1 | =head1 NAME
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2 |
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3 | perlnumber - semantics of numbers and numeric operations in Perl
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4 |
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5 | =head1 SYNOPSIS
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6 |
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7 | $n = 1234; # decimal integer
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8 | $n = 0b1110011; # binary integer
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9 | $n = 01234; # octal integer
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10 | $n = 0x1234; # hexadecimal integer
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11 | $n = 12.34e-56; # exponential notation
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12 | $n = "-12.34e56"; # number specified as a string
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13 | $n = "1234"; # number specified as a string
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14 |
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15 | =head1 DESCRIPTION
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16 |
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17 | This document describes how Perl internally handles numeric values.
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18 |
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19 | Perl's operator overloading facility is completely ignored here. Operator
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20 | overloading allows user-defined behaviors for numbers, such as operations
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21 | over arbitrarily large integers, floating points numbers with arbitrary
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22 | precision, operations over "exotic" numbers such as modular arithmetic or
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23 | p-adic arithmetic, and so on. See L<overload> for details.
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24 |
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25 | =head1 Storing numbers
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26 |
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27 | Perl can internally represent numbers in 3 different ways: as native
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28 | integers, as native floating point numbers, and as decimal strings.
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29 | Decimal strings may have an exponential notation part, as in C<"12.34e-56">.
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30 | I<Native> here means "a format supported by the C compiler which was used
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31 | to build perl".
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32 |
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33 | The term "native" does not mean quite as much when we talk about native
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34 | integers, as it does when native floating point numbers are involved.
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35 | The only implication of the term "native" on integers is that the limits for
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36 | the maximal and the minimal supported true integral quantities are close to
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37 | powers of 2. However, "native" floats have a most fundamental
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38 | restriction: they may represent only those numbers which have a relatively
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39 | "short" representation when converted to a binary fraction. For example,
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40 | 0.9 cannot be represented by a native float, since the binary fraction
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41 | for 0.9 is infinite:
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42 |
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43 | binary0.1110011001100...
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44 |
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45 | with the sequence C<1100> repeating again and again. In addition to this
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46 | limitation, the exponent of the binary number is also restricted when it
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47 | is represented as a floating point number. On typical hardware, floating
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48 | point values can store numbers with up to 53 binary digits, and with binary
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49 | exponents between -1024 and 1024. In decimal representation this is close
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50 | to 16 decimal digits and decimal exponents in the range of -304..304.
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51 | The upshot of all this is that Perl cannot store a number like
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52 | 12345678901234567 as a floating point number on such architectures without
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53 | loss of information.
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54 |
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55 | Similarly, decimal strings can represent only those numbers which have a
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56 | finite decimal expansion. Being strings, and thus of arbitrary length, there
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57 | is no practical limit for the exponent or number of decimal digits for these
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58 | numbers. (But realize that what we are discussing the rules for just the
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59 | I<storage> of these numbers. The fact that you can store such "large" numbers
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60 | does not mean that the I<operations> over these numbers will use all
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61 | of the significant digits.
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62 | See L<"Numeric operators and numeric conversions"> for details.)
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63 |
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64 | In fact numbers stored in the native integer format may be stored either
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65 | in the signed native form, or in the unsigned native form. Thus the limits
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66 | for Perl numbers stored as native integers would typically be -2**31..2**32-1,
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67 | with appropriate modifications in the case of 64-bit integers. Again, this
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68 | does not mean that Perl can do operations only over integers in this range:
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69 | it is possible to store many more integers in floating point format.
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70 |
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71 | Summing up, Perl numeric values can store only those numbers which have
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72 | a finite decimal expansion or a "short" binary expansion.
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73 |
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74 | =head1 Numeric operators and numeric conversions
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75 |
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76 | As mentioned earlier, Perl can store a number in any one of three formats,
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77 | but most operators typically understand only one of those formats. When
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78 | a numeric value is passed as an argument to such an operator, it will be
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79 | converted to the format understood by the operator.
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80 |
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81 | Six such conversions are possible:
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82 |
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83 | native integer --> native floating point (*)
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84 | native integer --> decimal string
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85 | native floating_point --> native integer (*)
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86 | native floating_point --> decimal string (*)
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87 | decimal string --> native integer
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88 | decimal string --> native floating point (*)
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89 |
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90 | These conversions are governed by the following general rules:
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91 |
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92 | =over 4
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93 |
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94 | =item *
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95 |
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96 | If the source number can be represented in the target form, that
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97 | representation is used.
