> For the complete documentation index, see [llms.txt](https://trizen.gitbook.io/perl6-rosettacode/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://trizen.gitbook.io/perl6-rosettacode/programming_tasks/a/arithmetic/rational.md).

# Rational

Raku supports rational arithmetic natively.

```perl
(2..2**19).hyper.map: -> $candidate {
    my $sum = 1 / $candidate;
    for 2 .. ceiling(sqrt($candidate)) -> $factor {
        if $candidate %% $factor {
            $sum += 1 / $factor + 1 / ($candidate / $factor);
        }
    }
    if $sum.nude[1] == 1 {
        say "Sum of reciprocal factors of $candidate = $sum exactly", ($sum == 1 ?? ", perfect!" !! ".");
    }
}
```

Note also that ordinary decimal literals are stored as Rats, so the following loop always stops exactly on 10 despite 0.1 not being exactly representable in floating point:

```perl
for 1.0, 1.1, 1.2 ... 10 { .say }
```

The arithmetic is all done in rationals, which are converted to floating-point just before display so that people don't have to puzzle out what 53/10 means.