4. Introducing symbolic algebra
4.1. Wouldn’t it be nice…
You might have struggled with handling some long gnarly mathematical expressions, and you might have wondered either:
Wouldn’t it be nice if they just gave me the numbers, so I can punch them into a calculator, and I don’t have to figure out the general answer with all the \(x\) and \(y\) variables…
For example, if my boss asks me “find all the values of x for which this equation is true”:
I wish I could just say: “sure, boss, the solutions are \((2.64575131107, -2, 4, -2.4, 9.42477796077)\)”
Or wouldn’t it be nice if there were a computer program which could do all those algebra or calculus manipulations for me preserving the variables without plugging numbers in – all those messy expressions they had me do in algebra class…
For example, if my boss asked me: “find me the derivative of \(\sin(7 x^3) e^{-x^2}\)”, I wish I could just say (with no effort):
The first of these wishes is the subject of numerical analysis, the second is the subject of symbolic math.
In this working group we will be working with a lot of algebraic expressions, and one way in which we will enhance this beyond what is done in school will be to use symbolic math.
We will conclude every segment with an interactive use of the sympy symbolic algebra system to have the computer redo our hard-work calculations.
4.2. Preparing your computer to use sympy
4.2.1. On the web
I recommend that at some point you learn to use sympy on your computer with a full python environment. But for the purpose of this course we will use the public web installation called sympygamma – https://sympygamma.com/ and enter expressions there.
More complex expressions, but with more prep required, can be used at https://live.sympy.org/ - this will look more like what you do on a computer, showing you the python instructions they use to start.
4.2.2. On a desktop or laptop computer
On a debian-based linux distribution you can run:
$ sudo apt install python3-sympy
Alternatively, on just about any computer you can run:
$ pip3 install sympy
You should now be able to type python3 and then in the python
interpreter:
>>> import sympy
>>> from sympy import *
>>> init_printing()
>>> x, y, z, t = symbols('x y z t')
>>> k, m, n = symbols('k m n', integer=True)
>>> f, g, h = symbols('f g h', cls=Function)
This will allow you to use \(x, y, z, t, k, m, n, f, g, h\) as symbolic variables.
4.3. Getting started with a tutorial
We will work with the simpler sympygamma, which has a collection of example expressions. They are well chosen, so we will work through them, discussing what it has done after we do each expression.
First we can warm up by putting in simple arithmetic, but we won’t do more than one or two expressions, like:
22 / 7
355 / 113
Then we will expand some expressions:
(a + b)**2
expand((a + b)**2)
expand((a + b)**3)
expand((a + b)**4)
expand((a + b)**5)
expand((a + b)**6)
expand((a + b)**7)
expand((a + b)**8)
Then let us follow their algebra examples:
x
(x+2)/((x+3)(x-4))
simplify((x**2 - 4)/((x+3)(x-2)))
If you are curious about how that expression simplified one you can factor \(x^2 - 4\) in your head, or you can put into sympy:
factor(x**2 - 4)
Now some polynomial and rational functions. First put the raw polynomials that will come up into desmos:
x^4 / 2 + 5 x^3 / 12 - x^2 / 3
x^2 + 4 x y + 4 y^2 = 0
x^2 + 4 x + 181
x^3 + 4 x + 181
(Note that the last two have that big y offset, so you will need to hunt for them in the graph.)
And then the expressions to put in to sympygamma:
div(x**2 - 4 + x, x-2)
# my comment for div(): polynomial division with remainders is
# not interesting
gcd(2*x**2 + 6*x, 12*x)
lcm(2*x**2 + 6*x, 12*x)
# my comment for gcd() and lcm(): this mostly goes to show that
# polynomials have a lot in common with integers - I make a brief
# passing mention of "rings" as an algebraic structure.
factor(x**4/2 + 5*x**3/12 - x**2/3)
factor(x**2 + 4*x*y + 4*y**2)
solve(x**2 + 4*x*y + 4*y**2)
solve(x**2 + 4*x*y + 4*y**2, y)
solve(x**2 + 4*x + 181, x)
solve(x**3 + 4*x + 181, x)
solve_poly_system([y**2 - x**3 + 1, y*x], x, y)
That last expression corresponds to the system of polynomial equations:
The sympy system is well documented. You can start from https://sympy.org/ and follow their documentation link to reach the tutorial page at https://docs.sympy.org/latest/tutorials/index.html#tutorials
The class now moves jumps away from this chapter to their tutorial, starting with the examples at:
https://docs.sympy.org/latest/tutorials/intro-tutorial/intro.html
and continue to their examples of simplification and factoring:
https://docs.sympy.org/latest/tutorials/intro-tutorial/simplification.html
We can stop now, since we will return to sympy examples as we cover those topics.
4.4. [unsorted] Some expressions we will do
Simple systems of two linear equations with two unknows, starting with two equations in the form \(y = m x + b\), rewriting them as \(m x - y + b = 0\), and then putting them into sympy. Have the students pick all the coefficients. For example:
\begin{eqnarray} \begin{cases} y \; = & \frac{1}{2} x + 4.2 \cr y \; = & - 3 x - \frac{1}{2} \cr \end{cases} \end{eqnarray}System of a linear equation and a quadratic equation. Have the students pick all the coefficients. You could keep one of the lines from before. For example:
\begin{eqnarray} \begin{cases} y \; = & \frac{1}{3} x^2 - 2 x - 1 \cr y \; = & \frac{1}{2} x + 4.2 \cr \end{cases} \end{eqnarray}The famous gaussian integral:
\[\int_{-\infty}^\infty e^{-x^2} dx\]
solve([2*x+a*y-z-1, -7*x+y+2*z+1, x-b*y-z+3], x, y, z)