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class="toctree-l2"><a class="reference internal" href="../../devdocs/cartesian/">Base.Cartesian</a></li><li class="toctree-l2"><a class="reference internal" href="../../devdocs/sysimg/">System Image Building</a></li></ul></li><li class="toctree-l1"><a class="reference internal" href="../../devdocs/C/">Developing/debugging Julia’s C code</a><ul><li class="toctree-l2"><a class="reference internal" href="../../devdocs/backtraces/">Reporting and analyzing crashes (segfaults)</a></li><li class="toctree-l2"><a class="reference internal" href="../../devdocs/debuggingtips/">gdb debugging tips</a></li></ul></li></ul></div> </nav><section class="wy-nav-content-wrap" data-toggle="wy-nav-shift"><nav aria-label="top navigation" class="wy-nav-top" role="navigation"><a href="../../">Julia Language</a></nav><div class="wy-nav-content"><div class="rst-content"><div aria-label="breadcrumbs navigation" role="navigation"><ul class="wy-breadcrumbs"><li><a href="../../">Docs</a> »</li><li>Complex and Rational Numbers</li><li class="wy-breadcrumbs-aside"><a href="../../_sources/manual/complex-and-rational-numbers.txt" rel="nofollow"> View page source</a></li></ul><hr/></div><div class="document" role="main"><div class="section" id="complex-and-rational-numbers"><h1>Complex and Rational Numbers<a class="headerlink" href="#complex-and-rational-numbers" title="Permalink to this headline">¶</a></h1>Julia ships with predefined types representing both complex and rational numbers, and supports all <a class="reference internal" href="../mathematical-operations/#man-mathematical-operations">standard mathematical operations</a> on them. <a class="reference internal" href="../conversion-and-promotion/#man-conversion-and-promotion">Conversion and Promotion</a> are defined so that operations on any combination of predefined numeric types, whether primitive or composite, behave as expected.<div class="section" id="complex-numbers"><h2>Complex Numbers<a class="headerlink" href="#complex-numbers" title="Permalink to this headline">¶</a></h2>The global constant <a class="reference internal" href="../../stdlib/numbers/#Base.im" title="Base.im"><tt class="xref jl jl-const docutils literal">im</tt></a> is bound to the complex number i, representing the principal square root of -1. It was deemed harmful to co-opt the name <tt class="docutils literal">i</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/#man-numeric-literal-coefficients">juxtaposed with identifiers as coefficients</a>, this binding suffices to provide convenient syntax for complex numbers, similar to the traditional mathematical notation:<div class="highlight-julia"><div class="highlight"><pre>julia>1+2im1+2im</pre></div></div>You can perform all the standard arithmetic operations with complex numbers:<div class="highlight-julia"><div class="highlight"><pre>julia>(1+2im)*(2-3im)8+1imjulia>(1+2im)/(1-2im)-0.6+0.8imjulia>(1+2im)+(1-2im)2+0imjulia>(-3+2im)-(5-1im)-8+3imjulia>(-1+2im)^2-3-4imjulia>(-1+2im)^2.52.7296244647840084-6.960664459571898imjulia>(-1+2im)^(1+1im)-0.27910381075826657+0.08708053414102428imjulia>3(2-5im)6-15imjulia>3(2-5im)^2-63-60imjulia>3(2-5im)^-1.00.20689655172413796+0.5172413793103449im</pre></div></div>The promotion mechanism ensures that combinations of operands of different types just work:<div class="highlight-julia"><div class="highlight"><pre>julia>2(1-1im)2-2imjulia>(2+3im)-11+3imjulia>(1+2im)+0.51.5+2.0imjulia>(2+3im)-0.5im2.0+2.5imjulia>0.75(1+2im)0.75+1.5imjulia>(2+3im)/21.0+1.5imjulia>(1-3im)/(2+2im)-0.5-1.0imjulia>2im^2-2+0imjulia>1+3/4im1.0-0.75im</pre></div></div>Note that <tt class="docutils literal">3/4im==3/(4*im)==-(3/4*im)</tt>, since a literal coefficient binds more tightly than division.Standard functions to manipulate complex values are provided:<div class="highlight-julia"><div