Exact Fraction of a length

Question

I want to place some elements on my page for which I need to calculate their size.

For my example lets say three of those elements should exactly occupy a \linewidth when put side by side without space in between.

The easiest possibility might be to just give their width as 0.3\linewidth -- but that might be a bit too small generating a gap somewhere. It is however possible to just use 0.3333333333333\linewidth -- but that is much to write for a seemingly simple fraction [and it strikes my pedanticism as it's not exactly one third].

If I want to get an exact value of one third, I may use

\newlength\onethirdlinewidth
\onethirdlinewidth=\linewidth
\divide\onethirdlinewidth by 3

that might be the best way if I use this length multiple times but might be a bit much to type for a one-shot use.

My question is: Is there any simple possibility to get a length of one third (or seven eighths) of a given length?

Answer 1

I described how TeX inputs dimensions and handles units in https://tex.stackexchange.com/a/231281 and Why pdf file cannot be reproduced? and possibly at other locations, including some comments which are not always read.

I am using Plain TeX but of course it works exactly the same in LaTeX.

\newdimen\fixed

\fixed 1pt

\newdimen\testA

\newdimen\testB

\testA 0.33333587646484374\fixed

\testB 0.33333587646484375\fixed

\ifdim\testA = \testB
  The two dimensions are equal
\else
  The two dimensions are not equal
\fi

\bye

Outputs:

One needs 17 fractional digits to be certain that the dimension stabilizes (of course you get only 1sp possible difference after 5 fractional digits, because 1/10^5 < 1/65536, here in this example where one multiplies 1pt). And some things are counterintuitive, for example 0.33333 is enough but 0.22222 is not although it looks closer to 0.222222 than 0.33333 was to 0.333333.

It goes without saying that Knuth has programmed it exactly to fetch 17 fractional digits and not one more, because the theorem is that it will never change after that.

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As another random example consider this

\number\dimexpr 0.824440000pt\relax

\number\dimexpr 0.824440003pt\relax

\bye

which produces

showing that 0.824440003 gives distinct result from 0.82444.

We can confirm this also in a rôle as <factor> :

\documentclass[a4paper]{article}
\usepackage{geometry}

\newlength{\mylength}
\begin{document}

\setlength{\mylength}{0.82444\linewidth}

\verb|0.82444\linewidth| gives \the\mylength.

\setlength{\mylength}{0.824440003\linewidth}

\verb|0.824440003\linewidth| gives \the\mylength.

These two things differ!

I hope this will dispel some misunderstandings\\
about ``five fractional digits suffice''. Wrong.
\end{document}

Notice that above \linewidth is 418.25368pt so 0.000000003\linewidth is in truth 0.00000125476104pt well below the TeX "error".

THE DIFFERENCE IS AMPLIFIED BY A FACTOR GREATER THAN 5000 !

As per the actual question, here is my comment

the most precise way to multiply by a fraction is \dimexpr\numexpr A*<dimen>/B sp\relax (where <dimen> is like \linewidth but not 10pt then use \dimexpr10pt\relax in place of <dimen>)