Foundations — SymPy — symbolic algebra, calculus, ODE solving

Before you can read a single line of the parent note, you must own a handful of tiny ideas. This page builds every one of them from nothing — no notation is used before it is drawn and explained. We go slow on purpose.

1. Number vs. symbol — the deepest split

The picture. Imagine a number line. A number is a pin stuck at one exact spot. A symbol is a slider that can sit anywhere — you keep it as a slider precisely so you can reason about all positions at once.

WHY the topic needs this. NumPy (see NumPy — numerical arrays) pins everything down to a number immediately. SymPy refuses: it keeps the slider, so it can factor, differentiate and solve before any value is chosen. This is the entire reason SymPy exists.

2. Operation — the verb that joins symbols

The picture. Think of $+$ as a machine with two funnels on top and one spout below. Pour two things in the funnels, one thing drops out. The funnels can be fed numbers, symbols, or the output of another machine — that stacking is what makes trees possible.

3. Expression tree — the shape of an idea

Reading the tree. The outermost machine is $+$. Its two funnels are fed by $x^2$ (a power machine) and $2x$ (a multiply machine) and $1$ (a plain number). Draw it top-down and you get a branching diagram — hence "tree".

WHY the topic needs this. Every SymPy command is a rule for rewriting this tree:

• expand pushes multiplications inward (opens brackets),
• factor pulls common structure outward,
• diff walks the tree applying the derivative rule at each node.

If you picture the tree, these stop being magic.

4. Equation, and "set to zero"

The picture. Plot the expression $a-b$ as a curve. The places where the curve crosses the horizontal axis (height zero) are exactly the solutions. Solving = finding the crossings.

5. The three calculus symbols you'll meet

You do not need to master calculus here — the parent note derives it. You only need to recognise the three symbols so they don't ambush you.

WHY these three. The parent's Section 3 recovers the derivative from the limit, and pairs it with the integral. The Taylor and Maclaurin Series section then uses repeated derivatives. Recognising the symbols is enough for now.

6. Function object $y(x)$ and the arbitrary constant $C_1$

WHY the topic needs this. dsolve returns Eq(y(x), C1*sin(x) + C2*cos(x)) — the whole family. Initial conditions (ics) turn the dials to a single curve. If you didn't know $y(x)$ was a shape and $C_1$ a dial, that output would look like noise.

7. Exact vs. decimal — the last gate

Picture sqrt(2) as a sealed envelope: SymPy carries it around unopened, doing algebra on the label. .evalf() is the act of tearing it open to read the (rounded) contents. The bridge to actual plotting numbers is Lambdify — bridging SymPy to NumPy for plotting.

Prerequisite map

Equipment checklist

Test yourself — cover the right side, answer aloud, then reveal.