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CSE115: Introduction to Computer Science I

This presentation focuses on the concept of integral types in computer science, specifically signed and unsigned integers. It introduces the two’s complement representation used for signed integral values, explains the differences in representation sizes among various types like byte, short, int, and long, and discusses the operations available for integers. Key properties of two’s complement, such as unique representation of zero and conversion methods between positive and negative values, are also examined. Understanding these representations is crucial for working effectively with digital systems.

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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: • Signed integral types • two’s complement representation

  3. int • values: 0, 1, -1, 2, -2, … • maximum int: 2147483647 = +2(32-1)-1 • minimum int: -2147483648 = -2(32-1) • operations: + - * / % 5+2 = 7 +: (int,int)  int 5-2 = 3 -: (int,int)  int 5*2 = 10 *: (int,int)  int 5/2 = 2 (quotient) /: (int,int)  int 5%2 = 1 (remainder) %: (int,int)  int

  4. integral types’ representations • representation used differs according to whether type is signed (byte, short, int, long) or unsigned (char): • signed integral values are represented using “two’s complement” representation • unsigned integral values are represented using “binary” representation • size of representation differs according to type: • byte is 1 byte wide (1 byte = 8 bits) • short is 2 bytes wide • int is 4 bytes wide • long is 8 bytes wide • main point: values of different types have different representations – you can’t “mix and match”!

  5. types of operators for type int • Notice that all of these operators take two int arguments, and produce an int result. • There is hardware circuitry to perform these operations.

  6. 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

  7. 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

  8. -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 • find the two’s complement of the result • do this by adding one to the one’s complement, ignoring any overflow carry

  9. Example • Using a 3-bit wide representation, find the representation of -3: • representation: 011 • one’s complement: 100 • two’s complement: 101 • Representation of -3 is 101 • Exercise: verify that the desirable properties hold!

  10. Extra pattern? • since -0 = 0, there is one leftover “negative” bit string • let that represent a negative number, -4 in the case of a 3-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 3-bit wide representation: -4 to +3

  11. Rollover • What happens when you add 1 to 3 in the 3-bit wide scheme? • 011 + 001 = 100 • The answer is -4 (!) • Adding one to the largest magnitude positive number yields the largest magnitude negative number.

  12. Extra negative number • The negative of the largest magnitude negative number is itself. • In the 3-bit wide scheme, -(-4) is -4.

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

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