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How can one determine the length of the different units (measured 1ex, 1em, 1in, 1pt, 1bp, 1dd, 1pc) in mm?

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1  
What is the question? –  topskip Jan 4 '11 at 16:48
    
@Patrick, he needs to convert 1ex, 1em, etc to a length in mm. –  xport Jan 4 '11 at 17:05
4  
Please work on your accept rate. You've asked several questions which have been answered; go back and click the checkmark next to whichever answer is best. –  Antal S-Z Jan 4 '11 at 17:46
1  
@Antal, I believe Regis doesn't really understand English. –  Caramdir Jan 4 '11 at 18:04
4  
Se você estiver satisfeito com a resposta, não se esqueça de colocar uma marca de seleção verde. –  xport Jan 5 '11 at 17:26

8 Answers 8

up vote 60 down vote accepted

Here is a variant on Herbert's answer, using \dimexpr instead (inspired from the thread \ifnum for real numbers of comp.text.tex), which allows to do conversions in a purely expandable way. The syntax is \convertto{mm}{1pt} to convert 1pt in mm:

\makeatletter
\def\convertto#1#2{\strip@pt\dimexpr #2*65536/\number\dimexpr 1#1}
\makeatother

The results are not quite the same as with the printlen package, probably due to the fact that \dimexpr performs arithmetic slightly differently from TeX. Here's a table showing all the converted lengths (I omitted sp to avoid arithmetic overflows):

\documentclass[a4paper]{article}

\usepackage{array}
\usepackage[hmargin=2cm]{geometry}

\makeatletter
%http://groups.google.com/group/comp.text.tex/msg/7e812e5d6e67fcc5
\def\convertto#1#2{\strip@pt\dimexpr #2*65536/\number\dimexpr 1#1}
\makeatother

\begin{document}

\begin{center}\begin{tabular}
  {>{\def\colunit{pt}}l<{\convertto{\rowunit}{1\colunit}}
   >{\def\colunit{mm}}l<{\convertto{\rowunit}{1\colunit}}
   >{\def\colunit{cm}}l<{\convertto{\rowunit}{1\colunit}}
   >{\def\colunit{ex}}l<{\convertto{\rowunit}{1\colunit}}
   >{\def\colunit{em}}l<{\convertto{\rowunit}{1\colunit}}
   >{\def\colunit{bp}}l<{\convertto{\rowunit}{1\colunit}}
   >{\def\colunit{dd}}l<{\convertto{\rowunit}{1\colunit}}
   >{\def\colunit{pc}}l<{\convertto{\rowunit}{1\colunit}}
   >{\def\colunit{in}}l<{\convertto{\rowunit}{1\colunit}}
   >{\bfseries}l}
\multicolumn{1}{l}{\bfseries 1pt} & \multicolumn{1}{l}{\bfseries 1mm} & \multicolumn{1}{l}{\bfseries 1cm} & \multicolumn{1}{l}{\bfseries 1ex} & \multicolumn{1}{l}{\bfseries 1em} & \multicolumn{1}{l}{\bfseries 1bp} & \multicolumn{1}{l}{\bfseries 1dd} & \multicolumn{1}{l}{\bfseries 1pc} & \multicolumn{1}{l}{\bfseries 1in} & \\
\gdef\rowunit{pt} & & & & & & & & & \rowunit\\
\gdef\rowunit{mm} & & & & & & & & & \rowunit\\
\gdef\rowunit{cm} & & & & & & & & & \rowunit\\
\gdef\rowunit{ex} & & & & & & & & & \rowunit\\
\gdef\rowunit{em} & & & & & & & & & \rowunit\\
\gdef\rowunit{bp} & & & & & & & & & \rowunit\\
\gdef\rowunit{dd} & & & & & & & & & \rowunit\\
\gdef\rowunit{pc} & & & & & & & & & \rowunit\\
\gdef\rowunit{in} & & & & & & & & & \rowunit\\
\end{tabular}\end{center}

\end{document}
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alt text

And TeX's nice arithmetic can be seen .. ;-)

