Bitwise operations (Cambridge International AS Computer Science)

Binary shifts

Spec point — 4.3.1

What are binary shifts?

• A binary shift is an operation that moves all the bits in a binary number left or right by a certain number of positions

• Often used in programming and computer systems for fast multiplication or division by powers of 2, or for manipulating individual bits

• There are three main types of binary shift:

  • Logical
  • Arithmetic
  • Cyclic

Logical

• Moves bits left or right and fills the gap with 0s
• Used for unsigned binary numbers or raw bit manipulation

Left

• The following number is shifted by two places to the left

• Original number: 0000 1110 = 14

• Left shift (2) result: 0011 1000* = 56

• Each left shift has doubled the number:

  • Original value = 14
  • Left shift 1 - Doubled the number to 28
  • Left shift 2 - Doubled the number to 56

Right

• The following number is shifted by three places to the right

• Original number: *1100 1000 *= 200

• Right shift (3) result: 0001 1001* = 25

• Each right shift has halved the number:

  • Original value = 200
  • Right shift 1 - Halved the number to 100
  • Right shift 2 - Halved the number to 50
  • Right shift 3 - Halved the number to 25

Arithmetic

• Moves bits, but preserves the sign bit (used for signed binary numbers in two's complement)

Right

• The following number is shifted by three places to the right

• Original number: 1110 1000 = −24

• Right shift (3) result: *1111 1101 *= −3

• Each arithmetic right shift divides the number by 2, rounding towards negative infinity (preserving the sign bit)

  • Original value = −24
  • Right shift 1 → *1111 0100 *= −12
  • Right shift 2 → *1111 1010 *= −6
  • Right shift 3 → *1111 1101 *= −3

Cyclic

• Rotates the bits around, nothing is lost, no 0s added
• The bit that falls off one end is reused at the other end

• The following number is shifted by three places to the left and right

Left and Right

• The 3 leftmost bits *101 *wrap around to the right side
• The 3 rightmost bits *001 *wrap around to the left side
• Original: *1011 0001 *= 177
• Cyclic left (3): *1000 1101 *= 141
• Cyclic right (3): *0011 0110 *= 54

• Cyclic shifts do not preserve the sign bit and are not equivalent to multiplying or dividing by powers of two

Device control & bit masking

What is device control?

• In computer systems and embedded devices, each bit in a byte can be used to represent the state of a device or feature (e.g. turning a light on, checking if a sensor is active, or flagging an error)
• You can control or monitor these bits using bitwise operations and bit masking

What is bit masking?

• Bit masking uses binary patterns (masks) to test, set, clear, or toggle specific bits without affecting the others

Bitwise AND operation (&)

• Used to test if a specific bit is set to 1
• Only returns 1 when both the binary value and the mask have 1 in the same position
• Useful for checking the status of a device

• Check if a heater (Bit 5) and light (Bit 4) are both ON
• AND the value with a mask that has 1s at those positions: 00110000
• If the result equals the mask (00110000), both devices are ON
• If the result is 0, both are OFF; any other result means only one of them is ON

Bitwise OR operation (|)

• Used to set specific bits to 1
• Returns 1 if either the binary value or the mask has 1
• This is useful to turn on a device (set a bit to 1) without changing the others

• Turn ON the alarm system (Bit 6) and door lock (Bit 5) by ORing with a mask that has 1s at those positions

Bitwise XOR operation (^)

• Used to toggle (flip) specific bits
• Returns 1 if the bits are different
• Can be used for switching a device from ON to OFF or vice versa

• Toggle the fan (Bit 5) and heating system (Bit 4) with a mask that has 1s in those positions

Summary