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NAME

       perlnumber - semantics of numbers and numeric operations in Perl

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

DESCRIPTION

       This document describes how Perl internally handles numeric values.

       Perl's operator overloading facility is completely ignored here.  Operator overloading
       allows user-defined behaviors for numbers, such as operations over arbitrarily large
       integers, floating points numbers with arbitrary precision, operations over "exotic"
       numbers such as modular arithmetic or p-adic arithmetic, and so on.  See overload for
       details.

Storing numbers

       Perl can internally represent numbers in 3 different ways: as native integers, as native
       floating point numbers, and as decimal strings.  Decimal strings may have an exponential
       notation part, as in "12.34e-56".  Native here means "a format supported by the C compiler
       which was used to build perl".

       The term "native" does not mean quite as much when we talk about native integers, as it
       does when native floating point numbers are involved.  The only implication of the term
       "native" on integers is that the limits for the maximal and the minimal supported true
       integral quantities are close to powers of 2.  However, "native" floats have a most
       fundamental restriction: they may represent only those numbers which have a relatively
       "short" representation when converted to a binary fraction.  For example, 0.9 cannot be
       represented by a native float, since the binary fraction for 0.9 is infinite:

         binary0.1110011001100...

       with the sequence 1100 repeating again and again.  In addition to this limitation,  the
       exponent of the binary number is also restricted when it is represented as a floating
       point number.  On typical hardware, floating point values can store numbers with up to 53
       binary digits, and with binary exponents between -1024 and 1024.  In decimal
       representation this is close to 16 decimal digits and decimal exponents in the range of
       -304..304.  The upshot of all this is that Perl cannot store a number like
       12345678901234567 as a floating point number on such architectures without loss of
       information.

       Similarly, decimal strings can represent only those numbers which have a finite decimal
       expansion.  Being strings, and thus of arbitrary length, there is no practical limit for
       the exponent or number of decimal digits for these numbers.  (But realize that what we are
       discussing the rules for just the storage of these numbers.  The fact that you can store
       such "large" numbers does not mean that the operations over these numbers will use all of
       the significant digits.  See "Numeric operators and numeric conversions" for details.)

       In fact numbers stored in the native integer format may be stored either in the signed
       native form, or in the unsigned native form.  Thus the limits for Perl numbers stored as
       native integers would typically be -2**31..2**32-1, with appropriate modifications in the
       case of 64-bit integers.  Again, this does not mean that Perl can do operations only over
       integers in this range: it is possible to store many more integers in floating point
       format.

       Summing up, Perl numeric values can store only those numbers which have a finite decimal
       expansion or a "short" binary expansion.

Numeric operators and numeric conversions

       As mentioned earlier, Perl can store a number in any one of three formats, but most
       operators typically understand only one of those formats.  When a numeric value is passed
       as an argument to such an operator, it will be converted to the format understood by the
       operator.

       Six such conversions are possible:

         native integer        --> native floating point       (*)
         native integer        --> decimal string
         native floating_point --> native integer              (*)
         native floating_point --> decimal string              (*)
         decimal string        --> native integer
         decimal string        --> native floating point       (*)

       These conversions are governed by the following general rules:

       ·   If the source number can be represented in the target form, that representation is
           used.

       ·   If the source number is outside of the limits representable in the target form, a
           representation of the closest limit is used.  (Loss of information)

       ·   If the source number is between two numbers representable in the target form, a
           representation of one of these numbers is used.  (Loss of information)

       ·   In "native floating point --> native integer" conversions the magnitude of the result
           is less than or equal to the magnitude of the source.  ("Rounding to zero".)

       ·   If the "decimal string --> native integer" conversion cannot be done without loss of
           information, the result is compatible with the conversion sequence "decimal_string -->
           native_floating_point --> native_integer".  In particular, rounding is strongly biased
           to 0, though a number like "0.99999999999999999999" has a chance of being rounded to
           1.

       RESTRICTION: The conversions marked with "(*)" above involve steps performed by the C
       compiler.  In particular, bugs/features of the compiler used may lead to breakage of some
       of the above rules.

Flavors of Perl numeric operations

       Perl operations which take a numeric argument treat that argument in one of four different
       ways: they may force it to one of the integer/floating/ string formats, or they may behave
       differently depending on the format of the operand.  Forcing a numeric value to a
       particular format does not change the number stored in the value.

       All the operators which need an argument in the integer format treat the argument as in
       modular arithmetic, e.g., "mod 2**32" on a 32-bit architecture.  "sprintf "%u", -1"
       therefore provides the same result as "sprintf "%u", ~0".

       Arithmetic operators
           The binary operators "+" "-" "*" "/" "%" "==" "!=" ">" "<" ">=" "<=" and the unary
           operators "-" "abs" and "--" will attempt to convert arguments to integers.  If both
           conversions are possible without loss of precision, and the operation can be performed
           without loss of precision then the integer result is used.  Otherwise arguments are
           converted to floating point format and the floating point result is used.  The caching
           of conversions (as described above) means that the integer conversion does not throw
           away fractional parts on floating point numbers.

       ++  "++" behaves as the other operators above, except that if it is a string matching the
           format "/^[a-zA-Z]*[0-9]*\z/" the string increment described in perlop is used.

       Arithmetic operators during "use integer"
           In scopes where "use integer;" is in force, nearly all the operators listed above will
           force their argument(s) into integer format, and return an integer result.  The
           exceptions, "abs", "++" and "--", do not change their behavior with "use integer;"

       Other mathematical operators
           Operators such as "**", "sin" and "exp" force arguments to floating point format.

       Bitwise operators
           Arguments are forced into the integer format if not strings.

       Bitwise operators during "use integer"
           forces arguments to integer format. Also shift operations internally use signed
           integers rather than the default unsigned.

       Operators which expect an integer
           force the argument into the integer format.  This is applicable to the third and
           fourth arguments of "sysread", for example.

       Operators which expect a string
           force the argument into the string format.  For example, this is applicable to "printf
           "%s", $value".

       Though forcing an argument into a particular form does not change the stored number, Perl
       remembers the result of such conversions.  In particular, though the first such conversion
       may be time-consuming, repeated operations will not need to redo the conversion.

AUTHOR

       Ilya Zakharevich "ilya@math.ohio-state.edu"

       Editorial adjustments by Gurusamy Sarathy <gsar@ActiveState.com>

       Updates for 5.8.0 by Nicholas Clark <nick@ccl4.org>

SEE ALSO

       overload, perlop