\documentclass{article}
\usepackage{printlen}
\parindent=0pt

\newlength\Length \Length=1cm
\begin{document}

\tabular{p{2cm}p{2cm}}
\mbox{--- 1cm ---}

\uselengthunit{cm}\printlength{\Length}\\
\uselengthunit{mm}\printlength{\Length}\\
\uselengthunit{in}\printlength{\Length}\\
\uselengthunit{pt}\printlength{\Length}\\
\uselengthunit{bp}\printlength{\Length}\\
\uselengthunit{sp}\printlength{\Length}\\
\uselengthunit{pc}\printlength{\Length}\\
& 
\mbox{--- 1em ---}
\Length=1em
\uselengthunit{cm}\printlength{\Length}\\
\uselengthunit{mm}\printlength{\Length}\\
\uselengthunit{in}\printlength{\Length}\\
\uselengthunit{pt}\printlength{\Length}\\
\uselengthunit{bp}\printlength{\Length}\\
\uselengthunit{sp}\printlength{\Length}\\
\uselengthunit{pc}\printlength{\Length}
\endtabular
\end{document}
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4  
Heh! I like how the centimeter didn't come out right... –  SamB Jan 4 '11 at 19:18
2  
1ex is supposed to be the height of 'x', and 1em is supposed to the width of 'm' -- so they should vary with the font and font size. Do they? –  frabjous Jan 5 '11 at 2:27
2  
@frabjous: 1em is the width of M and sure they vary –  Herbert Jan 5 '11 at 6:57
1  
Thanks, Herbert. That's an important thing to note for the question asked. –  frabjous Jan 5 '11 at 19:31
    
@frabjous: This isn't quite right for fonts with narrow M's, see en.wikipedia.org/wiki/Em_%28typography%29 –  Caramdir Jan 5 '11 at 23:33
  • An inch is 25.4 mm.
  • For TeX, 1 pt is 1/72.27 in, which is 0.351459804 mm.
  • For most other software, 1 pt is 1/72 in, which is 0.352777778 mm. Also called Postscript Point, in TeX this is called a big point (bp).
  • ex and em are not a fixed length, as they depend on the fontsize. (See this question.)

Edit: Herberts answer shows you how to find the length of 1 ex and 1em in your document.

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You have a small typo here. 1 TeX point is 1/72.27 inch (not 1/72.72 inch). BTW I think it is not widely known that Google's calculator supports both points. Exemplary searches: 1 TeX point - 1/72.27 inch, 1 PS point - 1/72 inch, both obviously returning 0. –  przemoc Jun 27 '11 at 20:37
    
@przemoc: Indeed, you're right, thanks. I don't think I'll edit the post though, that'll bring the question to the top of the list again, and the correct value is in the accepted answer anyway. –  Torbjørn T. Jun 27 '11 at 20:54
    
You shouldn't care about such small detail as moving your answer to the top, because keeping the mistake is definitely bad. Moreover, Ctrl+F on 1/72.27 returns only my comment, which is pretty bad, because it is formal definition of point in TeX and should be clearly mentioned somewhere, i.e. in the main text of answer, not in a comment. –  przemoc Jun 27 '11 at 21:06
    
@przemoc: That is a good point, I'll fix it. –  Torbjørn T. Jun 27 '11 at 22:44

Here is another type of table. It gives the exact irreducible conversion factors between units (em and ex are the special font dependent cases; for them I use \dimexpr but this seems not to be a too good idea, it would perhaps be better to get them from the the suitable \fontdimen parameters).

After the exact table I also give a table with values rounded to five decimal places.

Update: Following a suggestion done in a comment, the tables first. There seems to be something fishy with the em: it seems to be exactly 10pt+1sp and not the more intuitively reasonable 10pt. I would have guessed it should be exactly 10pt in that case with the default CM fonts. But I just tested with \the\fontdimen6\font and it also gave to my surprise 10.00002pt, (compiled with pdftex).

Update: before turning off the internet for a while, I checked the em for bold, slanted and teletype: respectively 11.5pt-4sp, 10pt+1sp and 10.5pt-7sp. Perhaps some font expert could explain what is the mechanism? perhaps an underlying conversion from big points at some stage in the font creation process?

Update: I have played a bit with continued fractions, and a good approximation to 1dd is 107/100pt which is clear also from the decimal expansion in the second table, and a very good fraction approximation to 1dd in millimeters is 44/117 (the previous centered convergent is 3/8=0.375 which is already quite good compared to the exact 0.3760650274... value of 1dd in mm).

