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| 22 | <title>Integer Representations (GNU C Language Manual)</title>
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| 31 | <link href="index.html#SEC_Contents" rel="contents" title="Table of Contents">
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| 32 | <link href="Integers-in-Depth.html" rel="up" title="Integers in Depth">
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| 55 | </head>
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| 56 |
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| 57 | <body lang="en">
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| 58 | <span id="Integer-Representations"></span><div class="header">
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| 59 | <p>
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| 60 | Next: <a href="Maximum-and-Minimum-Values.html" accesskey="n" rel="next">Maximum and Minimum Values</a>, Up: <a href="Integers-in-Depth.html" accesskey="u" rel="up">Integers in Depth</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Symbol-Index.html" title="Index" rel="index">Index</a>]</p>
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| 61 | </div>
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| 62 | <hr>
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| 63 | <span id="Integer-Representations-1"></span><h3 class="section">27.1 Integer Representations</h3>
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| 64 |
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| 65 | <span id="index-integer-representations"></span>
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| 66 | <span id="index-representation-of-integers"></span>
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| 67 |
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| 68 | <p>Modern computers store integer values as binary (base-2) numbers that
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| 69 | occupy a single unit of storage, typically either as an 8-bit
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| 70 | <code>char</code>, a 16-bit <code>short int</code>, a 32-bit <code>int</code>, or
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| 71 | possibly, a 64-bit <code>long long int</code>. Whether a <code>long int</code> is
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| 72 | a 32-bit or a 64-bit value is system dependent.<a id="DOCF11" href="#FOOT11"><sup>11</sup></a>
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| 73 | </p>
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| 74 | <span id="index-CHAR_005fBIT"></span>
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| 75 | <p>The macro <code>CHAR_BIT</code>, defined in <samp>limits.h</samp>, gives the number
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| 76 | of bits in type <code>char</code>. On any real operating system, the value
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| 77 | is 8.
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| 78 | </p>
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| 79 | <p>The fixed sizes of numeric types necessarily limits their <em>range
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| 80 | of values</em>, and the particular encoding of integers decides what that
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| 81 | range is.
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| 82 | </p>
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| 83 | <span id="index-two_0027s_002dcomplement-representation"></span>
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| 84 | <p>For unsigned integers, the entire space is used to represent a
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| 85 | nonnegative value. Signed integers are stored using
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| 86 | <em>two’s-complement representation</em>: a signed integer with <var>n</var>
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| 87 | bits has a range from <em>-2<sup>(<var>n</var> - 1)</sup></em> to -1 to 0
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| 88 | to 1 to <em>+2<sup>(<var>n</var> - 1)</sup> - 1</em>, inclusive. The leftmost, or
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| 89 | high-order, bit is called the <em>sign bit</em>.
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| 90 | </p>
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| 91 |
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| 92 | <p>There is only one value that means zero, and the most negative number
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| 93 | lacks a positive counterpart. As a result, negating that number
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| 94 | causes overflow; in practice, its result is that number back again.
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| 95 | For example, a two’s-complement signed 8-bit integer can represent all
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| 96 | decimal numbers from -128 to +127. We will revisit that
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| 97 | peculiarity shortly.
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| 98 | </p>
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| 99 | <p>Decades ago, there were computers that didn’t use two’s-complement
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| 100 | representation for integers (see <a href="Integers-in-Depth.html">Integers in Depth</a>), but they are
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| 101 | long gone and not worth any effort to support.
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| 102 | </p>
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| 103 |
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| 104 | <p>When an arithmetic operation produces a value that is too big to
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| 105 | represent, the operation is said to <em>overflow</em>. In C, integer
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| 106 | overflow does not interrupt the control flow or signal an error.
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| 107 | What it does depends on signedness.
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| 108 | </p>
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| 109 | <p>For unsigned arithmetic, the result of an operation that overflows is
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| 110 | the <var>n</var> low-order bits of the correct value. If the correct value
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| 111 | is representable in <var>n</var> bits, that is always the result;
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| 112 | thus we often say that “integer arithmetic is exact,” omitting the
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| 113 | crucial qualifying phrase “as long as the exact result is
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| 114 | representable.”
