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<div class="section" id="complex-and-rational-numbers">
<span id="man-complex-and-rational-numbers"></span><h1>Complex and Rational Numbers<a class="headerlink" href="#complex-and-rational-numbers" title="Permalink to this headline">¶</a></h1>
<p>Julia ships with predefined types representing both complex and rational
numbers, and supports all <a class="reference internal" href="mathematical-operations.html#man-mathematical-operations"><em>standard mathematical operations</em></a> on them. <a class="reference internal" href="conversion-and-promotion.html#man-conversion-and-promotion"><em>Conversion and Promotion</em></a>
are defined so that operations on any combination of
predefined numeric types, whether primitive or composite, behave as
expected.</p>
<div class="section" id="complex-numbers">
<span id="man-complex-numbers"></span><h2>Complex Numbers<a class="headerlink" href="#complex-numbers" title="Permalink to this headline">¶</a></h2>
<p>The global constant <tt class="docutils literal"><span class="pre">im</span></tt> is bound to the complex number <em>i</em>,
representing the principal square root of -1. It was deemed harmful to
co-opt the name <tt class="docutils literal"><span class="pre">i</span></tt> for a global constant, since it is such a popular
index variable name. Since Julia allows numeric literals to be
<a class="reference internal" href="integers-and-floating-point-numbers.html#man-numeric-literal-coefficients"><em>juxtaposed with identifiers as coefficients</em></a>,
this binding suffices to provide convenient syntax for complex numbers,
similar to the traditional mathematical notation:</p>
<div class="highlight-julia"><div class="highlight"><pre><span class="n">julia</span><span class="o">></span> <span class="mi">1</span> <span class="o">+</span> <span class="mi">2</span><span class="nb">im</span>
<span class="mi">1</span> <span class="o">+</span> <span class="mi">2</span><span class="nb">im</span>
</pre></div>
</div>
<p>You can perform all the standard arithmetic operations with complex
numbers:</p>
<div class="highlight-julia"><div class="highlight"><pre><span class="n">julia</span><span class="o">></span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="mi">2</span><span class="nb">im</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">3</span><span class="nb">im</span><span class="p">)</span>
<span class="mi">8</span> <span class="o">+</span> <span class="mi">1</span><span class="nb">im</span>
<span class="n">julia</span><span class="o">></span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="mi">2</span><span class="nb">im</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="mi">2</span><span class="nb">im</span><span class="p">)</span>
<span class="o">-</span><span class="mf">0.6</span> <span class="o">+</span> <span class="mf">0.8</span><span class="nb">im</span>
<span class="n">julia</span><span class="o">></span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="mi">2</span><span class="nb">im</span><span class="p">)</span> <span class="o">+</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="mi">2</span><span class="nb">im</span><span class="p">)</span>
<span class="mi">2</span> <span class="o">+</span> <span class="mi">0</span><span class="nb">im</span>
<span class="n">julia</span><span class="o">></span> <span class="p">(</span><span class="o">-</span><span class="mi">3</span> <span class="o">+</span> <span class="mi">2</span><span class="nb">im</span><span class="p">)</span> <span class="o">-</span> <span class="p">(</span><span class="mi">5</span> <span class="o">-</span> <span class="mi">1</span><span class="nb">im</span><span class="p">)</span>
<span class="o">-</span><span class="mi">8</span> <span class="o">+</span> <span class="mi">3</span><span class="nb">im</span>
<span class="n">julia</span><span class="o">></span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span> <span class="o">+</span> <span class="mi">2</span><span class="nb">im</span><span class="p">)</span><span class="o">^</span><span class="mi">2</span>
<span class="o">-</span><span class="mi">3</span> <span class="o">-</span> <span class="mi">4</span><span class="nb">im</span>
<span class="n">julia</span><span class="o">></span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span> <span class="o">+</span> <span class="mi">2</span><span class="nb">im</span><span class="p">)</span><span class="o">^</span><span class="mf">2.5</span>
<span class="mf">2.7296244647840084</span> <span class="o">-</span> <span class="mf">6.960664459571898</span><span class="nb">im</span>