Exact conversion table

Rounded to 5 places

\documentclass{article}

\usepackage{xintfrac} 
% http://www.ctan.org/tex-archive/macros/generic/xint
% This code was compiled with the version 1.06a of the xint
% bundle dated 2013/05/09, which should appear soon on CTAN.
% The current CTAN version 1.06 should be also OK for this.

\usepackage{array}
\usepackage[hmargin=.5cm]{geometry}

% Conversions to the basic dimension, chosen to be the centimeter
% Base dimension: 1cm
\def\onecm {1}
\def\onemm {1/10}
\def\onein {2.54}
%% \def\onept {\xintMul {1/72.27}{\onein}}
%% simpler:
\def\onept {2.54/72.27}
%% \def\onebp {\xintMul {1/72}{\onein}}
%% simpler:
\def\onebp {2.54/72}
\def\onepc {\xintMul {12}{\onept}}
\def\oneex {\xintMul {\the\numexpr\dimexpr 1ex\relax\relax}{\onesp}}
\def\oneem {\xintMul {\the\numexpr\dimexpr 1em\relax\relax}{\onesp}} 
\def\onedd {\xintMul {1238/1157}{\onept}} 
% 1157 dd = 1238 pt I take this conversion factor from the TeXBook 
% Wikipedia has other conversion factors, but of course here we
% have to do it the TeX way.
\def\onecc {\xintMul {12}{\onedd}}
\def\onesp {\xintMul {1/65536}{\onept}}


% Routines with delimited arguments, the good old TeX way
% (completely expandable)
\makeatletter

% exact conversion to an irreducible fraction:
% example: \convertexactly 126.2772pt\to {bp}
% and `pt' may be a macro expanding to it.
% idem for {bp} which may be a macro expanding to bp
\def\convertexactly #1\to #2%
    {\xintIrr{\convertexactly@ #1\to {#2}}}%

% Variant with rounding at a number of decimal places
% given by first argument.
\def\convertwithrounding #1#2\to #3%
    {\xintRound {#1}{\convertexactly@ #2\to {#3}}}%

% routines doing the job:

\def\convertexactly@ #1\to
{%
    \romannumeral0%
    \expandafter\expandafter\expandafter
    \convertexactly@a
    \xintReverseOrder {#1}\Z
}%
\def\convertexactly@a #1%
{%
    \ifcat\noexpand #1\relax
       \expandafter \convertexactly@b
    \else
       \expandafter \convertexactly@c
    \fi #1%
}%
\def\convertexactly@b #1#2\Z #3%
{%
    \xintdiv {\xintMul {\xintReverseOrder{#2}}{\csname one#1\endcsname}}
             {\csname one#3\endcsname}%
}%
\def\convertexactly@c #1#2#3\Z #4%
{%
    \xintdiv {\xintMul {\xintReverseOrder{#3}}{\csname one#2#1\endcsname}}
             {\csname one#4\endcsname}%
}%

\makeatother

\def\bigstrut {\vbox to 24pt{}\vbox to 12pt{}}%

\begin{document}

Testing:

72.27pt is exactly \convertexactly 72.27pt\to {bp}bp

1/2.54in is exactly \convertexactly 1/2.54in\to {mm}mm

10pt is exactly (for this font) \convertexactly 10pt\to {ex}ex, 
or approximately \convertwithrounding{20}10pt\to {ex}ex

10pt is exactly (for this font) \convertexactly 10pt\to {em}em, 
or approximately \convertwithrounding{20}10pt\to {em}em

1em is exactly (for this font) \convertexactly
1em\to {pt}pt, or approximately
\convertwithrounding{20}1em\to {pt}pt. 

And indeed
\verb+\the\dimexpr 1em\relax+ gives \the\dimexpr 1em\relax{} and
\verb+\the\fontdimen6\font+ gives \the\fontdimen6\font

1ex is exactly (for this font) \convertexactly 1ex\to {pt}pt, or approximately
\convertwithrounding{20}1ex\to {pt}pt. 

And indeed
\verb+\the\dimexpr 1ex\relax+ gives \the\dimexpr 1ex\relax.

