With that reasoning, there are no binary numbers. There are no octal numbers. There are no decimal numbers. Yet, anyone understands what you mean if you talk about 'binary numbers' or 'octal numbers'.

Admit it, it's much easier to talk about "binary numbers" than "binary representations of numbers". Even in the PODs we speak of binary numbers when we mean the representation.

       Binary number > 0b11111111111111111111111111111111 non-
       portable
           (W portable) The binary number you specified is larger
           than 2**32-1 (4294967295) and therefore non-portable
           between systems.  See perlport for more on portability
           concerns.

       Illegal binary digit %s ignored
           (W digit) You may have tried to use a digit other than
           0 or 1 in a binary number.  Interpretation of the
           binary number stopped before the offending digit.

       Integer overflow in %s number
           (W overflow) The hexadecimal, octal or binary number
           you have specified either as a literal or as an argu-
           ment to hex() or oct() is too big for your architec-
           ture, and has been converted to a floating point num-
           ber.  On a 32-bit architecture the largest hexadeci-
           mal, octal or binary number representable without
           overflow is 0xFFFFFFFF, 037777777777, or
           0b11111111111111111111111111111111 respectively.  Note
           that Perl transparently promotes all numbers to a
           floating point representation internally--subject to
           loss of precision errors in subsequent operations.

                    -- perldiag

SYNOPSIS
           $n = 1234;              # decimal integer
           $n = 0b1110011;         # binary integer
           $n = 01234;             # octal integer
           $n = 0x1234;            # hexadecimal integer
           $n = 12.34e-56;         # exponential notation
           $n = "-12.34e56";       # number specified as a string
           $n = "1234";            # number specified as a string

                    -- perlnumber

       grok_bin
               converts a string representing a binary number to
               numeric form.


               The hex number may optionally be prefixed with
               "0b" or "b" unless "PERL_SCAN_DISALLOW_PREFIX" is
               set in *flags on entry. If "PERL_SCAN_ALLOW_UNDER-
               SCORES" is set in *flags then the binary number
               may use '_' characters to separate digits.


                    -- perlapi
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Each time it mentions 'binary number', it means 'the representation of a number in binary'.

Abigail