<span class="n">julia</span><span class="o">></span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span> <span class="o">+</span> <span class="mi">2</span><span class="nb">im</span><span class="p">)</span><span class="o">^</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="mi">1</span><span class="nb">im</span><span class="p">)</span>
<span class="o">-</span><span class="mf">0.27910381075826657</span> <span class="o">+</span> <span class="mf">0.08708053414102428</span><span class="nb">im</span>
<span class="n">julia</span><span class="o">></span> <span class="mi">3</span><span class="p">(</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">5</span><span class="nb">im</span><span class="p">)</span>
<span class="mi">6</span> <span class="o">-</span> <span class="mi">15</span><span class="nb">im</span>
<span class="n">julia</span><span class="o">></span> <span class="mi">3</span><span class="p">(</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">5</span><span class="nb">im</span><span class="p">)</span><span class="o">^</span><span class="mi">2</span>
<span class="o">-</span><span class="mi">63</span> <span class="o">-</span> <span class="mi">60</span><span class="nb">im</span>
<span class="n">julia</span><span class="o">></span> <span class="mi">3</span><span class="p">(</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">5</span><span class="nb">im</span><span class="p">)</span><span class="o">^-</span><span class="mf">1.0</span>
<span class="mf">0.20689655172413796</span> <span class="o">+</span> <span class="mf">0.5172413793103449</span><span class="nb">im</span>
</pre></div>
</div>
<p>The promotion mechanism ensures that combinations of operands of
different types just work:</p>
<div class="highlight-julia"><div class="highlight"><pre><span class="n">julia</span><span class="o">></span> <span class="mi">2</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="mi">1</span><span class="nb">im</span><span class="p">)</span>
<span class="mi">2</span> <span class="o">-</span> <span class="mi">2</span><span class="nb">im</span>
<span class="n">julia</span><span class="o">></span> <span class="p">(</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">3</span><span class="nb">im</span><span class="p">)</span> <span class="o">-</span> <span class="mi">1</span>
<span class="mi">1</span> <span class="o">+</span> <span class="mi">3</span><span class="nb">im</span>
<span class="n">julia</span><span class="o">></span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="mi">2</span><span class="nb">im</span><span class="p">)</span> <span class="o">+</span> <span class="mf">0.5</span>
<span class="mf">1.5</span> <span class="o">+</span> <span class="mf">2.0</span><span class="nb">im</span>
<span class="n">julia</span><span class="o">></span> <span class="p">(</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">3</span><span class="nb">im</span><span class="p">)</span> <span class="o">-</span> <span class="mf">0.5</span><span class="nb">im</span>
<span class="mf">2.0</span> <span class="o">+</span> <span class="mf">2.5</span><span class="nb">im</span>
<span class="n">julia</span><span class="o">></span> <span class="mf">0.75</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="mi">2</span><span class="nb">im</span><span class="p">)</span>
<span class="mf">0.75</span> <span class="o">+</span> <span class="mf">1.5</span><span class="nb">im</span>
<span class="n">julia</span><span class="o">></span> <span class="p">(</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">3</span><span class="nb">im</span><span class="p">)</span> <span class="o">/</span> <span class="mi">2</span>
<span class="mf">1.0</span> <span class="o">+</span> <span class="mf">1.5</span><span class="nb">im</span>
<span class="n">julia</span><span class="o">></span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="mi">3</span><span class="nb">im</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="nb">im</span><span class="p">)</span>
<span class="o">-</span><span class="mf">0.5</span> <span class="o">-</span> <span class="mf">1.0</span><span class="nb">im</span>
<span class="n">julia</span><span class="o">></span> <span class="mi">2</span><span class="nb">im</span><span class="o">^</span><span class="mi">2</span>
<span class="o">-</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">0</span><span class="nb">im</span>
<span class="n">julia</span><span class="o">></span> <span class="mi">1</span> <span class="o">+</span> <span class="mi">3</span><span class="o">/</span><span class="mi">4</span><span class="nb">im</span>