\def\tableentry{$\displaystyle\xintFrac{\convertexactly 1\colunit\to\rowunit}$\bigstrut}

\begin{center}\begin{tabular}
  {>{\def\colunit{pt}}l<{\tableentry}
   >{\def\colunit{mm}}l<{\tableentry}
   >{\def\colunit{cm}}l<{\tableentry}
   >{\def\colunit{ex}}l<{\tableentry}
   >{\def\colunit{em}}l<{\tableentry}
   >{\def\colunit{bp}}l<{\tableentry}
   >{\def\colunit{dd}}l<{\tableentry}
   >{\def\colunit{pc}}l<{\tableentry}
   >{\def\colunit{in}}l<{\tableentry}
   >{\bfseries}l}
\multicolumn{1}{l}{\bfseries 1pt} & \multicolumn{1}{l}{\bfseries 1mm} &
\multicolumn{1}{l}{\bfseries 1cm} & \multicolumn{1}{l}{\bfseries 1ex} &
\multicolumn{1}{l}{\bfseries 1em} & \multicolumn{1}{l}{\bfseries 1bp} &
\multicolumn{1}{l}{\bfseries 1dd} & \multicolumn{1}{l}{\bfseries 1pc} &
\multicolumn{1}{l}{\bfseries 1in} & \\
\gdef\rowunit{pt} & & & & & & & & & \rowunit\\
\gdef\rowunit{mm} & & & & & & & & & \rowunit\\
\gdef\rowunit{cm} & & & & & & & & & \rowunit\\
\gdef\rowunit{ex} & & & & & & & & & \rowunit\\
\gdef\rowunit{em} & & & & & & & & & \rowunit\\
\gdef\rowunit{bp} & & & & & & & & & \rowunit\\
\gdef\rowunit{dd} & & & & & & & & & \rowunit\\
\gdef\rowunit{pc} & & & & & & & & & \rowunit\\
\gdef\rowunit{in} & & & & & & & & & \rowunit\\
\end{tabular}\end{center}

\clearpage

\def\tableentry{\convertwithrounding {5}1\colunit\to\rowunit}

\begin{center}\begin{tabular}
  {>{\def\colunit{pt}}l<{\tableentry}
   >{\def\colunit{mm}}l<{\tableentry}
   >{\def\colunit{cm}}l<{\tableentry}
   >{\def\colunit{ex}}l<{\tableentry}
   >{\def\colunit{em}}l<{\tableentry}
   >{\def\colunit{bp}}l<{\tableentry}
   >{\def\colunit{dd}}l<{\tableentry}
   >{\def\colunit{pc}}l<{\tableentry}
   >{\def\colunit{in}}l<{\tableentry}
   >{\bfseries}l}
\multicolumn{1}{l}{\bfseries 1pt} & \multicolumn{1}{l}{\bfseries 1mm} &
\multicolumn{1}{l}{\bfseries 1cm} & \multicolumn{1}{l}{\bfseries 1ex} &
\multicolumn{1}{l}{\bfseries 1em} & \multicolumn{1}{l}{\bfseries 1bp} &
\multicolumn{1}{l}{\bfseries 1dd} & \multicolumn{1}{l}{\bfseries 1pc} &
\multicolumn{1}{l}{\bfseries 1in} & \\
\gdef\rowunit{pt} & & & & & & & & & \rowunit\\
\gdef\rowunit{mm} & & & & & & & & & \rowunit\\
\gdef\rowunit{cm} & & & & & & & & & \rowunit\\
\gdef\rowunit{ex} & & & & & & & & & \rowunit\\
\gdef\rowunit{em} & & & & & & & & & \rowunit\\
\gdef\rowunit{bp} & & & & & & & & & \rowunit\\
\gdef\rowunit{dd} & & & & & & & & & \rowunit\\
\gdef\rowunit{pc} & & & & & & & & & \rowunit\\
\gdef\rowunit{in} & & & & & & & & & \rowunit\\
\end{tabular}\end{center}

\end{document}

testing the conversion macro

Actually the two approximations, given with 20 places after the decimal mark, for 1em and 1ex, are exact, the denominators are powers of 2, the complete decimal expansion only has zeros after those shown. I still do not quite understand why 1em turns out to be 655361sp and not 655360sp=10pt in the case of the CM font. This 10pt+1sp is strange.

