Just out of curiosity, a couple more variations. Note that Scalar::Util::looks_like_number() thinks things like  +1  -1  2e3 look like numbers that you may not want to accept.

c:\@Work\Perl\monks>perl -wMstrict -le
"use Scalar::Util qw(looks_like_number);
 ;;
 use Data::Dump qw(pp);
 ;;
 for my $n (
   undef, '', '.', '..', '.1.', '1..1',
   0, '0', '0.', '0.0', '.0', '000.000',
   1, '1', '1.', '1.0', '001.100',
   '+1', '-1', '2e3',
   ) {
   printf qq{%s %s like num \n}, pp($n),
     looks_like_number($n) ? 'looks' : 'does not look';
   }
"
undef does not look like num
"" does not look like num
"." does not look like num
".." does not look like num
".1." does not look like num
"1..1" does not look like num
0 looks like num
0 looks like num
"0." looks like num
"0.0" looks like num
".0" looks like num
"000.000" looks like num
1 looks like num
1 looks like num
"1." looks like num
"1.0" looks like num
"001.100" looks like num
"+1" looks like num
-1 looks like num
"2e3" looks like num

c:\@Work\Perl\monks>perl -wMstrict -le
"sub unsigned_real {
   return defined($_[0])
       && $_[0] =~ m{ \A (?: \d+ (?: [.] \d*)? | \d* [.] \d+) \z }xms
        ;
   }
 ;;
 use Data::Dump qw(pp);
 ;;
 for my $n (
   undef, '', '.', '..', '.1.', '1..1',
   '+1', '-1', '2e3',
   0, '0', '0.', '0.0', '.0', '000.000',
   1, '1', '1.', '1.0', '001.100',
   ) {
   printf qq{%s %s real \n}, pp($n),
     unsigned_real($n) ? 'looks' : 'does not look';
   }
"
undef does not look real
"" does not look real
"." does not look real
".." does not look real
".1." does not look real
"1..1" does not look real
"+1" does not look real
-1 does not look real
"2e3" does not look real
0 looks real
0 looks real
"0." looks real
"0.0" looks real
".0" looks real
"000.000" looks real
1 looks real
1 looks real
"1." looks real
"1.0" looks real
"001.100" looks real
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