < B ? Easy! Compute A - B and examine carry But – to compare two signed numbers? is A < B ? Most common mistake: –Compute R = A - B, then look at sign of R. –If R < 0 then A < B (N-bit) Not good enough!

Computer Engineering page 30 To compare two signed numbers: What about –A = - 4 –B = 3 – – – c 001 “If R neg then A < B” – We conclude A ≥ B, that is - 4 ≥ 3 Wrong!

Computer Engineering page 31 Some examples A (-1) (-1)

Computer Engineering page 32 Some examples B (-1) C < +1? No! 3 < -1? No! (-1) C < -1? No! 1 < -1? No! C C < +1? Yes! -4 < 1? Yes!

Computer Engineering page 33 Some examples C (-1) C < +1? No! 3 < -1? No! N = 0, V = 0 N = 1, V = (-1) C < -1? No! 1 < -1? No! N = 0, V = 0 N = 0, V = C C < +1? Yes! -4 < 1? Yes! N = 1, V = 0 N = 0, V = 1

Computer Engineering page 34 To compare signed numbers: Compute R = A - B A < B true if N and V are different A<B = exor(N,V) after computation