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+1, neat! Two suggestions: Use the parskip package to get some space between the “sentences”. Put the picture first in your answer so that it’s closer to your remarks; unlike in most other answers, the output seems more important than the code here. –  doncherry May 9 '13 at 21:02
    
@doncherry, done for the tables. –  jfbu May 9 '13 at 21:16
    
see also the section "Dimensions" in the xint manual, which perhaps explain better than here how TeX converts a dimension expressed in a given unit to an integer number of scaled points. The case of the Didot point is interesting. I think I understood less things when I wrote the answer above. –  jfbu Mar 26 at 17:21
    
Sorry. What is the exact range in pt? –  Please don't touch Jul 6 at 9:41

The same as Philippe Goutet's, but using the fp module of LaTeX3; however the syntax of the \convertto command is different:

\convertto{1in}{pt}

returns 72.26999, just the number. With \convertto*{1in}{pt} we'd get 72.26999pt, with the unit.

\documentclass[a4paper]{article}

\usepackage{array}
\usepackage[hmargin=2cm]{geometry}

\usepackage{xparse}
\ExplSyntaxOn
\DeclareExpandableDocumentCommand{\convertto}{smm}
 {
  \egreg_convertto:nn {#2}{#3}
  \IfBooleanT{#1}{#2}
 }

\cs_new:Npn \egreg_convertto:nn #1 #2
 {
  \fp_eval:n { round( \dim_to_fp:n { #1 } / 1#2 , 5 ) }
 }
\ExplSyntaxOff

\begin{document}

\begin{center}\begin{tabular}
  {>{\def\colunit{pt}}l<{\convertto{1\colunit}{\rowunit}}
   >{\def\colunit{mm}}l<{\convertto{1\colunit}{\rowunit}}
   >{\def\colunit{cm}}l<{\convertto{1\colunit}{\rowunit}}
   >{\def\colunit{ex}}l<{\convertto{1\colunit}{\rowunit}}
   >{\def\colunit{em}}l<{\convertto{1\colunit}{\rowunit}}
   >{\def\colunit{bp}}l<{\convertto{1\colunit}{\rowunit}}
   >{\def\colunit{dd}}l<{\convertto{1\colunit}{\rowunit}}
   >{\def\colunit{pc}}l<{\convertto{1\colunit}{\rowunit}}
   >{\def\colunit{in}}l<{\convertto{1\colunit}{\rowunit}}
   >{\bfseries}l}
\multicolumn{1}{l}{\bfseries 1pt} & \multicolumn{1}{l}{\bfseries 1mm} &
\multicolumn{1}{l}{\bfseries 1cm} & \multicolumn{1}{l}{\bfseries 1ex} &
\multicolumn{1}{l}{\bfseries 1em} & \multicolumn{1}{l}{\bfseries 1bp} &
\multicolumn{1}{l}{\bfseries 1dd} & \multicolumn{1}{l}{\bfseries 1pc} &
\multicolumn{1}{l}{\bfseries 1in} & \\
\gdef\rowunit{pt} & & & & & & & & & \rowunit\\
\gdef\rowunit{mm} & & & & & & & & & \rowunit\\
\gdef\rowunit{cm} & & & & & & & & & \rowunit\\
\gdef\rowunit{ex} & & & & & & & & & \rowunit\\
\gdef\rowunit{em} & & & & & & & & & \rowunit\\
\gdef\rowunit{bp} & & & & & & & & & \rowunit\\
\gdef\rowunit{dd} & & & & & & & & & \rowunit\\
\gdef\rowunit{pc} & & & & & & & & & \rowunit\\
\gdef\rowunit{in} & & & & & & & & & \rowunit\\
\end{tabular}\end{center}

\end{document}

Both \convertto and \convertto* can be used in an expandable context.

enter image description here

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It seems that TeX's memory you used is more important. For Philippe :777 strings out of 493481 10679 string characters out of 3140965 67406 words of memory out of 3000000 For you 7172 strings out of 493481 128969 string characters out of 3140965 174995 words of memory out of 3000000 –  Alain Matthes May 9 '13 at 8:16
1  
@AlainMatthes Loading expl3 on top of LaTeX2e has a big impact. If I load xparse over Philippe's macros I get 7166 strings out of 493481, 128879 string characters out of 3140966, 175547 words of memory out of 3000000. LaTeX3 will surely consume more memory than LaTeX2e, at the advantage of a cleaner approach. –  egreg May 9 '13 at 8:52
    
@AlainMatthes There's a reason that the LaTeX3 approach didn't fly at all when it was first developed (early 1990s). As egreg says, there is a cost to having a more structured approach, but the idea is that you enhance code readability and logic. –  Joseph Wright May 10 '13 at 6:50

My posting just crossed the other ones. Best to view it, in action.