<span class="mf">1.0</span> <span class="o">-</span> <span class="mf">0.75</span><span class="nb">im</span>
</pre></div>
</div>
<p>Note that <tt class="docutils literal"><span class="pre">3/4im</span> <span class="pre">==</span> <span class="pre">3/(4*im)</span> <span class="pre">==</span> <span class="pre">-(3/4*im)</span></tt>, since a literal
coefficient binds more tightly than division.</p>
<p>Standard functions to manipulate complex values are provided:</p>
<div class="highlight-julia"><div class="highlight"><pre><span class="n">julia</span><span class="o">></span> <span class="n">real</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="mi">2</span><span class="nb">im</span><span class="p">)</span>
<span class="mi">1</span>
<span class="n">julia</span><span class="o">></span> <span class="n">imag</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="mi">2</span><span class="nb">im</span><span class="p">)</span>
<span class="mi">2</span>
<span class="n">julia</span><span class="o">></span> <span class="n">conj</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="mi">2</span><span class="nb">im</span><span class="p">)</span>
<span class="mi">1</span> <span class="o">-</span> <span class="mi">2</span><span class="nb">im</span>
<span class="n">julia</span><span class="o">></span> <span class="n">abs</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="mi">2</span><span class="nb">im</span><span class="p">)</span>
<span class="mf">2.23606797749979</span>
<span class="n">julia</span><span class="o">></span> <span class="n">abs2</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="mi">2</span><span class="nb">im</span><span class="p">)</span>
<span class="mi">5</span>
<span class="n">julia</span><span class="o">></span> <span class="n">angle</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="mi">2</span><span class="nb">im</span><span class="p">)</span>
<span class="mf">1.1071487177940904</span>
</pre></div>
</div>
<p>As usual, the absolute value (<tt class="docutils literal"><span class="pre">abs</span></tt>) of a complex number is its
distance from zero. The <tt class="docutils literal"><span class="pre">abs2</span></tt> function gives the square of the
absolute value, and is of particular use for complex numbers where it
avoids taking a square root. The <tt class="docutils literal"><span class="pre">angle</span></tt> function returns the phase
angle in radians (also known as the <em>argument</em> or <em>arg</em> function). The
full gamut of other <a class="reference internal" href="mathematical-operations.html#man-elementary-functions"><em>Elementary Functions</em></a> is also defined
for complex numbers:</p>
<div class="highlight-julia"><div class="highlight"><pre><span class="n">julia</span><span class="o">></span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span><span class="nb">im</span><span class="p">)</span>
<span class="mf">0.7071067811865476</span> <span class="o">+</span> <span class="mf">0.7071067811865475</span><span class="nb">im</span>
<span class="n">julia</span><span class="o">></span> <span class="n">sqrt</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="mi">2</span><span class="nb">im</span><span class="p">)</span>
<span class="mf">1.272019649514069</span> <span class="o">+</span> <span class="mf">0.7861513777574233</span><span class="nb">im</span>
<span class="n">julia</span><span class="o">></span> <span class="n">cos</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="mi">2</span><span class="nb">im</span><span class="p">)</span>
<span class="mf">2.0327230070196656</span> <span class="o">-</span> <span class="mf">3.0518977991518</span><span class="nb">im</span>
<span class="n">julia</span><span class="o">></span> <span class="n">exp</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="mi">2</span><span class="nb">im</span><span class="p">)</span>
<span class="o">-</span><span class="mf">1.1312043837568135</span> <span class="o">+</span> <span class="mf">2.4717266720048188</span><span class="nb">im</span>
<span class="n">julia</span><span class="o">></span> <span class="n">sinh</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="mi">2</span><span class="nb">im</span><span class="p">)</span>
<span class="o">-</span><span class="mf">0.4890562590412937</span> <span class="o">+</span> <span class="mf">1.4031192506220405</span><span class="nb">im</span>
</pre></div>
</div>
<p>Note that mathematical functions typically return real values when applied
to real numbers and complex values when applied to complex numbers.
