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Consider the following code:
\documentclass{article}
\begin{document}
\noindent
Can I make \LaTeX{} reduce a fraction automatically?\\[\baselineskip]
For example, I would like the fraction
\[ \frac{278\,922}{74\,088} \]
to be reduced to
\[ \frac{6641}{1764} \]
\end{document}
P.S. The numerator and denominator are always both natural numbers in my case.
An expl3 implementation:
\nonstopmode \input expl3-generic \relax \ExplSyntaxOn % -*- expl3 -*-
\cs_new:Nn \svend_gcd:nn
{
\int_compare:nNnTF {#2} = { 0 } {#1}
{ \svend_gcd:ff {#2} { \int_mod:nn {#1} {#2} } }
}
\cs_generate_variant:Nn \svend_gcd:nn { ff }
\int_new:N \l__svend_tmp_int
\cs_new:Nn \svend_reduced:nn
{
\int_set:Nn \l__svend_tmp_int { \svend_gcd:nn {#1} {#2} }
{ \int_eval:n { #1 / \l__svend_tmp_int } }
\over
{ \int_eval:n { #2 / \l__svend_tmp_int } }
}
$$ \svend_reduced:nn {278922} {74088} $$
\bye
LaTeX version:
\documentclass{article}
\usepackage{xparse}
\ExplSyntaxOn
% ... code code code
\msg_new:nnn { svend } { malformed-fraction }
{
The~input~you~have~provided~is~malformed.~
Please~provide~input~in~the~form~of~`p/q'.
}
\NewDocumentCommand \ReducedFraction { > { \SplitList { / } } m }
{
\int_compare:nTF { \tl_count:n {#1} = 2 }
{ \svend_reduced:nn #1 }
{ \msg_error:nn { svend } { malformed-fraction } }
}
\ExplSyntaxOff
\begin{document}
\[ \ReducedFraction{278922/74088} \]
\end{document}
\documentclass{article}
\usepackage{xparse}
\ExplSyntaxOn
\cs_new:Nn \svend_gcd:nn
{
\int_compare:nNnTF {#2} = { 0 } {#1}
{ \svend_gcd:ff {#2} { \int_mod:nn {#1} {#2} } }
}
\cs_generate_variant:Nn \svend_gcd:nn { ff }
\int_new:N \l__svend_tmp_int
\cs_new:Nn \svend_reduced:nn
{
\int_set:Nn \l__svend_tmp_int { \svend_gcd:nn {#1} {#2} }
\frac { \svend_reduced_wrap:n { \int_eval:n { #1 / \l__svend_tmp_int } } }
{ \svend_reduced_wrap:n { \int_eval:n { #2 / \l__svend_tmp_int } } }
}
\cs_new:Nn \svend_reduced_use_wrapper:N
{ \cs_set_eq:NN \svend_reduced_wrap:n #1 }
\svend_reduced_use_wrapper:N \use:n
%%% Interface
\msg_new:nnn { svend } { malformed-fraction }
{
The~input~you~have~provided~is~malformed.~
Please~provide~input~in~the~form~of~`p/q'.
}
\NewDocumentCommand \ReducedFractionWrapper { m }
{ \svend_reduced_use_wrapper:N #1 }
\NewDocumentCommand \ReducedFraction { o > { \SplitList { / } } m }
{
\group_begin:
\IfValueT{#1}{\ReducedFractionWrapper{#1}}
\int_compare:nTF { \tl_count:n {#2} = 2 }
{ \svend_reduced:nn #2 }
{ \msg_error:nn { svend } { malformed-fraction } }
\group_end:
}
\ExplSyntaxOff
\usepackage{siunitx}
\begin{document}
\[ \ReducedFraction[\num]{278922/74088} \]
\ReducedFractionWrapper{\num}
\[ \ReducedFraction{27892/74088} \]
\end{document}
\svend_gcd:ff in the inner calls (define a variant), or the arguments will grow up at each recursion. At one stage, the second argument to \svend_gcd:nn in your example is \int_mod:nn {\int_mod:nn {278922}{74088}}{\int_mod:nn {74088}{\int_mod:nn { 278922}{74088}}}
texdoc interface3 is also invaluable). When it comes right down to it, it's still TeX.
