Building on everyone else's comments and leveraging Math::Calc::Parser, here is a simple yet reasonably secure solution.

#!/usr/bin/env perl
use strict;
use warnings;
use autodie;
use feature 'say';

use Math::Calc::Parser 'calc';

open my $in,  '<', 'numbers.txt';
open my $out, '>', 'sum.txt';
while (<$in>) {
    say $out calc $_;
}
close $out;
close $in;
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