Integer Representations

• This section describe 2 different ways bits can be used to encode integers:
  • Unsigned encodings: present non-negative numbers.
  • Two’s-complements encoding: present negative, zero and positive numbers.

Unsigned encodings:

B2U₄([0001]) = 0 * 2³ + 0 * 2² + 0 * 2¹ + 1 * 2⁰ = 0 + 0 + 0 + 1 = 1

Note 1

B2U_w: binary to unsigned (w bits)

Two’s-complements encodings:

• Use the most significant bit is called the sign bit.

  • If sign bit = 1, the represented value is negative
  • If sign bit = 0, the represented value is non-negative

• Examples:

  • B2T₄([0001]) = - 0 * 2³ + 0 * 2² + 0 * 2¹ + 1 * 2⁰ = 0 + 0 + 0 + 1 = 1
    
  • B2T₄([0101]) = - 0 * 2³ + 0 * 2² + 0 * 2¹ + 1 * 2⁰ = 0 + 4 + 0 + 1 = 5
    
  • B2T₄([1011]) = - 1 * 2³ + 0 * 2² + 0 * 2¹ + 1 * 2⁰ = -8 + 0 + 2 + 1 = -5
    
  • B2T₄([1111]) = - 1 * 2³ + 0 * 2² + 0 * 2¹ + 1 * 2⁰ = -8 + 4 + 2 + 1 = -1
    

Note 1

B2T_w: binary to two’s complement (w bits)

Expanding bits

Convert an unsigned number to a larger unsigned number:

• Add leading zeros the the representation (zero-extension)

Convert a signed number to a larger signed number:

• Add copies of the MSB to the representation (sign-extension)

Convert a signed number to a larger unsigned number (or reverse)

• First changes the sign

• Changes the type

• Example:

  short sx = -12345; // 2's complement: 1100 1111 1100 1111 (CFCF)
  unsigned uy = sx; // equivalent to (unsigned) (int) sx;

  • Convert to a larger type: 1100 1111 1100 1111 => 1111 1111 1111 1111 1100 1111 1100 1111 (sign extension)
  • Change the type: simply convert the binary to decimal => result = 4294954951