Jul 6, 2015 at 12:44
Here is a flat LaTeX2e implementation.
\documentclass{article}
\usepackage{amsmath}
\newcount{\numerator}
\newcount{\denominator}
\newcount{\gcd}
% compute \gcd and returns reduced \numerator and \denominator
\newcommand{\reduce}[2]% #1=numerator, #2=denominator
{\numerator=#1\relax
\denominator=#2\relax
\loop
\ifnum\numerator<\denominator
\advance\denominator by -\numerator
\gcd=\denominator
\else
\advance\numerator by -\denominator
\gcd=\numerator% swap
\fi
\ifnum\gcd>1 \repeat
\ifnum\gcd=0 \gcd=\denominator\fi
\numerator=#1\relax
\divide\numerator by \gcd
\denominator=#2\relax
\divide\denominator by \gcd
}
\begin{document}
For example, I would like the fraction
\begin{equation*}
\frac{278922}{74088}
\end{equation*}
to be reduced to\reduce{278922}{74088}
\begin{equation*}
\frac{\the\numerator}{\the\denominator} =
\frac{6641}{1764}
\end{equation*}
\end{document}
If you are not bound to expl3 (in which case you “just” need to implement the algorithm):
\documentclass{scrartcl}
\usepackage{xintgcd,xintfrac}
\newcommand*\reducedfrac[2]
{\begingroup
\edef\gcd{\xintGCD{#1}{#2}}%
\frac{\xintNum{\xintDiv{#1}{\gcd}}}{\xintNum{\xintDiv{#2}{\gcd}}}%
\endgroup}
\begin{document}
\[
\frac{278922}{74088} = \reducedfrac{278922}{74088}
\]
\end{document}
\xintIrr from xintfrac (no need then to load extra package xintgcd) does the job: \xintIrr{278922/74088} expands to 6641/1764. To typeset as a TeX fraction, there is typesetting macro \xintFrac (using LaTeX's \frac under the hood), thus \newcommand*\reducedfrac [2]{\xintFrac{\xintIrr{#1/#2}}} should do the job.
\frac{\xintNum{#1/\gcd}}{\xintNum{#2/\gcd}}, and, again assuming that #1 and #2 are guaranteed to be given as integers, \xintiiGCD has less overhead than \xintGCD (the latter accepts fractional inputs which it truncates to integers).
An option using Lua+LaTeX.
Made small improvement. Made a Lua function to be called as a LaTeX command, with the numerator and denominator passed as arguments, instead of hardcoding the values in as before. The command is \simplify{a}{b}:
\documentclass{article}
\usepackage{luacode}
\usepackage{amsmath}
%------------------------
\begin{luacode}
function simplify(a,b)
local function gcd(a,b)
if b ~= 0 then
return gcd(b, a % b)
else
return math.abs(a)
end
end
t = gcd(a, b)
tex.print("\\frac{"..a/t.."}{"..b/t.."}")
end
\end{luacode}
\newcommand\simplify[2]{\directlua{simplify(#1,#2) }}%
%-------------------
\begin{document}
\noindent Can I make \LaTeX{} reduce a fraction automatically?\\[\baselineskip]
For example, I would like the fraction
\begin{equation*}
\frac{278\,922}{74\,088}
\end{equation*}
to be reduced to
\begin{equation*}
\simplify{278922}{74088}
\end{equation*}
\end{document}
Original answer
\documentclass{article}
\usepackage{luacode}
\usepackage{amsmath}
\begin{document}
\noindent Can I make \LaTeX{} reduce a fraction automatically?\\[\baselineskip]
For example, I would like the fraction
\begin{equation*}
\frac{278\,922}{74\,088}
\end{equation*}
to be reduced to
%------------------------------------
\begin{luacode*}
function gcd(a,b)
if b ~= 0 then
return gcd(b, a % b)
else
return math.abs(a)
end
end
u = 278922
v = 74088
t = gcd(v, u)
tex.print("\\begin{equation*}")
tex.print(" \\frac{"..u/t.."}{"..v/t.."}")
tex.print("\\end{equation*}")
\end{luacode*}
%------------------------------------
\end{document}
lualatex foo.tex gives
references http://rosettacode.org/wiki/Greatest_common_divisor#Lua and http://www.lua.org/manual/5.3/manual.html
Here is a solution using R for the computations and the R-package knitr to link back to the LaTeX file.
