# 5.2: Bitwise operators


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.

5.2: Bitwise operators is shared under a CC BY-NC license and was authored, remixed, and/or curated by Allen B. Downey (Green Tea Press) .