SymPy is a pure Python package for symbolic mathematics.

First, we import SymPy, and enable rich display LaTeX-based printing in the IPython notebook (using the MathJax Javascript library).

In [ ]:

```
from sympy import *
init_printing()
```

With NumPy and the other packages we have been using so far, we were dealing with numbers and numerical arrays. With SymPy, we deal with symbolic variables. It's a radically different shift of paradigm, which mathematicians may be more familiar with.

To deal with symbolic variables, we need to declare them.

In [ ]:

```
var('x y')
```

The var function creates symbols and injects them into the namespace. This function should only be used in interactive mode. In a Python module, it is better to use the symbol function which returns the symbols.

In [ ]:

```
x, y = symbols('x y')
```

We can create mathematical expressions with these symbols.

In [ ]:

```
expr1 = (x + 1)**2
expr2 = x**2 + 2*x + 1
```

Are these expressions equal?

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```
expr1 == expr2
```

These expressions are mathematically equal, but not syntactically identical. To test whether they are equal, we can ask SymPy to simplify the difference algebraically.

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```
simplify(expr1-expr2)
```

A very common operation with symbolic expressions is substitution of a symbol by another symbol, expression, or a number.

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```
expr1.subs(x, expr1)
```

In [ ]:

```
expr1.subs(x, pi)
```

A rational number cannot be written simply as "1/2" as this Python expression evaluates to 0. A possibility is to use a SymPy object for 1, for example using the function S.

In [ ]:

```
expr1.subs(x, S(1)/2)
```

Exactly-represented numbers can be evaluated numerically with evalf:

In [ ]:

```
_.evalf()
```

We can transform this *symbolic* function into an actual Python function that can be evaluated on NumPy arrays, using the `lambdify`

function.

In [ ]:

```
f = lambdify(x, expr1)
```

In [ ]:

```
import numpy as np
f(np.linspace(-2., 2., 5))
```

You'll find all the explanations, figures, references, and much more in the book (to be released later this summer).

IPython Cookbook, by Cyrille Rossant, Packt Publishing, 2014 (500 pages).