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| 115 | </p>
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| 116 | <p>In principle, a C program should be written so that overflow never
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| 117 | occurs for signed integers, but in GNU C you can specify various ways
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| 118 | of handling such overflow (see <a href="Integer-Overflow.html">Integer Overflow</a>).
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| 119 | </p>
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| 120 | <p>Integer representations are best understood by looking at a table for
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| 121 | a tiny integer size; here are the possible values for an integer with
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| 122 | three bits:
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| 123 | </p>
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| 124 | <table>
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| 125 | <thead><tr><th width="25%">Unsigned</th><th width="25%">Signed</th><th width="25%">Bits</th><th width="25%">2s Complement</th></tr></thead>
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| 126 | <tr><td width="25%">0</td><td width="25%">0</td><td width="25%">000</td><td width="25%">000 (0)</td></tr>
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| 127 | <tr><td width="25%">1</td><td width="25%">1</td><td width="25%">001</td><td width="25%">111 (-1)</td></tr>
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| 128 | <tr><td width="25%">2</td><td width="25%">2</td><td width="25%">010</td><td width="25%">110 (-2)</td></tr>
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| 129 | <tr><td width="25%">3</td><td width="25%">3</td><td width="25%">011</td><td width="25%">101 (-3)</td></tr>
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| 130 | <tr><td width="25%">4</td><td width="25%">-4</td><td width="25%">100</td><td width="25%">100 (-4)</td></tr>
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| 131 | <tr><td width="25%">5</td><td width="25%">-3</td><td width="25%">101</td><td width="25%">011 (3)</td></tr>
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| 132 | <tr><td width="25%">6</td><td width="25%">-2</td><td width="25%">110</td><td width="25%">010 (2)</td></tr>
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| 133 | <tr><td width="25%">7</td><td width="25%">-1</td><td width="25%">111</td><td width="25%">001 (1)</td></tr>
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| 134 | </table>
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| 135 |
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| 136 | <p>The parenthesized decimal numbers in the last column represent the
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| 137 | signed meanings of the two’s-complement of the line’s value. Recall
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| 138 | that, in two’s-complement encoding, the high-order bit is 0 when
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| 139 | the number is nonnegative.
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| 140 | </p>
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| 141 | <p>We can now understand the peculiar behavior of negation of the
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| 142 | most negative two’s-complement integer: start with 0b100,
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| 143 | invert the bits to get 0b011, and add 1: we get
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| 144 | 0b100, the value we started with.
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| 145 | </p>
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| 146 | <p>We can also see overflow behavior in two’s-complement:
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| 147 | </p>
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| 148 | <div class="example">
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| 149 | <pre class="example">3 + 1 = 0b011 + 0b001 = 0b100 = (-4)
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| 150 | 3 + 2 = 0b011 + 0b010 = 0b101 = (-3)
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| 151 | 3 + 3 = 0b011 + 0b011 = 0b110 = (-2)
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| 152 | </pre></div>
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| 153 |
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| 154 | <p>A sum of two nonnegative signed values that overflows has a 1 in the
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| 155 | sign bit, so the exact positive result is truncated to a negative
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| 156 | value.
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| 157 | </p>
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| 158 |
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| 159 | <div class="footnote">
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| 160 | <hr>
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| 161 | <h4 class="footnotes-heading">Footnotes</h4>
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| 162 |
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| 163 | <h5><a id="FOOT11" href="#DOCF11">(11)</a></h3>
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| 164 | <p>In theory,
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| 165 | any of these types could have some other size, bit it’s not worth even
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| 166 | a minute to cater to that possibility. It never happens on
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| 167 | GNU/Linux.</p>
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| 168 | </div>
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| 169 | <hr>
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| 170 | <div class="header">
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| 171 | <p>
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| 172 | Next: <a href="Maximum-and-Minimum-Values.html" accesskey="n" rel="next">Maximum and Minimum Values</a>, Up: <a href="Integers-in-Depth.html" accesskey="u" rel="up">Integers in Depth</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Symbol-Index.html" title="Index" rel="index">Index</a>]</p>
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| 173 | </div>
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| 174 |
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| 175 |
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| 176 |
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| 177 | </body>
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| 178 | </html>
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