How to show carries in long addition?

Question

I am doing binary addition in LaTeX, and I cannot figure out how to show carried numbers above my work. (That is, 1 + 1 = 0 with carry 1, etc.)

Here is a minimal example of my addition without carries:

$\begin{array}[t]{r}
    1100\ 1001 \\
+ \ 1111\ 1111 \\ \hline
    1100\ 1000
\end{array}$ \

How can I show my carried numbers above the top line (in a smaller typeface)?

Answer 1

Do not use inline math ($ … $) for big arrays. I’d use displayed math (\[ … \]). If you are using the amsmath package (which is required for \overset and always a good idea), the unnumbered equation* environment may be used.

References

• Are \( and \) preferable to dollar signs for math mode?
• Why is \[ ... \] preferable to $$ ... $$?

\carry macro

The \carry[<num>] macro oversets the following group with <num> or 1 if the optional argument is not used.

B column type

I also introduced a new column type with the help of the array package, that automatically repeats the right amount of r columns and inserts a appropriate horizontal space (that can be universally changed). If more than nine columns are needed the number has to be grouped: B{12}.

Code

\documentclass{article}
\usepackage{array,mathtools}
\newcommand*{\carry}[1][1]{\overset{#1}}
\newcolumntype{B}[1]{r*{#1}{@{\,}r}}
\begin{document}
\begin{equation*}
\begin{array}{B3}
    \carry 0 & \carry 1\carry 1\carry 0\carry 0 & \carry 1\carry 0\carry 01 \\
      {} + 0 &                             1111 &                      1111 \\ \hline
           1 &                             1100 &                      1000 \\
\end{array}
\end{equation*}

\begin{equation*}
\begin{array}{@{}B2}
    \carry 2\carry 5 & \carry 4\carry[2] 68 \\
                  24 &                  389 \\
             {} + 31 &                  436 \\ \hline
                  81 &                  293 \\
\end{array}
\end{equation*}
\end{document}

Output

Answer 2

Every year I teach a introductory course on computer architecture and I always begin the course with a review of decimal arithmetic to demonstrate the general principles of radix representation and arithmetic.

I also wanted to represent the carry in binary addition but my solution uses a Tikz matrix of math nodes. A minimal example below shows my solution.

\documentclass{article}

\usepackage{lmodern}
\renewcommand*\familydefault{\sfdefault}

\usepackage{amsmath}
\usepackage{tikz}
\usetikzlibrary{matrix}

\newcommand{\addend}{\text{\textsl{\color{gray}{Addend}}}}
\newcommand{\augend}{\text{\textsl{\color{gray}{Augend}}}}
\newcommand{\sumOut}{\text{\textsl{\color{gray}{Sum}}}}

\begin{document}
Calculate the binary sum $0001\;0011\;1101+0000\;1011\;0111$.
\begin{equation}
\begin{tikzpicture}[
    row 1/.style={font=\textsl,font=\scriptsize,black!85, anchor=west,
        inner sep=1.5pt},
    every node/.style={column sep=.5mm,row sep=1mm}]
    \matrix (m) [matrix of math nodes,
        nodes in empty cells,
        %nodes=draw
    ] 
    {
        &   &   &   &   &   & 1 & 1 & 1 & 1 & 1 & 1 &   &                   \\
        & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 1 &[10mm]     \addend \\
    +   & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 1 &           \augend \\ 
        & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 0 & 0 &           \sumOut \\                                                  
    };

    \draw[-,color=black,semithick] (m-3-2.south west) -- (m-3-13.south east);

\end{tikzpicture}
\label{binary_integer_addition}
\end{equation}
\end{document}

Which outputs:

Because each digit is a node it is possible to enhance the presentation of the sum by drawing colored arrows between the nodes (which can be useful in explaining borrowing in in binary subtraction for example).