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+        <h1>15.1. Diving into symbolic computing with SymPy</h1>
+    </header>
+
+    <section id="page">
+        <p><a href="/"><img src="https://raw.githubusercontent.com/ipython-books/cookbook-2nd/master/cover-cookbook-2nd.png" align="left" alt="IPython Cookbook, Second Edition" height="130" style="margin-right: 20px; margin-bottom: 10px;" /></a> <em>This is one of the 100+ free recipes of the <a href="/">IPython Cookbook, Second Edition</a>, by <a href="http://cyrille.rossant.net">Cyrille Rossant</a>, a guide to numerical computing and data science in the Jupyter Notebook. The ebook and printed book are available for purchase at <a href="https://www.packtpub.com/big-data-and-business-intelligence/ipython-interactive-computing-and-visualization-cookbook-second-e">Packt Publishing</a>.</em></p>
+<p>▶&nbsp;&nbsp;<em><a href="https://github.com/ipython-books/cookbook-2nd">Text on GitHub</a> with a <a href="https://creativecommons.org/licenses/by-nc-nd/3.0/us/legalcode">CC-BY-NC-ND license</a></em><br />
+▶&nbsp;&nbsp;<em><a href="https://github.com/ipython-books/cookbook-2nd-code">Code on GitHub</a> with a <a href="https://opensource.org/licenses/MIT">MIT license</a></em></p>
+<p>▶&nbsp;&nbsp;<a href="http://ipython-books.github.io/chapter-15-symbolic-and-numerical-mathematics/"><strong><em>Go to</em></strong> <em>Chapter 15 : Symbolic and Numerical Mathematics</em></a><br />
+▶&nbsp;&nbsp;<a href="https://github.com/ipython-books/cookbook-2nd-code/blob/master/chapter15_symbolic/01_sympy_intro.ipynb"><em><strong>Get</strong> the Jupyter notebook</em></a>  </p>
+<p>In this recipe, we will give a brief introduction to symbolic computing with SymPy. We will see more advanced features of SymPy in the next recipes.</p>
+<h2>Getting ready</h2>
+<p>Anaconda should come with SymPy by default, but you can always install it with <code>conda install sympy</code>.</p>
+<h2>How to do it...</h2>
+<p>SymPy can be used from a Python module, or interactively in Jupyter/IPython. In the Notebook, all mathematical expressions are displayed with LaTeX, thanks to the MathJax JavaScript library.</p>
+<p>Here is an introduction to SymPy:</p>
+<p><strong>1.&nbsp;</strong> First, we import SymPy and enable LaTeX printing in the Jupyter Notebook:</p>
+<div class="highlight highlight-python"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sympy</span> <span class="kn">import</span> <span class="o">*</span>
+<span class="n">init_printing</span><span class="p">()</span>
+</pre></div>
+</div>
+
+
+<p><strong>2.&nbsp;</strong> To deal with symbolic variables, we first need to declare them:</p>
+<div class="highlight highlight-python"><div class="highlight"><pre><span></span><span class="n">var</span><span class="p">(</span><span class="s1">&#39;x y&#39;</span><span class="p">)</span>
+</pre></div>
+</div>
+
+
+<p><img alt="(x, y)" src="http://ipython-books.github.io/pages/chapter15_symbolic/01_sympy_intro_files/01_sympy_intro_10_0.png" /></p>
+<p><strong>3.&nbsp;</strong> The <code>var()</code> function creates symbols and injects them into the namespace. This function should only be used in the interactive mode. In a Python module, it is better to use the <code>symbols()</code> function that returns the symbols:</p>
+<div class="highlight highlight-python"><div class="highlight"><pre><span></span><span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">symbols</span><span class="p">(</span><span class="s1">&#39;x y&#39;</span><span class="p">)</span>
+</pre></div>
+</div>
+
+
+<p><strong>4.&nbsp;</strong> We can create mathematical expressions with these symbols:</p>
+<div class="highlight highlight-python"><div class="highlight"><pre><span></span><span class="n">expr1</span> <span class="o">=</span> <span class="p">(</span><span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span>
+<span class="n">expr2</span> <span class="o">=</span> <span class="n">x</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="n">x</span> <span class="o">+</span> <span class="mi">1</span>
+</pre></div>
+</div>
+
+
+<p><strong>5.&nbsp;</strong> Are these expressions equal?</p>
+<div class="highlight highlight-python"><div class="highlight"><pre><span></span><span class="n">expr1</span> <span class="o">==</span> <span class="n">expr2</span>
+</pre></div>
+</div>
+
+
+<div class="highlight highlight-text"><div class="highlight"><pre><span></span>False
+</pre></div>
+</div>
+
+
+<p><strong>6.&nbsp;</strong> These expressions are mathematically equal, but not syntactically identical. To test whether they are mathematically equal, we can ask SymPy to simplify the difference algebraically:</p>
