1. Number Systems

From naturals to reals — building the number systems from axioms, and telling countable infinities from uncountable ones.

• Natural Numbers and Induction — The natural numbers start from the Peano axioms: $P(1)$ plus $P(k) \Rightarrow P(k+1)$ yields $P(n)$ for every natural number $n$ — addition and multiplication are also defined recursively, and this one principle proves infinitely many statements at once.
• The Binomial Theorem — The binomial theorem $(x+y)^n = \sum_{k=0}^{n}\binom{n}{k}x^{n-k}y^k$ proved by induction — Pascal's identity $\binom{n}{k}+\binom{n}{k-1}=\binom{n+1}{k}$ is the inductive step, and row $n$ of Pascal's triangle is drawn live.
• Integers and Rationals — Integers arise as equivalence classes of natural pairs with $(a,b)\sim(c,d)\iff a+d=b+c$, and rationals as classes of integer pairs with $(a,b)\sim(c,d)\iff ad=bc$. The "numbers as equivalence classes" trick returns when the reals are built from the rationals.
• Sets and Functions — See set operations $A \cup B$, $A \cap B$, $A^c$, De Morgan's laws, and injective / surjective / bijective maps as pictures.
• The Supremum Axiom — Every bounded-above set has a least upper bound $\sup S$ — the axiom filling the gaps in the rationals, twin of the rational-holes scene.
• Countable vs Uncountable — The rationals fit into a single list; the reals do not — verified by Cantor's diagonal argument.
• Groups vs Fields — A group is reversible moves (associativity, identity, inverses); a field is a world where all four operations work. Includes the subgroup test and $(\mathbb{Z},+)$ vs $(\mathbb{Q},+,\times)$.
• Vector Spaces and Isomorphism — A vector space is an abelian group over a field with scalar multiplication — $\mathbb{R}^n$, sequence spaces, and spaces of continuous functions are the same structure in different packaging. Move $(a, b)$ and the line $y = a+bx$ and the point $(a, b)$ move together.
• Dimension and Rank — If isomorphism means "same structure", dimension and rank measure its size. Watch the grid get squeezed by a matrix and verify $\dim\ker T + \mathrm{rank}\,T = \dim V$.