For example, <tt class="docutils literal"><span class="pre">sqrt</span></tt> behaves differently when applied to <tt class="docutils literal"><span class="pre">-1</span></tt>
versus <tt class="docutils literal"><span class="pre">-1</span> <span class="pre">+</span> <span class="pre">0im</span></tt> even though <tt class="docutils literal"><span class="pre">-1</span> <span class="pre">==</span> <span class="pre">-1</span> <span class="pre">+</span> <span class="pre">0im</span></tt>:</p>
<div class="highlight-julia"><div class="highlight"><pre><span class="n">julia</span><span class="o">></span> <span class="n">sqrt</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="n">ERROR</span><span class="p">:</span> <span class="n">DomainError</span>
<span class="n">sqrt</span> <span class="n">will</span> <span class="n">only</span> <span class="k">return</span> <span class="n">a</span> <span class="n">complex</span> <span class="n">result</span> <span class="k">if</span> <span class="n">called</span> <span class="n">with</span> <span class="n">a</span> <span class="n">complex</span> <span class="n">argument</span><span class="o">.</span>
<span class="k">try</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">complex</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
<span class="k">in</span> <span class="n">sqrt</span> <span class="n">at</span> <span class="n">math</span><span class="o">.</span><span class="n">jl</span><span class="p">:</span><span class="mi">131</span>
<span class="n">julia</span><span class="o">></span> <span class="n">sqrt</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span> <span class="o">+</span> <span class="mi">0</span><span class="nb">im</span><span class="p">)</span>
<span class="mf">0.0</span> <span class="o">+</span> <span class="mf">1.0</span><span class="nb">im</span>
</pre></div>
</div>
<p>The <a class="reference internal" href="integers-and-floating-point-numbers.html#man-numeric-literal-coefficients"><em>literal numeric coefficient notation</em></a>
does not work when constructing complex number from variables. Instead, the
multiplication must be explicitly written out:</p>
<div class="highlight-julia"><div class="highlight"><pre><span class="n">julia</span><span class="o">></span> <span class="n">a</span> <span class="o">=</span> <span class="mi">1</span><span class="p">;</span> <span class="n">b</span> <span class="o">=</span> <span class="mi">2</span><span class="p">;</span> <span class="n">a</span> <span class="o">+</span> <span class="n">b</span><span class="o">*</span><span class="nb">im</span>
<span class="mi">1</span> <span class="o">+</span> <span class="mi">2</span><span class="nb">im</span>
</pre></div>
</div>
<p>However, this is <em>not</em> recommended; Use the <tt class="docutils literal"><span class="pre">complex</span></tt> function instead to
construct a complex value directly from its real and imaginary parts.:</p>
<div class="highlight-julia"><div class="highlight"><pre><span class="n">julia</span><span class="o">></span> <span class="n">complex</span><span class="p">(</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">)</span>
<span class="mi">1</span> <span class="o">+</span> <span class="mi">2</span><span class="nb">im</span>
</pre></div>
</div>
<p>This construction avoids the multiplication and addition operations.</p>
<p><tt class="docutils literal"><span class="pre">Inf</span></tt> and <tt class="docutils literal"><span class="pre">NaN</span></tt> propagate through complex numbers in the real
and imaginary parts of a complex number as described in the
<a class="reference internal" href="integers-and-floating-point-numbers.html#man-special-floats"><em>Special floating-point values</em></a> section:</p>
<div class="highlight-julia"><div class="highlight"><pre><span class="n">julia</span><span class="o">></span> <span class="mi">1</span> <span class="o">+</span> <span class="nb">Inf</span><span class="o">*</span><span class="nb">im</span>
<span class="mf">1.0</span> <span class="o">+</span> <span class="nb">Inf</span><span class="o">*</span><span class="nb">im</span>
<span class="n">julia</span><span class="o">></span> <span class="mi">1</span> <span class="o">+</span> <span class="n">NaN</span><span class="o">*</span><span class="nb">im</span>
<span class="mf">1.0</span> <span class="o">+</span> <span class="n">NaN</span><span class="o">*</span><span class="nb">im</span>
</pre></div>
</div>
</div>
<div class="section" id="rational-numbers">
<span id="man-rational-numbers"></span><h2>Rational Numbers<a class="headerlink" href="#rational-numbers" title="Permalink to this headline">¶</a></h2>
<p>Julia has a rational number type to represent exact ratios of integers.