+<div class="highlight highlight-python"><div class="highlight"><pre><span></span><span class="n">simplify</span><span class="p">(</span><span class="n">expr1</span> <span class="o">-</span> <span class="n">expr2</span><span class="p">)</span>
+</pre></div>
+</div>
+
+
+<p><img alt="0" src="http://ipython-books.github.io/pages/chapter15_symbolic/01_sympy_intro_files/01_sympy_intro_18_0.png" /></p>
+<p><strong>7.&nbsp;</strong> A very common operation with symbolic expressions is the substitution of a symbol by another symbol, expression, or a number, using the subs() method of a symbolic expression:</p>
+<div class="highlight highlight-python"><div class="highlight"><pre><span></span><span class="n">expr1</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">expr1</span><span class="p">)</span>
+</pre></div>
+</div>
+
+
+<p><img alt="Output" src="http://ipython-books.github.io/pages/chapter15_symbolic/01_sympy_intro_files/01_sympy_intro_20_0.png" /></p>
+<div class="highlight highlight-python"><div class="highlight"><pre><span></span><span class="n">expr1</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">pi</span><span class="p">)</span>
+</pre></div>
+</div>
+
+
+<p><img alt="Output" src="http://ipython-books.github.io/pages/chapter15_symbolic/01_sympy_intro_files/01_sympy_intro_21_0.png" /></p>
+<p><strong>8.&nbsp;</strong> A rational number cannot be written simply as <code>1/2</code> as this Python expression evaluates to 0.5. A possibility is to convert the number <code>1</code> into a SymPy integer object, for example by using the <code>S()</code> function:</p>
+<div class="highlight highlight-python"><div class="highlight"><pre><span></span><span class="n">expr1</span><span class="o">.</span><span class="n">subs</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span> <span class="o">/</span> <span class="mi">2</span><span class="p">)</span>
+</pre></div>
+</div>
+
+
+<p><img alt="9/4" src="http://ipython-books.github.io/pages/chapter15_symbolic/01_sympy_intro_files/01_sympy_intro_23_0.png" /></p>
+<p><strong>9.&nbsp;</strong> Exactly represented numbers can be evaluated numerically with <code>evalf()</code>:</p>
+<div class="highlight highlight-python"><div class="highlight"><pre><span></span><span class="n">_</span><span class="o">.</span><span class="n">evalf</span><span class="p">()</span>
+</pre></div>
+</div>
+
+
+<p><img alt="2.25000000000000" src="http://ipython-books.github.io/pages/chapter15_symbolic/01_sympy_intro_files/01_sympy_intro_25_0.png" /></p>
+<p><strong>10.&nbsp;</strong> We can easily create a Python function from a SymPy symbolic expression using the <code>lambdify()</code> function. The resulting function can notably be evaluated on NumPy arrays. This is quite convenient when we need to go from the symbolic world to the numerical world:</p>
+<div class="highlight highlight-python"><div class="highlight"><pre><span></span><span class="n">f</span> <span class="o">=</span> <span class="n">lambdify</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">expr1</span><span class="p">)</span>
+</pre></div>
+</div>
+
+
+<div class="highlight highlight-python"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span>
+<span class="n">f</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mf">2.</span><span class="p">,</span> <span class="mf">2.</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>
+</pre></div>
+</div>
+
+
+<div class="highlight highlight-text"><div class="highlight"><pre><span></span>array([ 1.,  0.,  1.,  4.,  9.])
+</pre></div>
+</div>
+
+
+<h2>How it works...</h2>
+<p>A core idea in SymPy is to use the standard Python syntax to manipulate exact expressions. Although this is very convenient and natural, there are a few caveats. Symbols such as <code>x</code>, which represent mathematical variables, cannot be used in Python before being instantiated (otherwise, a <code>NameError</code> exception is thrown by the interpreter). This is in contrast to most other computer algebra systems. For this reason, SymPy offers ways to declare symbolic variables beforehand.</p>
+<p>Another example is integer division; as <code>1/2</code> evaluates to <code>0.5</code> (or 0 in Python 2), SymPy has no way to know that the user intended to write a fraction instead. We need to convert the numerical integer 1 to the symbolic integer <code>1</code> before dividing it by 2.</p>
+<p>Also, the Python equality refers to the equality between syntax trees rather than between mathematical expressions.</p>
+<h2>See also</h2>
+<ul>
+<li>Solving equations and inequalities</li>
+<li>Getting started with Sage</li>
+</ul>
+    </section>
+
+            </div>
+        </section>
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