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1 | =head1 NAME |
2 | |
3 | perlnumber - semantics of numbers and numeric operations in Perl |
4 | |
5 | =head1 SYNOPSIS |
6 | |
7 | $n = 1234; # decimal integer |
8 | $n = 0b1110011; # binary integer |
9 | $n = 01234; # octal integer |
10 | $n = 0x1234; # hexadecimal integer |
11 | $n = 12.34e-56; # exponential notation |
12 | $n = "-12.34e56"; # number specified as a string |
13 | $n = "1234"; # number specified as a string |
14 | $n = v49.50.51.52; # number specified as a string, which in |
15 | # turn is specified in terms of numbers :-) |
16 | |
17 | =head1 DESCRIPTION |
18 | |
19 | This document describes how Perl internally handles numeric values. |
20 | |
21 | Perl's operator overloading facility is completely ignored here. Operator |
22 | overloading allows user-defined behaviors for numbers, such as operations |
23 | over arbitrarily large integers, floating points numbers with arbitrary |
24 | precision, operations over "exotic" numbers such as modular arithmetic or |
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25 | p-adic arithmetic, and so on. See L<overload> for details. |
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26 | |
27 | =head1 Storing numbers |
28 | |
29 | Perl can internally represents numbers in 3 different ways: as native |
30 | integers, as native floating point numbers, and as decimal strings. |
31 | Decimal strings may have an exponential notation part, as in C<"12.34e-56">. |
32 | I<Native> here means "a format supported by the C compiler which was used |
33 | to build perl". |
34 | |
35 | The term "native" does not mean quite as much when we talk about native |
36 | integers, as it does when native floating point numbers are involved. |
37 | The only implication of the term "native" on integers is that the limits for |
38 | the maximal and the minimal supported true integral quantities are close to |
39 | powers of 2. However, for "native" floats have a most fundamental |
40 | restriction: they may represent only those numbers which have a relatively |
41 | "short" representation when converted to a binary fraction. For example, |
42 | 0.9 cannot be respresented by a native float, since the binary fraction |
43 | for 0.9 is infinite: |
44 | |
45 | binary0.1110011001100... |
46 | |
47 | with the sequence C<1100> repeating again and again. In addition to this |
48 | limitation, the exponent of the binary number is also restricted when it |
49 | is represented as a floating point number. On typical hardware, floating |
50 | point values can store numbers with up to 53 binary digits, and with binary |
51 | exponents between -1024 and 1024. In decimal representation this is close |
52 | to 16 decimal digits and decimal exponents in the range of -304..304. |
53 | The upshot of all this is that Perl cannot store a number like |
54 | 12345678901234567 as a floating point number on such architectures without |
55 | loss of information. |
56 | |
57 | Similarly, decimal strings may represent only those numbers which have a |
58 | finite decimal expansion. Being strings, and thus of arbitrary length, there |
59 | is no practical limit for the exponent or number of decimal digits for these |
60 | numbers. (But realize that what we are discussing the rules for just the |
61 | I<storage> of these numbers. The fact that you can store such "large" numbers |
62 | does not mean that that the I<operations> over these numbers will use all |
63 | of the significant digits. |
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64 | See L<"Numeric operators and numeric conversions"> for details.) |
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65 | |
66 | In fact numbers stored in the native integer format may be stored either |
67 | in the signed native form, or in the unsigned native form. Thus the limits |
68 | for Perl numbers stored as native integers would typically be -2**31..2**32-1, |
69 | with appropriate modifications in the case of 64-bit integers. Again, this |
70 | does not mean that Perl can do operations only over integers in this range: |
71 | it is possible to store many more integers in floating point format. |
72 | |
73 | Summing up, Perl numeric values can store only those numbers which have |
74 | a finite decimal expansion or a "short" binary expansion. |
75 | |
76 | =head1 Numeric operators and numeric conversions |
