use warnings;

# Write the warnings handler. It only handles runtime errors.
# To have it handle compile-time errors, place it in a 
# BEGIN{} block
$SIG{__WARN__} = sub {
    for my $w (@_) {
       if($w =~ /isn't numeric/) {die "Treated a string as a number"}
       else {print "Warning received: $w"}
    }
};

# Generates a compile-time warning ("...used only once...")
# that the handler doesn't deal with.
$foo = 0;

#Generates a runtime warning ("... uninitialized value ...")
# that the handler *does* deal with.
my $bar;
print $bar, "\n";

$int = 7;
$string = '5foo';

if($int == 7) {print "Yep, \$int == 7\n"}

# Generates a warning that causes the handler to throw
# the error
if($string == 5) {print "Yep, \$string == 5\n"}
[download]

use Scalar::Util;
.
.
if(Scalar::Util::looks_like_number($x) &&
   $x > 0) {
   print "It's a positive number - not necessarily an integer";
}
[download]

use Scalar::Util;
.
.
if(Scalar::Util::looks_like_number($x) &&
   $x > 0 && $x == int($x)) {
   print "It's a positive integer";
}
[download]

use Scalar::Util;
use warnings;

$string = '5.' . '0' x 30 . '1';;

print $string, "\n";

if(Scalar::Util::looks_like_number($string) &&
  $string > 0 && $string == int($string) ) {
  print "looks like a positive integer\n";
}
else {print "not a positive integer\n"}

__END__
Outputs:
5.0000000000000000000000000000001
looks like a positive integer
[download]

looks_like_number() is good. But this is also dubious as you have pointed out.
And number comparison with string should return an error.