proof format 2 columns

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How to correctly format (and align) a LaTeX proof?

In my proofs I would like to have justifications of each step to the side of the numerical manipulation. For instance consider what I have been doing:

\begin{align*}
\lvert p-q \rvert &= \sqrt{(p-q)^2} \text{      by definition of square root}\\
&=\sqrt{p^2 -2pq +q^2}\text{        by multiplication}\\
&=\sqrt{p^2-2pq+q^2 +2pq -2pq} \text{by the additive identity}\\
&=\sqrt{p^2+2pq+q^2 -4pq}\text{     by grouping like terms}\\
&=\sqrt{(p+q)^2 -4n}\text{          by the distributive law}
\end{align*}

This is messy and more time consuming.

How can I achieve my justifications in some kind of defined format?
What are other best practices?

A tip: If you indent lines by 4 spaces or [enclose words in backticks ```](http://meta.tex.stackexchange.com/q/863), they'll be marked as code, as can be seen in my edit. You can also highlight the code and click the "code" button (with "{}" on it). — hpesoj626, Oct 28 '12 at 22:10

score 6 · Accepted Answer · edited Jun 08 '17 at 23:53

You can also use the alignat math environment as follows:

\documentclass[letterpaper]{article}
\usepackage{amsmath,amssymb}
\newcommand{\justif}[2]{&{#1}&\text{#2}}
\begin{document}
\begin{alignat*}{2}
\lvert p-q\rvert &=\sqrt{(p-q)^2}               &\quad &\text{by definition of square root}\\
                 &=\sqrt{p^2 -2pq +q^2}         &\quad &\text{by multiplication}\\
                 &=\sqrt{p^2-2pq+q^2 +2pq -2pq} &\quad &\text{by the additive identity}\\
                 &=\sqrt{p^2+2pq+q^2 -4pq}      &\quad &\text{by grouping like terms}\\
                 &=\sqrt{(p+q)^2 -4n}           &\quad &\text{by the distributive law}        
\end{alignat*}
%--------------------------------------------------------------------------------
\begin{alignat*}{2}
\lvert p-q\rvert &=\sqrt{(p-q)^2}               \justif{\quad}{by definition of square root}\\
                 &=\sqrt{p^2 -2pq +q^2}         \justif{\quad}{by multiplication}\\
                 &=\sqrt{p^2-2pq+q^2 +2pq -2pq} \justif{\quad}{by the additive identity}\\
                 &=\sqrt{p^2+2pq+q^2 -4pq}      \justif{\quad}{by grouping like terms}\\
                 &=\sqrt{(p+q)^2 -4n}           \justif{\quad}{by the distributive law}           
\end{alignat*}
\end{document}

enter image description here

Increase the space between the math expression and text by adjusting the quad part. Note @egreg suggested an intuitive way of simplifying the input of the comments by a using \justif command.

You only need one of the first &\quad& incantation; on subsequent lines && will be sufficient — Andrew Swann, May 14 '13 at 08:29

score 2 · Answer 2 · edited Jul 15 '17 at 16:20

2

In such a situation I personally use an align with multiple "equations" on the same line:

\begin{align*}
  (p-q) &= \sqrt{(p-q)^2} & & \text{by definition of square root}\\
  &=\sqrt{p^2 -2pq +q^2} & & \text{by multiplication}\\
  &=\sqrt{p^2-2pq+q^2 +2pq -2pq} & & \text{by the additive identity}\\
  &=\sqrt{p^2+2pq+q^2 -4pq} & & \text{by grouping like terms}\\
  &=\sqrt{(p+q)^2 -4n} & & \text{by the distributive law}
\end{align*}

edited Jul 15 '17 at 16:20

David Carlisle

757,742

answered Oct 28 '12 at 22:13

Marco Lombardi

1,415

Possibly adding a personal command (see my comment to Gonzalo's answer). – egreg Oct 28 '12 at 22:22

proof format 2 columns

2 Answers2

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