Rationals are constructed using the <tt class="docutils literal"><span class="pre">//</span></tt> operator:</p>
<div class="highlight-julia"><div class="highlight"><pre><span class="n">julia</span><span class="o">></span> <span class="mi">2</span><span class="o">//</span><span class="mi">3</span>
<span class="mi">2</span><span class="o">//</span><span class="mi">3</span>
</pre></div>
</div>
<p>If the numerator and denominator of a rational have common factors, they
are reduced to lowest terms such that the denominator is non-negative:</p>
<div class="highlight-julia"><div class="highlight"><pre><span class="n">julia</span><span class="o">></span> <span class="mi">6</span><span class="o">//</span><span class="mi">9</span>
<span class="mi">2</span><span class="o">//</span><span class="mi">3</span>
<span class="n">julia</span><span class="o">></span> <span class="o">-</span><span class="mi">4</span><span class="o">//</span><span class="mi">8</span>
<span class="o">-</span><span class="mi">1</span><span class="o">//</span><span class="mi">2</span>
<span class="n">julia</span><span class="o">></span> <span class="mi">5</span><span class="o">//-</span><span class="mi">15</span>
<span class="o">-</span><span class="mi">1</span><span class="o">//</span><span class="mi">3</span>
<span class="n">julia</span><span class="o">></span> <span class="o">-</span><span class="mi">4</span><span class="o">//-</span><span class="mi">12</span>
<span class="mi">1</span><span class="o">//</span><span class="mi">3</span>
</pre></div>
</div>
<p>This normalized form for a ratio of integers is unique, so equality of
rational values can be tested by checking for equality of the numerator
and denominator. The standardized numerator and denominator of a
rational value can be extracted using the <tt class="docutils literal"><span class="pre">num</span></tt> and <tt class="docutils literal"><span class="pre">den</span></tt> functions:</p>
<div class="highlight-julia"><div class="highlight"><pre><span class="n">julia</span><span class="o">></span> <span class="n">num</span><span class="p">(</span><span class="mi">2</span><span class="o">//</span><span class="mi">3</span><span class="p">)</span>
<span class="mi">2</span>
<span class="n">julia</span><span class="o">></span> <span class="n">den</span><span class="p">(</span><span class="mi">2</span><span class="o">//</span><span class="mi">3</span><span class="p">)</span>
<span class="mi">3</span>
</pre></div>
</div>
<p>Direct comparison of the numerator and denominator is generally not
necessary, since the standard arithmetic and comparison operations are
defined for rational values:</p>
<div class="highlight-julia"><div class="highlight"><pre><span class="n">julia</span><span class="o">></span> <span class="mi">2</span><span class="o">//</span><span class="mi">3</span> <span class="o">==</span> <span class="mi">6</span><span class="o">//</span><span class="mi">9</span>
<span class="n">true</span>
<span class="n">julia</span><span class="o">></span> <span class="mi">2</span><span class="o">//</span><span class="mi">3</span> <span class="o">==</span> <span class="mi">9</span><span class="o">//</span><span class="mi">27</span>
<span class="n">false</span>
<span class="n">julia</span><span class="o">></span> <span class="mi">3</span><span class="o">//</span><span class="mi">7</span> <span class="o"><</span> <span class="mi">1</span><span class="o">//</span><span class="mi">2</span>
<span class="n">true</span>
<span class="n">julia</span><span class="o">></span> <span class="mi">3</span><span class="o">//</span><span class="mi">4</span> <span class="o">></span> <span class="mi">2</span><span class="o">//</span><span class="mi">3</span>
<span class="n">true</span>
<span class="n">julia</span><span class="o">></span> <span class="mi">2</span><span class="o">//</span><span class="mi">4</span> <span class="o">+</span> <span class="mi">1</span><span class="o">//</span><span class="mi">6</span>
<span class="mi">2</span><span class="o">//</span><span class="mi">3</span>
