Integer Representation(Signed and Un-signed Numbers)

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  • Computers use a fixed number of bits to represent an integer. 
  • The commonly-used bit-lengths for integers are 8-bit, 16-bit, 32-bit or 64-bit. 
  • Besides bit-lengths, there are two representation schemes for integers:
    • Unsigned Integers: can represent zero and positive integers.
    • Signed Integers: can represent zero, positive and negative integers.

n-bit Unsigned Integers

  • Unsigned integers can represent zero and positive integers, but not negative integers.
  • Example 1: Suppose that n=8 and the binary pattern is 0100 0001B, the value of this unsigned integer is 1×2^0 + 1×2^6 = 65D.
  • Example 2: Suppose that n=16 and the binary pattern is 0001 0000 0000 1000B, the value of this unsigned integer is 1×2^3 + 1×2^12 = 4104D.

Signed Integers

Signed integers can represent zero, positive integers, as well as negative integers. Three representation schemes are available for signed integers:
    1. Sign-Magnitude representation
    2. 1's Complement representation
    3. 2's Complement representation
  • In all the above three schemes, the most-significant bit (msb) is called the sign bit. 
  • The sign bit is used to represent the sign of the integer - with 0 for positive integers and 1 for negative integers.

Sign-Magnitude representation

  • The most-significant bit (MSB) is the sign bit, with value of 0 representing positive integer and 1 representing negative integer.
  • The remaining n-1 bits represents the magnitude (absolute value) of the integer. 
  • The absolute value of the integer is interpreted as "the magnitude of the (n-1)-bit binary pattern".
  • Example 1: Suppose that n=8 and the binary representation is 0 100 0001B. 
    • Sign bit is 0 means positive Absolute value is 100 0001B = 65D 
    • Hence, the integer is +65D
  • Example 2: Suppose that n=8 and the binary representation is 1 000 0001B. 
    • Sign bit is 1 means negative Absolute value is 000 0001B = 1D 
    • Hence, the integer is -1D

n-bit Sign Integers in 1's Complement Representation

  • Again, the most significant bit (MSB) is the sign bit, with value of 0 representing positive integers and 1 representing negative integers.
  • The remaining n-1 bits represents the magnitude of the integer, as follows:
  • for positive integers, the absolute value of the integer is equal to "the magnitude of the (n-1)-bit binary pattern".
  • for negative integers, the absolute value of the integer is equal to "the magnitude of the complement (inverse) of the (n-1)-bit binary pattern" (hence called 1's complement).
  • Example 1: Suppose that n=8 and the binary representation 0 100 0001B. 
    • Sign bit is 0 means positive Absolute value is 100 0001B = 65D 
    • Hence, the integer is +65D
  • Example 2: Suppose that n=8 and the binary representation 1 000 0001B. 
    • Sign bit is 1 means negative Absolute value is the complement of 000 0001B, 
    • i.e., 111 1110B = 126D Hence, the integer is -126D

n-bit Sign Integers in 2's Complement Representation

  • Again, the most significant bit (msb) is the sign bit, with value of 0 representing positive integers and 1 representing negative integers.
  • The remaining n-1 bits represents the magnitude of the integer, as follows:
    • for positive integers, the absolute value of the integer is equal to "the magnitude of the (n-1)-bit binary pattern".
    • for negative integers, the absolute value of the integer is equal to "the magnitude of the complement of the (n-1)-bit binary pattern plus one" (hence called 2's complement).
  • Example 1: Suppose that n=8 and the binary representation 0 100 0001B. 
    • Sign bit is 0 means positive Absolute value is 100 0001B = 65D 
    • Hence, the integer is +65D
  • Example 2: Suppose that n=8 and the binary representation 1 000 0001B. 
    • Sign bit is 1 means negative 
    • Absolute value is the complement of 000 0001B plus 1, i.e., 111 1110B + 1B = 127D 
    • Hence, the integer is -127D

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