1% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved. 2% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk. 3\newcommand{\RomanNumeralXmpTitle}{RomanNumeral} 4\newcommand{\RomanNumeralXmpNumber}{9.72} 5% 6% ===================================================================== 7\begin{page}{RomanNumeralXmpPage}{9.72 RomanNumeral} 8% ===================================================================== 9\beginscroll 10 11% 12% 13% 14% 15% 16 17 18 19The Roman numeral package was added to \Language{} in MCMLXXXVI 20%-% \HDindex{Roman numerals}{RomanNumeralXmpPage}{9.72}{RomanNumeral} 21for use in denoting higher order derivatives. 22 23\xtc{ 24For example, let \spad{f} be a symbolic operator. 25}{ 26\spadpaste{f := operator 'f \bound{f}} 27} 28\xtc{ 29This is the seventh derivative of \spad{f} with respect to \spad{x}. 30}{ 31\spadpaste{D(f x,x,7) \free{f}} 32} 33\xtc{ 34You can have integers printed as Roman numerals by declaring variables to 35be of type \spadtype{RomanNumeral} (abbreviation \spadtype{ROMAN}). 36}{ 37\spadpaste{a := roman(1978 - 1965) \bound{a}} 38} 39 40This package now has a small but devoted group of followers that claim 41this domain has shown its efficacy in many other contexts. 42They claim that Roman numerals are every bit as useful as ordinary 43integers. 44\xtc{ 45In a sense, they are correct, because Roman numerals form a ring and you 46can therefore construct polynomials with Roman numeral coefficients, 47matrices over Roman numerals, etc.. 48}{ 49\spadpaste{x : UTS(ROMAN,'x,0) := x \bound{x}} 50} 51\xtc{ 52Was Fibonacci Italian or ROMAN? 53%-% \HDindex{Fibonacci numbers}{RomanNumeralXmpPage}{9.72}{RomanNumeral} 54}{ 55\spadpaste{recip(1 - x - x^2) \free{x}} 56} 57\xtc{ 58You can also construct fractions with Roman numeral numerators and 59denominators, as this matrix Hilberticus illustrates. 60}{ 61\spadpaste{m : MATRIX FRAC ROMAN \bound{m}} 62} 63\xtc{ 64}{ 65\spadpaste{m := matrix [[1/(i + j) for i in 1..3] for j in 1..3] \free{m} \bound{m1}} 66} 67\xtc{ 68Note that the inverse of the matrix has integral \spadtype{ROMAN} entries. 69}{ 70\spadpaste{inverse m \free{m1}} 71} 72\xtc{ 73Unfortunately, the spoil-sports say that the fun stops when 74the numbers get big---mostly 75because the Romans didn't establish conventions about representing 76very large numbers. 77}{ 78\spadpaste{y := factorial 10 \bound{y}} 79} 80\xtc{ 81You work it out! 82}{ 83\spadpaste{roman y \free{y}} 84} 85\showBlurb{RomanNumeral} 86\endscroll 87\autobuttons 88\end{page} 89% 90