<span class="n">julia</span><span class="o">></span> <span class="mi">5</span><span class="o">//</span><span class="mi">12</span> <span class="o">-</span> <span class="mi">1</span><span class="o">//</span><span class="mi">4</span>
<span class="mi">1</span><span class="o">//</span><span class="mi">6</span>
<span class="n">julia</span><span class="o">></span> <span class="mi">5</span><span class="o">//</span><span class="mi">8</span> <span class="o">*</span> <span class="mi">3</span><span class="o">//</span><span class="mi">12</span>
<span class="mi">5</span><span class="o">//</span><span class="mi">32</span>
<span class="n">julia</span><span class="o">></span> <span class="mi">6</span><span class="o">//</span><span class="mi">5</span> <span class="o">/</span> <span class="mi">10</span><span class="o">//</span><span class="mi">7</span>
<span class="mi">21</span><span class="o">//</span><span class="mi">25</span>
</pre></div>
</div>
<p>Rationals can be easily converted to floating-point numbers:</p>
<div class="highlight-julia"><div class="highlight"><pre><span class="n">julia</span><span class="o">></span> <span class="n">float</span><span class="p">(</span><span class="mi">3</span><span class="o">//</span><span class="mi">4</span><span class="p">)</span>
<span class="mf">0.75</span>
</pre></div>
</div>
<p>Conversion from rational to floating-point respects the following
identity for any integral values of <tt class="docutils literal"><span class="pre">a</span></tt> and <tt class="docutils literal"><span class="pre">b</span></tt>, with the exception
of the case <tt class="docutils literal"><span class="pre">a</span> <span class="pre">==</span> <span class="pre">0</span></tt> and <tt class="docutils literal"><span class="pre">b</span> <span class="pre">==</span> <span class="pre">0</span></tt>:</p>
<div class="highlight-julia"><div class="highlight"><pre><span class="n">julia</span><span class="o">></span> <span class="nb">isequal</span><span class="p">(</span><span class="n">float</span><span class="p">(</span><span class="n">a</span><span class="o">//</span><span class="n">b</span><span class="p">),</span> <span class="n">a</span><span class="o">/</span><span class="n">b</span><span class="p">)</span>
<span class="n">true</span>
</pre></div>
</div>
<p>Constructing infinite rational values is acceptable:</p>
<div class="highlight-julia"><div class="highlight"><pre><span class="n">julia</span><span class="o">></span> <span class="mi">5</span><span class="o">//</span><span class="mi">0</span>
<span class="mi">1</span><span class="o">//</span><span class="mi">0</span>
<span class="n">julia</span><span class="o">></span> <span class="o">-</span><span class="mi">3</span><span class="o">//</span><span class="mi">0</span>
<span class="o">-</span><span class="mi">1</span><span class="o">//</span><span class="mi">0</span>
<span class="n">julia</span><span class="o">></span> <span class="nb">typeof</span><span class="p">(</span><span class="n">ans</span><span class="p">)</span>
<span class="n">Rational</span><span class="p">{</span><span class="kt">Int64</span><span class="p">}</span> <span class="p">(</span><span class="n">constructor</span> <span class="n">with</span> <span class="mi">1</span> <span class="n">method</span><span class="p">)</span>
</pre></div>
</div>
<p>Trying to construct a <tt class="docutils literal"><span class="pre">NaN</span></tt> rational value, however, is not:</p>
<div class="highlight-julia"><div class="highlight"><pre><span class="n">julia</span><span class="o">></span> <span class="mi">0</span><span class="o">//</span><span class="mi">0</span>
<span class="n">ERROR</span><span class="p">:</span> <span class="n">invalid</span> <span class="n">rational</span><span class="p">:</span> <span class="mi">0</span><span class="o">//</span><span class="mi">0</span>
<span class="k">in</span> <span class="n">Rational</span> <span class="n">at</span> <span class="n">rational</span><span class="o">.</span><span class="n">jl</span><span class="p">:</span><span class="mi">6</span>