77 | |
78 | As mentioned earlier, Perl can store a number in any one of three formats, |
79 | but most operators typically understand only one of those formats. When |
80 | a numeric value is passed as an argument to such an operator, it will be |
81 | converted to the format understood by the operator. |
82 | |
83 | Six such conversions are possible: |
84 | |
85 | native integer --> native floating point (*) |
86 | native integer --> decimal string |
87 | native floating_point --> native integer (*) |
88 | native floating_point --> decimal string (*) |
89 | decimal string --> native integer |
90 | decimal string --> native floating point (*) |
91 | |
92 | These conversions are governed by the following general rules: |
93 | |
94 | =over |
95 | |
96 | =item * |
97 | |
98 | If the source number can be represented in the target form, that |
99 | representation is used. |
100 | |
101 | =item * |
102 | |
103 | If the source number is outside of the limits representable in the target form, |
104 | a representation of the closest limit is used. (I<Loss of information>) |
105 | |
106 | =item * |
107 | |
108 | If the source number is between two numbers representable in the target form, |
109 | a representation of one of these numbers is used. (I<Loss of information>) |
110 | |
111 | =item * |
112 | |
113 | In C<< native floating point --> native integer >> conversions the magnitude |
114 | of the result is less than or equal to the magnitude of the source. |
115 | (I<"Rounding to zero".>) |
116 | |
117 | =item * |
118 | |
119 | If the C<< decimal string --> native integer >> conversion cannot be done |
120 | without loss of information, the result is compatible with the conversion |
121 | sequence C<< decimal_string --> native_floating_point --> native_integer >>. |
122 | In particular, rounding is strongly biased to 0, though a number like |
123 | C<"0.99999999999999999999"> has a chance of being rounded to 1. |
124 | |
125 | =back |
126 | |
127 | B<RESTRICTION>: The conversions marked with C<(*)> above involve steps |
128 | performed by the C compiler. In particular, bugs/features of the compiler |
129 | used may lead to breakage of some of the above rules. |
130 | |
131 | =head1 Flavors of Perl numeric operations |
132 | |
133 | Perl operations which take a numeric argument treat that argument in one |
134 | of four different ways: they may force it to one of the integer/floating/ |
135 | string formats, or they may behave differently depending on the format of |
136 | the operand. Forcing a numeric value to a particular format does not |
137 | change the number stored in the value. |
138 | |
139 | All the operators which need an argument in the integer format treat the |
140 | argument as in modular arithmetic, e.g., C<mod 2**32> on a 32-bit |
141 | architecture. C<sprintf "%u", -1> therefore provides the same result as |
142 | C<sprintf "%u", ~0>. |
143 | |
144 | =over |
145 | |
146 | =item Arithmetic operators except, C<no integer> |
147 | |
148 | force the argument into the floating point format. |
149 | |
150 | =item Arithmetic operators except, C<use integer> |
151 | |
152 | =item Bitwise operators, C<no integer> |
153 | |
154 | force the argument into the integer format if it is not a string. |
155 | |
156 | =item Bitwise operators, C<use integer> |
157 | |
158 | force the argument into the integer format |
159 | |
160 | =item Operators which expect an integer |
161 | |
162 | force the argument into the integer format. This is applicable |
163 | to the third and fourth arguments of C<sysread>, for example. |
164 | |
165 | =item Operators which expect a string |
166 | |
167 | force the argument into the string format. For example, this is |
168 | applicable to C<printf "%s", $value>. |
169 | |
170 | =back |
171 | |
172 | Though forcing an argument into a particular form does not change the |
173 | stored number, Perl remembers the result of such conversions. In |
174 | particular, though the first such conversion may be time-consuming, |
175 | repeated operations will not need to redo the conversion. |
176 | |
177 | =head1 AUTHOR |
178 | |
179 | Ilya Zakharevich C<ilya@math.ohio-state.edu> |
180 | |
181 | Editorial adjustments by Gurusamy Sarathy <gsar@ActiveState.com> |
182 | |
183 | =head1 SEE ALSO |
184 | |
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185 | L<overload> |