class="highlight"><pre>julia>real(1+2im)1julia>imag(1+2im)2julia>conj(1+2im)1-2imjulia>abs(1+2im)2.23606797749979julia>abs2(1+2im)5julia>angle(1+2im)1.1071487177940904</pre></div></div>As usual, the absolute value (<a class="reference internal" href="../../stdlib/math/#Base.abs" title="Base.abs"><tt class="xref jl jl-func docutils literal">abs()</tt></a>) of a complex number is its distance from zero. <a class="reference internal" href="../../stdlib/math/#Base.abs2" title="Base.abs2"><tt class="xref jl jl-func docutils literal">abs2()</tt></a> gives the square of the absolute value, and is of particular use for complex numbers where it avoids taking a square root. <a class="reference internal" href="../../stdlib/math/#Base.angle" title="Base.angle"><tt class="xref jl jl-func docutils literal">angle()</tt></a> returns the phase angle in radians (also known as the argument or arg function). The full gamut of other <a class="reference internal" href="../mathematical-operations/#man-elementary-functions">Elementary Functions</a> is also defined for complex numbers:<div class="highlight-julia"><div class="highlight"><pre>julia>sqrt(1im)0.7071067811865476+0.7071067811865475imjulia>sqrt(1+2im)1.272019649514069+0.7861513777574233imjulia>cos(1+2im)2.0327230070196656-3.0518977991518imjulia>exp(1+2im)-1.1312043837568135+2.4717266720048188imjulia>sinh(1+2im)-0.4890562590412937+1.4031192506220405im</pre></div></div>Note that mathematical functions typically return real values when applied to real numbers and complex values when applied to complex numbers. For example, <a class="reference internal" href="../../stdlib/math/#Base.sqrt" title="Base.sqrt"><tt class="xref jl jl-func docutils literal">sqrt()</tt></a> behaves differently when applied to <tt class="docutils literal">-1</tt> versus <tt class="docutils literal">-1+0im</tt> even though <tt class="docutils literal">-1==-1+0im</tt>:<div class="highlight-julia"><div class="highlight"><pre>julia>sqrt(-1)ERROR:DomainErrorsqrtwillonlyreturnacomplexresultifcalledwithacomplexargument.trysqrt(complex(x))insqrtatmath.jl:133julia>sqrt(-1+0im)0.0+1.0im</pre></div></div>The <a class="reference internal" href="../integers-and-floating-point-numbers/#man-numeric-literal-coefficients">literal numeric coefficient notation</a> does not work when constructing complex number from variables. Instead, the multiplication must be explicitly written out:<div class="highlight-julia"><div class="highlight"><pre>julia>a=1;b=2;a+b*im1+2im</pre></div></div>However, this is not recommended; Use the <a class="reference internal" href="../../stdlib/numbers/#Base.complex" title="Base.complex"><tt class="xref jl jl-func docutils literal">complex()</tt></a> function instead to construct a complex value directly from its real and imaginary parts.:<div class="highlight-julia"><div class="highlight"><pre>julia>complex(a,b)1+2im</pre></div></div>This construction avoids the multiplication and addition operations.<a class="reference internal" href="../../stdlib/numbers/#Base.Inf" title="Base.Inf"><tt class="xref jl jl-const docutils literal">Inf</tt></a> and <a class="reference internal" href="../../stdlib/numbers/#Base.NaN" title="Base.NaN"><tt class="xref jl jl-const docutils literal">NaN</tt></a> 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/#man-special-floats">Special floating-point values</a> section:<div class="highlight-julia"><div class="highlight"><pre>julia>1+Inf*im1.0+Inf*imjulia>1+NaN*im1.0+NaN*im</pre></div></div></div><div class="section" id="rational-numbers"><h2>Rational Numbers<a class="headerlink" href="#rational-numbers" title="Permalink to this headline">¶</a></h2>Julia has a rational number type to represent exact ratios of integers. Rationals are constructed using the <a class="reference internal" href="../../stdlib/math/#Base.//" title="Base.