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98 |
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99 | =item *
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100 |
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101 | If the source number is outside of the limits representable in the target form,
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102 | a representation of the closest limit is used. (I<Loss of information>)
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103 |
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104 | =item *
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105 |
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106 | If the source number is between two numbers representable in the target form,
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107 | a representation of one of these numbers is used. (I<Loss of information>)
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108 |
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109 | =item *
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110 |
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111 | In C<< native floating point --> native integer >> conversions the magnitude
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112 | of the result is less than or equal to the magnitude of the source.
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113 | (I<"Rounding to zero".>)
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114 |
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115 | =item *
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116 |
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117 | If the C<< decimal string --> native integer >> conversion cannot be done
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118 | without loss of information, the result is compatible with the conversion
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119 | sequence C<< decimal_string --> native_floating_point --> native_integer >>.
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120 | In particular, rounding is strongly biased to 0, though a number like
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121 | C<"0.99999999999999999999"> has a chance of being rounded to 1.
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122 |
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123 | =back
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124 |
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125 | B<RESTRICTION>: The conversions marked with C<(*)> above involve steps
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126 | performed by the C compiler. In particular, bugs/features of the compiler
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127 | used may lead to breakage of some of the above rules.
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128 |
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129 | =head1 Flavors of Perl numeric operations
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130 |
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131 | Perl operations which take a numeric argument treat that argument in one
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132 | of four different ways: they may force it to one of the integer/floating/
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133 | string formats, or they may behave differently depending on the format of
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134 | the operand. Forcing a numeric value to a particular format does not
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135 | change the number stored in the value.
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136 |
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137 | All the operators which need an argument in the integer format treat the
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138 | argument as in modular arithmetic, e.g., C<mod 2**32> on a 32-bit
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139 | architecture. C<sprintf "%u", -1> therefore provides the same result as
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140 | C<sprintf "%u", ~0>.
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141 |
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142 | =over 4
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143 |
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144 | =item Arithmetic operators
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145 |
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146 | The binary operators C<+> C<-> C<*> C</> C<%> C<==> C<!=> C<E<gt>> C<E<lt>>
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147 | C<E<gt>=> C<E<lt>=> and the unary operators C<-> C<abs> and C<--> will
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148 | attempt to convert arguments to integers. If both conversions are possible
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149 | without loss of precision, and the operation can be performed without
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150 | loss of precision then the integer result is used. Otherwise arguments are
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151 | converted to floating point format and the floating point result is used.
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152 | The caching of conversions (as described above) means that the integer
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153 | conversion does not throw away fractional parts on floating point numbers.
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154 |
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155 | =item ++
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156 |
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157 | C<++> behaves as the other operators above, except that if it is a string
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158 | matching the format C</^[a-zA-Z]*[0-9]*\z/> the string increment described
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159 | in L<perlop> is used.
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160 |
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161 | =item Arithmetic operators during C<use integer>
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162 |
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163 | In scopes where C<use integer;> is in force, nearly all the operators listed
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164 | above will force their argument(s) into integer format, and return an integer
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165 | result. The exceptions, C<abs>, C<++> and C<-->, do not change their
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166 | behavior with C<use integer;>
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167 |
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168 | =item Other mathematical operators
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169 |
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170 | Operators such as C<**>, C<sin> and C<exp> force arguments to floating point
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171 | format.
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172 |
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173 | =item Bitwise operators
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174 |
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175 | Arguments are forced into the integer format if not strings.
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176 |
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177 | =item Bitwise operators during C<use integer>
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178 |
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179 | forces arguments to integer format. Also shift operations internally use
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180 | signed integers rather than the default unsigned.
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181 |
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182 | =item Operators which expect an integer
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183 |
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184 | force the argument into the integer format. This is applicable
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185 | to the third and fourth arguments of C<sysread>, for example.
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186 |
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187 | =item Operators which expect a string
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188 |
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189 | force the argument into the string format. For example, this is
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190 | applicable to C<printf "%s", $value>.
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191 |
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192 | =back
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193 |
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194 | Though forcing an argument into a particular form does not change the
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195 | stored number, Perl remembers the result of such conversions. In
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196 | particular, though the first such conversion may be time-consuming,
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197 | repeated operations will not need to redo the conversion.
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198 |
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199 | =head1 AUTHOR
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200 |
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201 | Ilya Zakharevich C<[email protected]>
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202 |
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203 | Editorial adjustments by Gurusamy Sarathy <[email protected]>
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204 |
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205 | Updates for 5.8.0 by Nicholas Clark <[email protected]>
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206 |
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207 | =head1 SEE ALSO
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208 |
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209 | L<overload>, L<perlop>
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