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5.2: Bitwise operators

  • Page ID
    40733
  • People learning C are sometimes confused about the bitwise operators & and |. These operators treat integers as bit vectors and compute logical operations on corresponding bits.

    For example, & computes the AND operation, which yields 1 if both operands are 1, and 0 otherwise. Here is an example of & applied to two 4-bit numbers:

      1100
    & 1010
      ----
      1000
    

    In C, this means that the expression 12 & 10 has the value 8.

    Similarly, | computes the OR operation, which yields 1 if either operand is 1, and 0 otherwise.

      1100
    | 1010
      ----
      1110
    

    So the expression 12 | 10 has the value 14.

    Finally, ^ computes the XOR operation, which yields 1 if either operand is 1, but not both.

      1100
    ^ 1010
      ----
      0110
    

    So the expression 12 ^ 10 has the value 6.

    Most commonly, & is used to clear a set of bits from a bit vector, | is used to set bits, and ^ is used to flip, or “toggle” bits. Here are the details:

    Clearing bits: For any value \( x \), \( x\ \& \ 0 \) is 0, and \( x\ \& \ 1 \) is \( x \). So if you AND a vector with 3, it selects only the two rightmost bits, and sets the rest to 0.

      xxxx
    & 0011
      ----
      00xx
    

    In this context, the value 3 is called a “mask” because it selects some bits and masks the rest.

    Setting bits: Similarly, for any \( x \), \( x\ | \ 0 \) is \( x \), and \( x\ | \ 1 \) is 1. So if you OR a vector with 3, it sets the rightmost bits, and leaves the rest alone:

      xxxx
    | 0011
      ----
      xx11
    

    Toggling bits: Finally, if you XOR a vector with 3, it flips the rightmost bits and leaves the rest alone. As an exercise, see if you can compute the two’s complement of 12 using ^. Hint: what’s the two’s complement representation of -1?

    C also provides shift operators, << and >>, which shift bits left and right. Each left shift doubles a number, so 5 << 1 is 10, and 5 << 2 is 20. Each right shift divides by two (rounding down), so 5 >> 1 is 2 and 2 >> 1 is 1.

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