Symbolic Calculators

Run the following to load the SymPy module for symbolic computations in Python.

The following code is a Python one-liner that creates a symbolic calculator.

Evaluate the cell repeatedly with Ctrl+Enter to try some calculations below using this calculator:

1. 2³ by entering 2**3;
2. $\frac23$ by entering 2/3;
3. $\left\lceil\frac32\right\rceil$ by entering 3//2;
4. $3\mod 2$ by entering 3%2;
5. $\sqrt{2}$ by entering 2**(1/2); and
6. $\sin(\pi/6)$ by entering sin(pi/6);

Unlike eval(input()), JupyterLab render the $\LaTeX$^[1] code of the expression using MathJax^[2]. For instance, for the SymPy expression of $1/2$, the $\LaTeX$ code and the rendering by Mathjax is as follows:

For this lab, you will see how arbitrary precision arithmetic can be carried out using symbols.

Symbolic Hypotenuse Calculator¶

We can define the following function to calculate the length c of the hypotenuse when given the lengths a and b of the other sides:

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def length_of_hypotenuse(a, b):
    c = (a**2 + b**2) ** S("1/2")
    return c

You can check your code against a few cases listed in the test cell below.

The tests use == instead of isclose because the compution should be exact. For instance, the hypotenuse of the last test case is:

If you are curious about the hidden test, it is like this:

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a, b = symbols('a, b', nonnegative=True)
assert length_of_hypotenuse(a, b) == sqrt(a**2 + b**2)

The first line assign the variables a and b to the symbols Symbol('a') and Symbol('b') respectively, both of which are assumed to be nonnegative.

Assumptions can affect the check for equivalence ==:

In other words,

• $\sqrt{\lvert a\rvert^2} \not\equiv a$ in general, but
• $\sqrt{\lvert a\rvert^2} = a \qquad \forall a\geq 0.$

There is a further subtlety that, in Mathematics, equality Eq is different from equivalence ==:

In the first case, the equality $\sqrt{\lvert a \rvert^2} = a$

• neither simplifies to True, as a_ may be negative,
• nor simplifies to False, as a_ may be non-negative.

In the second case, the equality holds True as a is non-negative.

Similarly, symbols with the same name but different assumptions may be equal but are not equivalent:

Calculus¶

Can we do complicated arithmetics with python. What about Calculus?

Try SymPy Gamma or SymPy Beta:

• Take a look at the different panels to learn about the solution: Steps, Plot, and Derivative.
• Try different random examples.

While web applications like SymPy Gamma or Beta offer a user-friendly and convenient interface for performing symbolic computations quickly without needing to install anything, using SymPy within a Python program provides significantly greater flexibility and control.

To compute the integration in Python, we first define a symbolic variable x:

The SymPy expression for $\tan(x)$ is:

To compute the integration:

To compute the derivative:

The answer can be simplified as expected:

To plot the functions:

The following test should plot your expression f in SymPy.