Bitwise Operations (Cambridge (CIE) A Level Computer Science) : Revision Note

Binary shifts

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 1010 = −6

• 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 0010 = 50

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

• Apply a mask with 1s at those positions

• If the result is not 0, the devices are 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