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<a name="Mathematical-Constants"></a>
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Previous: <a href="Coordinate-Transformations.html#Coordinate-Transformations" accesskey="p" rel="prev">Coordinate Transformations</a>, Up: <a href="Arithmetic.html#Arithmetic" accesskey="u" rel="up">Arithmetic</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
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<hr>
<a name="Mathematical-Constants-1"></a>
<h3 class="section">17.9 Mathematical Constants</h3>
<a name="XREFe"></a><dl>
<dt><a name="index-e"></a><em></em> <strong>e</strong></dt>
<dt><a name="index-e-1"></a><em></em> <strong>e</strong> <em>(<var>n</var>)</em></dt>
<dt><a name="index-e-2"></a><em></em> <strong>e</strong> <em>(<var>n</var>, <var>m</var>)</em></dt>
<dt><a name="index-e-3"></a><em></em> <strong>e</strong> <em>(<var>n</var>, <var>m</var>, <var>k</var>, …)</em></dt>
<dt><a name="index-e-4"></a><em></em> <strong>e</strong> <em>(…, <var>class</var>)</em></dt>
<dd><p>Return a scalar, matrix, or N-dimensional array whose elements are all equal
to the base of natural logarithms.
</p>
<p>The constant
‘<samp>e</samp>’ satisfies the equation <code>log</code> (e) = 1.
</p>
<p>When called with no arguments, return a scalar with the value <em>e</em>.
</p>
<p>When called with a single argument, return a square matrix with the
dimension specified.
</p>
<p>When called with more than one scalar argument the first two arguments are
taken as the number of rows and columns and any further arguments specify
additional matrix dimensions.
</p>
<p>The optional argument <var>class</var> specifies the return type and may be
either <code>"double"</code> or <code>"single"</code>.
</p>
<p><strong>See also:</strong> <a href="Exponents-and-Logarithms.html#XREFlog">log</a>, <a href="Exponents-and-Logarithms.html#XREFexp">exp</a>, <a href="#XREFpi">pi</a>, <a href="#XREFI">I</a>.
</p></dd></dl>
<a name="XREFpi"></a><dl>
<dt><a name="index-pi"></a><em></em> <strong>pi</strong></dt>
<dt><a name="index-pi-1"></a><em></em> <strong>pi</strong> <em>(<var>n</var>)</em></dt>
<dt><a name="index-pi-2"></a><em></em> <strong>pi</strong> <em>(<var>n</var>, <var>m</var>)</em></dt>
<dt><a name="index-pi-3"></a><em></em> <strong>pi</strong> <em>(<var>n</var>, <var>m</var>, <var>k</var>, …)</em></dt>
<dt><a name="index-pi-4"></a><em></em> <strong>pi</strong> <em>(…, <var>class</var>)</em></dt>
<dd><p>Return a scalar, matrix, or N-dimensional array whose elements are all equal
to the ratio of the circumference of a circle to its
diameter.
</p>
<p>Internally, <code>pi</code> is computed as ‘<samp>4.0 * atan (1.0)</samp>’.
</p>
<p>When called with no arguments, return a scalar with the value of
pi.
</p>
<p>When called with a single argument, return a square matrix with the
dimension specified.
</p>
<p>When called with more than one scalar argument the first two arguments are
taken as the number of rows and columns and any further arguments specify
additional matrix dimensions.
</p>
<p>The optional argument <var>class</var> specifies the return type and may be
either <code>"double"</code> or <code>"single"</code>.
</p>
<p><strong>See also:</strong> <a href="#XREFe">e</a>, <a href="#XREFI">I</a>.
</p></dd></dl>
<a name="XREFI"></a><a name="index-i"></a>
<a name="index-j"></a>
<a name="index-J"></a>
<dl>
<dt><a name="index-I"></a><em></em> <strong>I</strong></dt>
<dt><a name="index-I-1"></a><em></em> <strong>I</strong> <em>(<var>n</var>)</em></dt>
<dt><a name="index-I-2"></a><em></em> <strong>I</strong> <em>(<var>n</var>, <var>m</var>)</em></dt>
<dt><a name="index-I-3"></a><em></em> <strong>I</strong> <em>(<var>n</var>, <var>m</var>, <var>k</var>, …)</em></dt>
<dt><a name="index-I-4"></a><em></em> <strong>I</strong> <em>(…, <var>class</var>)</em></dt>
<dd><p>Return a scalar, matrix, or N-dimensional array whose elements are all equal
to the pure imaginary unit, defined as
<code>sqrt <span class="nolinebreak">(-1)</span></code><!-- /@w -->.
</p>
<p>I, and its equivalents i, j, and J, are functions so any of the names may
be reused for other purposes (such as i for a counter variable).
</p>
<p>When called with no arguments, return a scalar with the value <em>i</em>.
</p>
<p>When called with a single argument, return a square matrix with the
dimension specified.
