5.2: Bitwise operators
- Page ID
- 40733
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.