\documentclass{article}
\begin{document}
Using R with the package 'knitr' to reduce the fraction and then get both the reduced fraction but also the components of the fraction.
Note: This is a quick demo of linking R and LaTeX using the 'knitr' package. This has been run on Windows 8.1, with MikTeX 2.9, and TeXmaker 4.4.1 as the IDE. The following code is saved as \textbf{knit02.Rnw} (and this is case sensitive). With the package 'knitr' installed in R 3.1.3 you run the command: \emph{knit("knit02.Rnw")}. This will generate the file \textbf{knit02.tex} which you now compile with pdflatex and view as a pdf.
<<echo=FALSE>>=
library(MASS)
## This function is from http://stackoverflow.com/questions
## /14820029/getting-numerator-and-denominator-of-a-fraction-in-r
getfracs <- function(frac) {
tmp <- strsplit(attr(frac,"fracs"), "/")[[1]]
list(numerator=as.numeric(tmp[1]),denominator=as.numeric(tmp[2]))
}
dd<-278922
nn<-74088
x<- fractions(dd/nn)
fracs<-getfracs(x)
denom<-fracs$denominator
numer<-fracs$numerator
@
\medskip
The original fraction is $\displaystyle{\frac{\Sexpr{as.integer(dd)}}{\Sexpr{as.integer(nn)}}}$.
\medskip
The reduced fraction components are \Sexpr{fracs}.
\medskip
The reduced denominator is \Sexpr{denom}.
\medskip
The reduced numerator is \Sexpr{numer}.
\medskip
And the reduced fraction is $\displaystyle{\frac{\Sexpr{denom}}{\Sexpr{numer}}}$
\end{document}
And the output using pdflatex:
There's already another LuaTeX answer, but IMO it doesn't handle properly with locality and integer division. So here, and taking advantage of the fact Lua 5.3+ includes bitwise operators, a different approach using a binary GCD algorithm:
\documentclass{standalone}
%\usepackage{amsmath}
\usepackage{luacode}
\begin{luacode*}
userdata = userdata or {}
--https://xlinux.nist.gov/dads/HTML/binaryGCD.html
userdata.gcd = function (u,v)
--To handle with negative values
local u, v, g = math.abs(u), math.abs(v), 1
--Nice error message
assert(v~=0, "Denominator cannot be zero")
while u&1==0 and v&1==0 do
u=u>>1
v=v>>1
g=g<<1
end
while u>0 do
if u&1==0 then
u=u>>1
elseif v&1==0 then
v=v>>1
else
local t = math.abs(u-v)>>1
if u<v then v=t else u=t end
end
end
return v*g
end
userdata.simplified = function(u,v)
local gcd = userdata.gcd(u,v)
tex.sprint(("\\frac{%d}{%d}")
:format(u//gcd, v//gcd))
end
\end{luacode*}
\newcommand\simplified[2]{\directlua{userdata.simplified(#1,#2)}}
\begin{document}
$\displaystyle\simplified{278922}{74088}$
\end{document}
The binary algorithm is slightly faster, but that's only noticeable when simplified fractions are extensively used in a document.
If you are an Emacs user, just call Calc embedded (C-x * e) while the point is on
\[ \frac{278922}{74088} \]
and the expression is transformed to
\[ \frac{6641}{1764} \]
`C-x * e' a second time to come back to latex-mode.
Similarly Emacs Calc can perform reduction for rational fractions, for example
\[\frac{x^3 - x^2 - x - 2}{2 * x^3 - x^2 - x - 3}\]
"C- * e" to enter the calc embedded mode and the command "a f" product
\[\frac{x - 2}{2 x - 3}\]
C-x * (Type ? for a list of Calc options) typed e. Error: calc-embedded-make-info: Symbol’s value as variable is void: the-language
Mar 8, 2021 at 16:39