HG

Chapter 7: Integer Arithmetic Rotations

Shifts and Rotations

Overview of Integer Arithmetic

  • Discussed in class: Integer arithmetic, focused on shift and rotate operations.

  • Importance of shifts and rotates for efficient arithmetic operations.

  • Structure of the class: This session covers shifts; another session will cover multiplication and division.

Shift and Rotate Operations

  • Shift Operations: Involves moving bits horizontally either left or right in a binary number.

  • Left Shift: Multiplies the number by 2 for each shift.

  • Right Shift: Divides the number by 2 for each shift.

  • Rotation Operations: Circularly shifts bits, retaining the original bit count.

  • Rotating left brings the leftmost bit to the rightmost position.

  • Rotating right brings the rightmost bit to the leftmost position.

Detailed Explanation of Shifts

  • Left Shift (SHL):

  • Example with a byte: Starting with 0001 (1 in decimal), if shifted left, it becomes 0010 (2), then 0100 (4), then 1000 (8), etc.

  • Each left shift effectively multiplies by 2.

  • Right Shift (SHR):

  • Example with 0001 (1 in decimal): If shifted right, it becomes 0000 (0), if further right it remains 0000.

  • Each right shift effectively divides by 2, rounding down.

Types of Shifts

  • Logical Shift:

  • Ignoring the sign bit, treats all bits uniformly (0s filled in).

  • Used for unsigned numbers.

  • Arithmetic Shift:

  • Considers the sign of the number.

  • When right shifting, the sign bit is preserved to maintain the number's sign (e.g., shifting -8 (11111000) will fill with 1 on the left if it's a negative number).

Important Points on Shifting

  • Carry Flag:

  • When the leftmost or rightmost bit is shifted out, it typically stores that bit in the carry flag for later use.

  • Limits of Shifting:

  • Can only shift up to the number of bits in the register (e.g., 0-7 for an 8-bit register).

  • Shifting too far leads to loss of data and improper results.

Rotations

  • Left Rotate (ROL):

  • Involves taking the most significant bit (MSB) and moving it to the least significant bit (LSB) position, essentially wrapping around.

  • Right Rotate (ROR):

  • Takes the least significant bit (LSB) and places it in the MSB position, overturning the bits circularly.

  • No loss of information while rotating as each bit re-enters the number system, unlike shifting.