\documentclass[11pt]{article} % use larger type; default would be 10pt
\usepackage{xcolor}
\begin{document}

\newdimen\temp

\def\alength#1#2{
\temp#1
\nointerlineskip \baselineskip=2pt
\vbox{\hbox{\hskip-29pt\texttt{\textcolor{#2}{#1=\the\temp}}}
\hbox{\vtop{\color{#2}\hrule width 130pt\vspace{#1}\hrule width 130pt}}%
\vspace{0.9cm}}
}

\alength{1cm}{blue}
\alength{1pc}{red}
\alength{1mm}{orange}
\alength{1cc}{red}
\alength{1dd}{red}
\alength{100000sp}{purple}
\alength{1in}{black}
\alength{1bp}{brown}
\alength{1em}{black}
\alength{1ex}{black}
\end{document}
share|improve this answer
1  
It is better if you put all in one base line to make a comparison. –  xport Jan 4 '11 at 17:26
    
@xport The base is the relationship to points! I tried to give something to visualize the length (with the exception of the sp, which is smaller than the visible wavelength and hence I scaled that by100K. Unit conversion tables are normally done like this. –  Yiannis Lazarides Jan 4 '11 at 18:32
    
Thanks Yiannis Lazarides, but how to calc result in mm? 1pc=x mm 1pt=x mm 1ex=x mm 1em=x mm –  Regis da Silva Jan 5 '11 at 11:05

Some more examples with 1em

\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage{mathpazo}
\newsavebox\CBox
\newlength\Length \Length=1em
\begin{document}

\sbox\CBox{M}
\the\wd\CBox : \the\Length

\tiny\Length=1em
\sbox\CBox{M}
\the\wd\CBox : \the\Length

\Huge\Length=1em
\sbox\CBox{M}
\the\wd\CBox : \the\Length

\end{document}

alt text

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As already mentioned in other answers, the e-TeX \numexpr and dimexpr allow easy comparisons of dimensions, through conversions to integers. In particular applying \number to a \dimexpr expression returns the internal representation of the dimension in scaled points: there are 65536 scaled points in one pt. Using this we can see some funny effect of the internal truncations done by TeX:

big points

We see that 100bp are represented in an exact manner internally (exact meaning that the ratio 72.27/72 is exactly verified). The next to last line should read rather 100bp/100pt, and not 1bp/1pt, to illustrate that this exact ratio is obtained only for dimensions which are integer multiples of 100bp. As we can see indeed, it is not possible to represent exactly 10bp as an integer number of sp. We see that truncation, rather than rounding is used. Indeed 10bp would be more accurately represented by 657818sp and 1bp would be more accurately equated to 65782sp.

The code for producing the above:

1bp=\number\dimexpr 1bp\relax sp

10bp=\number\dimexpr 10bp\relax sp

100bp=\number\dimexpr 100bp\relax sp

1000bp=\number\dimexpr 1000bp\relax sp

10000bp=\number\dimexpr 10000bp\relax sp


1pt=\number\dimexpr 1pt\relax sp

10pt=\number\dimexpr 10pt\relax sp

100pt=\number\dimexpr 100pt\relax sp

1000pt=\number\dimexpr 1000pt\relax sp

10000pt=\number\dimexpr 10000pt\relax sp

\bye

The last lines were done with \input xintfrac.sty (package xint ) and

1bp/1pt=\xintIrr{\number\dimexpr 100bp\relax/\number\dimexpr 100pt\relax }

72.27/72=\xintIrr{72.27/72}

\bye

To get the e-TeX extensions the executable etex (or pdftex which is what I actually used) rather than tex must be used.

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