<span class="k">in</span> <span class="o">//</span> <span class="n">at</span> <span class="n">rational</span><span class="o">.</span><span class="n">jl</span><span class="p">:</span><span class="mi">15</span>
</pre></div>
</div>
<p>As usual, the promotion system makes interactions with other numeric
types effortless:</p>
<div class="highlight-julia"><div class="highlight"><pre><span class="n">julia</span><span class="o">></span> <span class="mi">3</span><span class="o">//</span><span class="mi">5</span> <span class="o">+</span> <span class="mi">1</span>
<span class="mi">8</span><span class="o">//</span><span class="mi">5</span>
<span class="n">julia</span><span class="o">></span> <span class="mi">3</span><span class="o">//</span><span class="mi">5</span> <span class="o">-</span> <span class="mf">0.5</span>
<span class="mf">0.09999999999999998</span>
<span class="n">julia</span><span class="o">></span> <span class="mi">2</span><span class="o">//</span><span class="mi">7</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="mi">2</span><span class="nb">im</span><span class="p">)</span>
<span class="mi">2</span><span class="o">//</span><span class="mi">7</span> <span class="o">+</span> <span class="mi">4</span><span class="o">//</span><span class="mi">7</span><span class="o">*</span><span class="nb">im</span>
<span class="n">julia</span><span class="o">></span> <span class="mi">2</span><span class="o">//</span><span class="mi">7</span> <span class="o">*</span> <span class="p">(</span><span class="mf">1.5</span> <span class="o">+</span> <span class="mi">2</span><span class="nb">im</span><span class="p">)</span>
<span class="mf">0.42857142857142855</span> <span class="o">+</span> <span class="mf">0.5714285714285714</span><span class="nb">im</span>
<span class="n">julia</span><span class="o">></span> <span class="mi">3</span><span class="o">//</span><span class="mi">2</span> <span class="o">/</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="mi">2</span><span class="nb">im</span><span class="p">)</span>
<span class="mi">3</span><span class="o">//</span><span class="mi">10</span> <span class="o">-</span> <span class="mi">3</span><span class="o">//</span><span class="mi">5</span><span class="o">*</span><span class="nb">im</span>
<span class="n">julia</span><span class="o">></span> <span class="mi">1</span><span class="o">//</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="nb">im</span>
<span class="mi">1</span><span class="o">//</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">2</span><span class="o">//</span><span class="mi">1</span><span class="o">*</span><span class="nb">im</span>
<span class="n">julia</span><span class="o">></span> <span class="mi">1</span> <span class="o">+</span> <span class="mi">2</span><span class="o">//</span><span class="mi">3</span><span class="nb">im</span>
<span class="mi">1</span><span class="o">//</span><span class="mi">1</span> <span class="o">-</span> <span class="mi">2</span><span class="o">//</span><span class="mi">3</span><span class="o">*</span><span class="nb">im</span>
<span class="n">julia</span><span class="o">></span> <span class="mf">0.5</span> <span class="o">==</span> <span class="mi">1</span><span class="o">//</span><span class="mi">2</span>
<span class="n">true</span>
<span class="n">julia</span><span class="o">></span> <span class="mf">0.33</span> <span class="o">==</span> <span class="mi">1</span><span class="o">//</span><span class="mi">3</span>
<span class="n">false</span>
<span class="n">julia</span><span class="o">></span> <span class="mf">0.33</span> <span class="o"><</span> <span class="mi">1</span><span class="o">//</span><span class="mi">3</span>
<span class="n">true</span>
<span class="n">julia</span><span class="o">></span> <span class="mi">1</span><span class="o">//</span><span class="mi">3</span> <span class="o">-</span> <span class="mf">0.33</span>
<span class="mf">0.0033333333333332993</span>
</pre></div>
</div>
</div>
</div>
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