</p>
<p>When called with more than one scalar argument the first two arguments are
taken as the number of rows and columns and any further arguments specify
additional matrix dimensions.
</p>
<p>The optional argument <var>class</var> specifies the return type and may be
either <code>"double"</code> or <code>"single"</code>.
</p>
<p><strong>See also:</strong> <a href="#XREFe">e</a>, <a href="#XREFpi">pi</a>, <a href="Exponents-and-Logarithms.html#XREFlog">log</a>, <a href="Exponents-and-Logarithms.html#XREFexp">exp</a>.
</p></dd></dl>
<a name="XREFInf"></a><a name="index-inf"></a>
<dl>
<dt><a name="index-Inf"></a><em></em> <strong>Inf</strong></dt>
<dt><a name="index-Inf-1"></a><em></em> <strong>Inf</strong> <em>(<var>n</var>)</em></dt>
<dt><a name="index-Inf-2"></a><em></em> <strong>Inf</strong> <em>(<var>n</var>, <var>m</var>)</em></dt>
<dt><a name="index-Inf-3"></a><em></em> <strong>Inf</strong> <em>(<var>n</var>, <var>m</var>, <var>k</var>, …)</em></dt>
<dt><a name="index-Inf-4"></a><em></em> <strong>Inf</strong> <em>(…, <var>class</var>)</em></dt>
<dd><p>Return a scalar, matrix or N-dimensional array whose elements are all equal
to the IEEE representation for positive infinity.
</p>
<p>Infinity is produced when results are too large to be represented using the
IEEE floating point format for numbers. Two common examples which produce
infinity are division by zero and overflow.
</p>
<div class="example">
<pre class="example">[ 1/0 e^800 ]
⇒ Inf Inf
</pre></div>
<p>When called with no arguments, return a scalar with the value ‘<samp>Inf</samp>’.
</p>
<p>When called with a single argument, return a square matrix with the
dimension specified.
</p>
<p>When called with more than one scalar argument the first two arguments are
taken as the number of rows and columns and any further arguments specify
additional matrix dimensions.
</p>
<p>The optional argument <var>class</var> specifies the return type and may be
either <code>"double"</code> or <code>"single"</code>.
</p>
<p><strong>See also:</strong> <a href="Finding-Elements-and-Checking-Conditions.html#XREFisinf">isinf</a>, <a href="#XREFNaN">NaN</a>.
</p></dd></dl>
<a name="XREFNaN"></a><a name="index-nan"></a>
<dl>
<dt><a name="index-NaN"></a><em></em> <strong>NaN</strong></dt>
<dt><a name="index-NaN-1"></a><em></em> <strong>NaN</strong> <em>(<var>n</var>)</em></dt>
<dt><a name="index-NaN-2"></a><em></em> <strong>NaN</strong> <em>(<var>n</var>, <var>m</var>)</em></dt>
<dt><a name="index-NaN-3"></a><em></em> <strong>NaN</strong> <em>(<var>n</var>, <var>m</var>, <var>k</var>, …)</em></dt>
<dt><a name="index-NaN-4"></a><em></em> <strong>NaN</strong> <em>(…, <var>class</var>)</em></dt>
<dd><p>Return a scalar, matrix, or N-dimensional array whose elements are all equal
to the IEEE symbol NaN (Not a Number).
</p>
<p>NaN is the result of operations which do not produce a well defined
numerical result. Common operations which produce a NaN are arithmetic
with infinity
(Inf - Inf), zero divided by zero (0/0),
and any operation involving another NaN value (5 + NaN).
</p>
<p>Note that NaN always compares not equal to NaN (NaN != NaN). This behavior
is specified by the IEEE standard for floating point arithmetic. To find
NaN values, use the <code>isnan</code> function.
</p>
<p>When called with no arguments, return a scalar with the value ‘<samp>NaN</samp>’.
</p>
<p>When called with a single argument, return a square matrix with the
dimension specified.
</p>
<p>When called with more than one scalar argument the first two arguments are
taken as the number of rows and columns and any further arguments specify
additional matrix dimensions.
</p>
<p>The optional argument <var>class</var> specifies the return type and may be
either <code>"double"</code> or <code>"single"</code>.
</p>
<p><strong>See also:</strong> <a href="Finding-Elements-and-Checking-Conditions.html#XREFisnan">isnan</a>, <a href="#XREFInf">Inf</a>.
</p></dd></dl>
<a name="XREFeps"></a><dl>
<dt><a name="index-eps"></a><em></em> <strong>eps</strong></dt>
<dt><a name="index-eps-1"></a><em></em> <strong>eps</strong> <em>(<var>x</var>)</em></dt>
<dt><a name="index-eps-2"></a><em></em> <strong>eps</strong> <em>(<var>n</var>, <var>m</var>)</em></dt>
<dt><a name="index-eps-3"></a><em></em> <strong>eps</strong> <em>(<var>n</var>, <var>m</var>, <var>k</var>, …)</em></dt>
<dt><a name="index-eps-4"></a><em></em> <strong>eps</strong> <em>(…, <var>class</var>)</em></dt>
<dd><p>Return a scalar, matrix or N-dimensional array whose elements are all eps,
the machine precision.
