Tuesday, May 31, 2016

Roman Calculus

Ever get one of those crazy ideas in your head that just won't leave you alone until you let it out? Last week I got the exceedingly odd idea of doing calculus with Roman numerals, and couldn't get the idea out of my mind until writing this post.

There are plenty of good reasons why this is a poor idea and why Arabic-Hindu numerals superseded the Roman variants in general usage, which I appreciate a lot more after trying to use Roman numerals. I figured I could do some simple integrals using integers and fractions, so I looked up Roman numerals on Wikipedia to see if they had a system for doing fractions.

Turns out they did, and it is (if possible) even more complicated than their integer system, being a duodecimal (base-12) system instead of the decimal (base-10) system used for integers. (This was apparently in order to more easily handle the common fractions 1/3 and 1/4.) Basically, it used dots to represent 1/12, and the letter S to represent 1/2, and they would just add dots and S's as necessary. (Fractions smaller than 1/12 involved a whole collection of different symbols, many of which I've never seen before, and some that won't even display on my computer.)

The article didn't mention how to actually write fractions and integers, so I used a system where fractions to the left of an integer mean to multiply, while fractions to the right of an integer represent improper fractions. I decided to use y as the variable of integration rather than x, to avoid confusion with X, the numeral, but otherwise used modern conventions such as the integral symbol, the concept of exponents, and using dy for the differential (I considered DY, but that would conflict with D the numeral). The actual integrals used are fairly arbitrary, chosen to be simple but still complicated enough to be interesting.

Anyway, take a look at the madness that results, and feel free to check my work.

\begin{equation}
\begin{split}
\int_\text{I}^\text{IV}y^\text{II}+y+\text{I}\,dy &=
\left[::y^\text{III}+\text{S}y^\text{II}+y\right]_\text{I}^\text{IV}\\
&=\left(::\text{IV}^\text{III}+\text{SIV}^\text{II}+\text{IV}\right)-\left(::\text{I}^\text{III}+\text{SI}^\text{II}+\text{I}\right)\\
&=\left(::\text{LXIV}+\text{SXVI}+\text{IV}\right)-\left(::+\text{S}+\text{I}\right)\\
&=\text{XXI}::+\text{VIII}+\text{IV}-::-\text{S}-\text{I}\\
&=\text{XXXIS}
\end{split}
\end{equation}
\begin{equation}
\begin{split}
\int_\text{II}^\text{IV}y^\text{III}-y^\text{II}\,dy&=
\left[\therefore y^\text{IV}-::y^\text{III}\right]_\text{II}^\text{IV}\\
&=\left(\therefore\text{IV}^\text{IV}-::\text{IV}^\text{III}\right)-
\left(\therefore\text{II}^\text{IV}-::\text{II}^\text{III}\right)\\
&=\therefore\text{CCLVI}-::\text{LXIV}-\therefore\text{XVI}+::\text{VIII}\\
&=\therefore\text{CCXL}-::\text{LVI}\\
&=\text{LX}-\text{XVIIIS}:\\
&=\text{XLI}::
\end{split}
\end{equation}

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