Raku Land

github:grondilu

# Modular arithmetics in Perl6

Inspired from P5´s Math::ModInt

``````use Modular;

my \$x = 6 Mod 10;
my \$y = 7 Mod 10;
say \$x + \$y;  # should display 3 「mod 10」

# using a verbose object notation
# the modulus must be passed as a named argument
my \$x = Modulo.new: 6, :modulus(10);
my \$x = Modulo.new: 7, :modulus(10);
say \$x * \$y;  # should display 2 「mod 10」
say \$x div \$y; # should display 8 「mod 10」
``````

I found that creating a Modular arithmetic module is not technically difficult (after all most of the operators are already defined in Perl), but it's not easy to chose the most convenient kind of notations about it. And there is also the vocabulary issue.

When talking about modular arithmetics, there are at least three different words that come to mind: "modular", "modulo" and "modulus". Here is what I chose. The name of the module (in the perl6 sense) is 'Modular', as in "Modular arithmetic". I chose this because this module exports a bunch of stuff that are related to modular arithmetic, and I wanted to keep the name short, so I kept only the adjective.

'Modular' defines a class called 'Modulo'. The constructor takes two integers as argument: a residue and a modulus, both of them being public instance variables.

In latin, modulus means 'measure' and modulo is the ablative form. Basically this means that the modulo is what you get from the measure. Hopefully with this etymological hindsight, the choice of names for the class makes sense.

## Purpose of this module

The purpose of this module is to provide a class implementing Real in order to define modular integers. Most modular operators are already defined in perl6, such as '%', 'mod' and 'expmod'. But as far as I know, there is no class to use them with generic functions.

For instance, say you defined the factorial function as such:

``````sub postfix:<!>(\$n) { [*] 1 .. \$n }
``````

You can use such a function to compute 100! (mod 101) efficiently only if you have defined a modular version of '*' and '..'. That is the purpose of this module: to use modular integers as if there were normal integers.