How#

How to create a symbolic numeric value#

To create a symbolic numerical value use sympy.S.

Tip

sympy.S(a)

For example:

import sympy

value = sympy.S(3)
value
\[\displaystyle 3\]

Attention

If we combine a symbolic value with a non symbolic value it will automatically give a symbolic value:

1 / value
\[\displaystyle \frac{1}{3}\]

How to get the numerical value of a symbolic expression#

We can get the numerical value of a symbolic value using float or int:

  • float will give the numeric approximation in \(\mathbb{R}\)

    Tip

    float(x)

  • int will give the integer value

    Tip

    int(x)

For example, let us create a symbolic numeric variable with value \(\frac{1}{5}\):

value = 1 / sympy.S(5)
value
\[\displaystyle \frac{1}{5}\]

To get the numerical value:

float(value)

0.2

If we wanted the integer value:

int(value)

0

Attention

This is not rounding to the nearest integer. It is returning the integer part.

How to factor an expression#

We use the sympy.factor tool to factor expressions.

Tip

sympy.factor(expression)

For example:

x = sympy.Symbol("x")
sympy.factor(x ** 2 - 9)
\[\displaystyle \left(x - 3\right) \left(x + 3\right)\]

How to expand an expression#

We use the sympy.expand tool to expand expressions.

Tip

sympy.expand(expression)

For example:

sympy.expand((x - 3) * (x + 3))
\[\displaystyle x^{2} - 9\]

How to simplify an expression#

We use the sympy.simplify tool to simplify an expression.

Tip

sympy.simplify(expression)

For example:

sympy.simplify((x - 3) * (x + 3))
\[\displaystyle x^{2} - 9\]

Attention

This will not always give the expected (or any) result. At times it could be more beneficial to use sympy.expand and/or sympy.factor.

How to solve an equation#

We use the sympy.solveset tool to solve an equation. It takes two values as inputs. The first is either:

  • An expression for which a root is to be found

  • An equation

The second is the variable we want to solve for.

Tip

sympy.solveset(equation, variable)

Here is how we can use sympy to obtain the roots of the general quadratic:

\[ a x ^ 2 + bx + c \]
a = sympy.Symbol("a")
b = sympy.Symbol("b")
c = sympy.Symbol("c")
quadratic = a * x ** 2 + b * x + c
sympy.solveset(quadratic, x)
\[\displaystyle \left\{- \frac{b}{2 a} - \frac{\sqrt{- 4 a c + b^{2}}}{2 a}, - \frac{b}{2 a} + \frac{\sqrt{- 4 a c + b^{2}}}{2 a}\right\}\]

Here is how we would solve the same equation but not for \(x\) but for \(b\):

sympy.solveset(quadratic, b)
\[\displaystyle \left\{- \frac{a x^{2} + c}{x}\right\}\]

It is however clearer to specifically write the equation that we want to solve:

equation = sympy.Eq(a * x ** 2 + b * x + c, 0)
sympy.solveset(equation, x)
\[\displaystyle \left\{- \frac{b}{2 a} - \frac{\sqrt{- 4 a c + b^{2}}}{2 a}, - \frac{b}{2 a} + \frac{\sqrt{- 4 a c + b^{2}}}{2 a}\right\}\]

How to substitute a value in to an expression#

Given a sympy expression it is possible to substitute values in to it using the .subs() tool.

Tip

expression.subs({variable: value})

Attention

It is possible to pass multiple variables at a time.

For example we can substitute the values for \(a, b, c\) in to our quadratic:

quadratic = a * x ** 2 + b * x + c
quadratic.subs({a: 1, b: sympy.S(7) / 8, c: 0})
\[\displaystyle x^{2} + \frac{7 x}{8}\]