</p>
<p>More precisely, <code>eps</code> is the relative spacing between any two adjacent
numbers in the machine’s floating point system. This number is obviously
system dependent. On machines that support IEEE floating point arithmetic,
<code>eps</code> is approximately
2.2204e-16 for double precision and 1.1921e-07
for single precision.
</p>
<p>When called with no arguments, return a scalar with the value
<code>eps (1.0)</code>.
</p>
<p>Given a single argument <var>x</var>, return the distance between <var>x</var> and the
next largest value.
</p>
<p>When called with more than one argument the first two arguments are taken as
the number of rows and columns and any further arguments specify additional
matrix dimensions. The optional argument <var>class</var> specifies the return
type and may be either <code>"double"</code> or <code>"single"</code>.
</p>
<p><strong>See also:</strong> <a href="#XREFrealmax">realmax</a>, <a href="#XREFrealmin">realmin</a>, <a href="Integer-Data-Types.html#XREFintmax">intmax</a>, <a href="Integer-Data-Types.html#XREFflintmax">flintmax</a>.
</p></dd></dl>
<a name="XREFrealmax"></a><dl>
<dt><a name="index-realmax"></a><em></em> <strong>realmax</strong></dt>
<dt><a name="index-realmax-1"></a><em></em> <strong>realmax</strong> <em>(<var>n</var>)</em></dt>
<dt><a name="index-realmax-2"></a><em></em> <strong>realmax</strong> <em>(<var>n</var>, <var>m</var>)</em></dt>
<dt><a name="index-realmax-3"></a><em></em> <strong>realmax</strong> <em>(<var>n</var>, <var>m</var>, <var>k</var>, …)</em></dt>
<dt><a name="index-realmax-4"></a><em></em> <strong>realmax</strong> <em>(…, <var>class</var>)</em></dt>
<dd><p>Return a scalar, matrix, or N-dimensional array whose elements are all equal
to the largest floating point number that is representable.
</p>
<p>The actual value is system dependent. On machines that support IEEE
floating point arithmetic, <code>realmax</code> is approximately
1.7977e+308 for double precision and 3.4028e+38
for single precision.
</p>
<p>When called with no arguments, return a scalar with the value
<code>realmax (<code>"double"</code>)</code>.
</p>
<p>When called with a single argument, return a square matrix with the
dimension specified.
</p>
<p>When called with more than one scalar argument the first two arguments are
taken as the number of rows and columns and any further arguments specify
additional matrix dimensions.
</p>
<p>The optional argument <var>class</var> specifies the return type and may be
either <code>"double"</code> or <code>"single"</code>.
</p>
<p><strong>See also:</strong> <a href="#XREFrealmin">realmin</a>, <a href="Integer-Data-Types.html#XREFintmax">intmax</a>, <a href="Integer-Data-Types.html#XREFflintmax">flintmax</a>, <a href="#XREFeps">eps</a>.
</p></dd></dl>
<a name="XREFrealmin"></a><dl>
<dt><a name="index-realmin"></a><em></em> <strong>realmin</strong></dt>
<dt><a name="index-realmin-1"></a><em></em> <strong>realmin</strong> <em>(<var>n</var>)</em></dt>
<dt><a name="index-realmin-2"></a><em></em> <strong>realmin</strong> <em>(<var>n</var>, <var>m</var>)</em></dt>
<dt><a name="index-realmin-3"></a><em></em> <strong>realmin</strong> <em>(<var>n</var>, <var>m</var>, <var>k</var>, …)</em></dt>
<dt><a name="index-realmin-4"></a><em></em> <strong>realmin</strong> <em>(…, <var>class</var>)</em></dt>
<dd><p>Return a scalar, matrix, or N-dimensional array whose elements are all equal
to the smallest normalized floating point number that is representable.
</p>
<p>The actual value is system dependent. On machines that support
IEEE floating point arithmetic, <code>realmin</code> is approximately
2.2251e-308 for double precision and 1.1755e-38
for single precision.
</p>
<p>When called with no arguments, return a scalar with the value
<code>realmin (<code>"double"</code>)</code>.
</p>
<p>When called with a single argument, return a square matrix with the
dimension specified.
</p>
<p>When called with more than one scalar argument the first two arguments are
taken as the number of rows and columns and any further arguments specify
additional matrix dimensions.
</p>
<p>The optional argument <var>class</var> specifies the return type and may be
either <code>"double"</code> or <code>"single"</code>.
</p>
<p><strong>See also:</strong> <a href="#XREFrealmax">realmax</a>, <a href="Integer-Data-Types.html#XREFintmin">intmin</a>, <a href="#XREFeps">eps</a>.
</p></dd></dl>
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