NAMSSN UI Chapter

1. First contact with SageMath

SageMath is most comfortable inside a notebook. Each notebook is made of cells. You write code in a cell, press “Run”, and the output appears just underneath. You can then adjust the code and run it again until it behaves the way you want.

Your very first cell does not need to be grand. A better goal is simply: “I can press Run and read the output calmly.”

# Simple arithmetic
2 + 3
2^10
(3^5 - 1) / 2

# A first factorisation
factor(2^16 - 1)

# A tiny loop (watch the indentation)
for n in range(2, 8):
    print(n, n^2 + n + 1)

When this runs without errors, you have taken the first step: the environment is alive, and you can begin to treat it like a partner in your calculations.

2. Sage integers, rationals, and the idea of “objects”

Sage is built on Python, but it adds mathematical objects such as integer rings, rational fields, and symbolic expressions. Instead of guessing types, you can ask Sage exactly what you are working with.

# Compare plain Python and Sage integers
n_python = 5           # plain Python int
n_sage   = ZZ(5)       # Sage integer in ZZ

print(n_python, type(n_python))
print(n_sage,   type(n_sage))

# Exact rational arithmetic
r1 = QQ(1) / 3
r2 = QQ(2) / 7
print("r1 =", r1)
print("r2 =", r2)
print("r1 + r2 =", r1 + r2)

# Ask Sage what kind of object r1 is
print("Parent of r1:", r1.parent())

The vague idea to keep in mind is: Sage cares about where numbers live. The ring or field (“parent”) matters for what operations make sense.

3. First symbolic variables and plots

Sage can manipulate expressions symbolically. To do this, you declare your variables first and then build expressions out of them.

# Declare x as a symbolic variable
var('x')

# Define a symbolic expression
f = sin(x) / x
print("f(x) =", f)

# Plot it on an interval (works in a full Sage notebook)
p = plot(f, (x, -10, 10))
p.show()

# Multiple curves on one plot
q = plot([sin(x), cos(x)], (x, 0, 2*pi))
q.show()

In the public SageCell (and in the LaTeX–like styling of your notebook), these plots give immediate feedback for calculus and analysis questions.

4. Organising your Sage workspace

Instead of one giant “playground” notebook, it is better to keep a small library of focused notebooks. This mirrors how real research is done: separate files, clear names, saved experiments.

On the main SageMath Lab page, the Sage scratchpad is designed as a temporary place to shape snippets. Once a snippet feels stable, paste it into a proper notebook file and add a short text cell explaining what it does.