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CSE115: Introduction to Computer Science I

CSE115: Introduction to Computer Science I. Dr. Carl Alphonce 219 Bell Hall Office hours: M-F 11:00-11:50 645-4739 alphonce@buffalo.edu. Agenda. Today: Two’s complement representation. Two’s complement. fixed-width encoding limited range of values

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CSE115: Introduction to Computer Science I

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  1. CSE115: Introduction to Computer Science I Dr. Carl Alphonce 219 Bell Hall Office hours: M-F 11:00-11:50 645-4739 alphonce@buffalo.edu

  2. Agenda • Today: • Two’s complement representation

  3. Two’s complement • fixed-width encoding • limited range of values • encodes both negative and non-negative values • familiar properties hold • unique representation of zero ( 0 = -0 ) • x = - ( -x ) • x + (-x) = 0 • x - y = x + ( -y ) • last property lets us use addition circuitry to perform subtraction

  4. Bit pattern interpretation • half of bit patterns (those with a zero in the leftmost bit) are for non-negative values, and are interpreted just as base 2 (binary) numbers are • the assignment of values to the remaining bit patterns is done as described on the following slides

  5. -x • To find representation of -x given the representation of x: • find the one’s complement of x • do this by flipping all the bits in the representation (1 becomes 0, 0 becomes 1) • find the two’s complement of the result • do this by adding one to the one’s complement, ignoring any overflow carry

  6. Example • Using a 4-bit wide representation, find the representation of -3: • representation: 0011 • one’s complement: 1100 • two’s complement: 1101 • Representation of -3 is 1101 • Exercise: verify that the desirable properties hold!

  7. Extra pattern? • since -0 = 0, there is one leftover “negative” bit string • let that represent a negative number, -8 in the case of a 4-bit wide representation • in general, range of values for a k-bit wide two’s complement representation is from -2(k-1) to +2(k-1)-1 • for 4-bit wide representation: -8 to +7

  8. Rollover • What happens when you add 1 to 7 in the 4-bit wide scheme? • 0111 + 0001 = 1000 • The answer is -8 (!) • Adding one to the largest magnitude positive number yields the largest magnitude negative number.

  9. Extra negative number • The negative of the largest magnitude negative number is itself. • In the 4-bit wide scheme, -(-8) is -8.

  10. Understand the representation! • It is important that you understand the limitations of working with fixed-width representations.

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