//"><tt class="xref jl jl-obj docutils literal">//</tt></a> operator:<div class="highlight-julia"><div class="highlight"><pre>julia>2//32//3</pre></div></div>If the numerator and denominator of a rational have common factors, they are reduced to lowest terms such that the denominator is non-negative:<div class="highlight-julia"><div class="highlight"><pre>julia>6//92//3julia>-4//8-1//2julia>5//-15-1//3julia>-4//-121//3</pre></div></div>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 <a class="reference internal" href="../../stdlib/math/#Base.num" title="Base.num"><tt class="xref jl jl-func docutils literal">num()</tt></a> and <a class="reference internal" href="../../stdlib/math/#Base.den" title="Base.den"><tt class="xref jl jl-func docutils literal">den()</tt></a> functions:<div class="highlight-julia"><div class="highlight"><pre>julia>num(2//3)2julia>den(2//3)3</pre></div></div>Direct comparison of the numerator and denominator is generally not necessary, since the standard arithmetic and comparison operations are defined for rational values:<div class="highlight-julia"><div class="highlight"><pre>julia>2//3==6//9truejulia>2//3==9//27falsejulia>3//7<1//2truejulia>3//4>2//3truejulia>2//4+1//62//3julia>5//12-1//41//6julia>5//8*3//125//32julia>6//5/10//721//25</pre></div></div>Rationals can be easily converted to floating-point numbers:<div class="highlight-julia"><div class="highlight"><pre>julia>float(3//4)0.75</pre></div></div>Conversion from rational to floating-point respects the following identity for any integral values of <tt class="docutils literal">a</tt> and <tt class="docutils literal">b</tt>, with the exception of the case <tt class="docutils literal">a==0</tt> and <tt class="docutils literal">b==0</tt>:<div class="highlight-julia"><div class="highlight"><pre>julia>isequal(float(a//b),a/b)true</pre></div></div>Constructing infinite rational values is acceptable:<div class="highlight-julia"><div class="highlight"><pre>julia>5//01//0julia>-3//0-1//0julia>typeof(ans)Rational{Int64}(constructorwith1method)</pre></div></div>Trying to construct a <a class="reference internal" href="../../stdlib/numbers/#Base.NaN" title="Base.NaN"><tt class="xref jl jl-const docutils literal">NaN</tt></a> rational value, however, is not:<div class="highlight-julia"><div class="highlight"><pre>julia>0//0ERROR:invalidrational:0//0inRationalatrational.jl:6in//atrational.jl:15</pre></div></div>As usual, the promotion system makes interactions with other numeric types effortless:<div class="highlight-julia"><div class="highlight"><pre>julia>3//5+18//5julia>3//5-0.50.09999999999999998julia>2//7*(1+2im)2//7+4//7*imjulia>2//7*(1.5+2im)0.42857142857142855+0.5714285714285714imjulia>3//2/(1+2im)3//10-3//5*imjulia>1//2+2im1//2+2//1*imjulia>1+2//3im1//1-2//3*imjulia>0.5==1//2truejulia>0.33==1//3falsejulia>0.33<1//3truejulia>1//3-0.330.0033333333333332993</pre></div></div></div></div></div><footer><div aria-label="footer navigation" class="rst-footer-buttons" role="navigation"><a class="btn btn-neutral float-right" href="../strings/" title="Strings">Next </a><a class="btn btn-neutral" href="../mathematical-operations/" title="Mathematical Operations and Elementary Functions"> Previous</a></div><hr/><div role="contentinfo"></div><a href="https://github.com/snide/sphinx_rtd_theme">Sphinx theme</a> provided by <a href="https://readthedocs.org">Read the Docs</a></footer></div></div></section></div><script type="text/javascript"> var DOCUMENTATION_OPTIONS = { URL_ROOT:'../../', VERSION:'0.3.13-pre', COLLAPSE_INDEX:false, FILE_SUFFIX:'', HAS_SOURCE: true }; </script><script src="../../_static/jquery.js" type="text/javascript"></script><script src="../../_static/underscore.js" type="text/javascript"></script><script src="../../_static/doctools.js" type="text/javascript"></script><script src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML" type="text/javascript"></script><script src="../../_static/js/theme.js" type="text/javascript"></script><script type="text/javascript"> jQuery(function () { SphinxRtdTheme.StickyNav.enable(); }); </script></body></HTML>