THE PATH_

Numbers, rebuilt from raw counting. You think you know this. You don't.

1. 1What Is a Number?Counting, the successor idea, and why ℕ exists before notation.
2. 2The Laws of ArithmeticCommutativity, associativity, distributivity — the axioms you've been using blind.
3. 3Zero & the NegativesAdditive identity and inverses; the integers ℤ and signed arithmetic.
4. 4Fractions & the RationalsMultiplicative inverses, equivalent fractions, and arithmetic in ℚ.
5. 5Exponents & RootsPowers as repeated multiplication; exponent laws derived; roots as the inverse question.
6. 6Order & InequalityThe number line as an ordered structure; rules for manipulating inequalities.
7. 7Irrationals & the Real Line√2 is not a fraction; the reals ℝ as the completed continuum.
8. 8Absolute Value & Distance|x| as distance from zero; solving equations and inequalities with |·|.

The machinery of certainty. Learn to argue so well reality has to agree.

1. 1Propositions & ConnectivesStatements, AND/OR/NOT, and truth tables.
2. 2ImplicationIf–then statements, converse, contrapositive, and vacuous truth.
3. 3Quantifiers∀ and ∃, nested quantifiers, and how to negate them.
4. 4Direct ProofYour first real proofs: definitions in, logic through, QED out.
5. 5Contrapositive & Iff ProofsProving P→Q via ¬Q→¬P, and proving 'if and only if' both ways.
6. 6Proof by ContradictionAssume the opposite, watch the universe break: √2 irrational, infinitude of primes.
7. 7InductionThe domino principle: prove it for 1, prove each step passes it on, own all of ℕ.
8. 8Strong Induction & Well-OrderingUse every previous domino; every nonempty set of naturals has a least element.

Everything is a set. EVERYTHING. The periodic table of mathematics.

1. 1Sets & MembershipSets, ∈, ⊆, set-builder notation, the empty set, and power sets.
2. 2Set OperationsUnion, intersection, difference, complement — and Venn diagrams done honestly.
3. 3Set Identities & Element ProofsDe Morgan's laws for sets; proving A=B by double inclusion.
4. 4Ordered Pairs & Cartesian Products(a,b) where order matters; A×B and why the plane is ℝ².
5. 5Relations & EquivalenceRelations as subsets of A×B; reflexive/symmetric/transitive; equivalence classes and partitions.
6. 6Functions as MappingsFunctions defined set-theoretically: domain, codomain, image, well-definedness.
7. 7Injections, Surjections, BijectionsOne-to-one, onto, and perfect pairings — with proofs of each.
8. 8Composition & Inverse Functionsg∘f, why composition order matters, and when an inverse exists.
9. 9Cardinality & InfinitySame size = bijection exists; ℕ, ℤ, ℚ are countable; ℝ is not. Cantor's diagonal.

Arithmetic with the masks on. Solve for the unknown, every time.

1. 1Variables & ExpressionsLetters as placeholders; building, evaluating, and simplifying expressions.
2. 2Linear EquationsSolving ax+b=c and friends by doing the same crime to both sides.
3. 3Linear InequalitiesSolving and graphing inequalities; compound inequalities.
4. 4The Coordinate PlanePlotting in ℝ²; distance and midpoint formulas.
5. 5Lines & SlopeSlope as rate; point-slope, slope-intercept, parallel and perpendicular.
6. 6Systems of Linear EquationsTwo equations, two unknowns: substitution, elimination, and the geometry of solutions.
7. 7Polynomial ArithmeticAdding, subtracting, multiplying polynomials; degree; special products.
8. 8FactoringRunning distributivity backwards: GCF, trinomials, difference of squares, grouping.
9. 9Quadratic EquationsCompleting the square, deriving the quadratic formula, and the discriminant.
10. 10Rational Expressions & EquationsAlgebraic fractions: simplifying, combining, solving — and domain landmines.

Machines that eat numbers. Curves, growth, and the art of the inverse.

1. 1Functions & Their Graphsf(x) notation meets the plane: reading graphs, domain/range, transformations.
2. 2Quadratic Functions & ParabolasVertex form, axis of symmetry, max/min, and graphs of y=ax²+bx+c.
3. 3Polynomial FunctionsEnd behavior, zeros and multiplicity, the Factor Theorem.
4. 4Exponential Functionsy=a·bˣ: growth and decay, the shape, and why exponentials beat polynomials.
5. 5Logarithmslog_b(x) as the inverse exponent question; log laws derived from exponent laws.
6. 6Inverse Functionsf⁻¹ in the wild: which functions are invertible, finding inverses, mirror graphs.
7. 7Sequences & SeriesArithmetic and geometric sequences, Σ-notation, and closed-form sums.
8. 8Counting & the Binomial TheoremPermutations, combinations, Pascal's triangle, and expanding (a+b)ⁿ.

Triangles and circles — the load-bearing shapes of the universe.

1. 1Angles, Triangles & SimilarityAngle facts, triangle angle sum, congruence vs similarity, and proportional sides.
2. 2The Pythagorean Theorema²+b²=c², proved at least twice, plus its converse and the distance formula payoff.
3. 3Trig Ratiossin, cos, tan as similarity invariants of right triangles; solving triangles.
4. 4The Unit Circle & RadiansAngles as arc length; sin and cos as coordinates; trig beyond 90°.
5. 5Trig IdentitiesPythagorean identity, sum/difference formulas, double angle — derived and deployed.
6. 6Circles in the Plane(x−h)²+(y−k)²=r² and completing the square to find centers.

Arrows and grids of numbers that move entire planes around.

1. 1VectorsArrows and tuples; addition, scalar multiplication, and length.
2. 2The Dot Productv·w two ways: components and ‖v‖‖w‖cos θ; orthogonality and projection.
3. 3Matrices & Their ArithmeticMatrices as data grids; addition, scalar multiplication, transpose.
4. 4Matrix MultiplicationRow-times-column: why THAT rule, and why AB ≠ BA.
5. 5Linear Transformations of the PlaneMatrices as machines that move ℝ²: rotations, scalings, shears — read from the columns.
6. 6Determinantsdet as area-scaling factor; 2×2 and 3×3 computation; det(AB)=det(A)det(B).
7. 7Matrix InversesA⁻¹ as the undo machine; the 2×2 formula; solving Ax=b.
8. 8Gaussian EliminationRow reduction to (R)REF: the industrial-strength solver for any size system.

Linear algebra with proofs. The cathedral. The spectral theorem lives at the top.

1. 1Vector SpacesThe axioms: any set with a +, a scalar ·, and ten laws is a vector space.
2. 2SubspacesSubsets that are spaces in their own right; the three-condition test.
3. 3Span & Linear CombinationsEverything reachable by scaling and adding; span as the smallest subspace containing your set.
4. 4Linear IndependenceNo vector in the list is a combination of the others; the c₁v₁+⋯+cₖvₖ=0 test.
5. 5Basis & DimensionIndependent spanning sets; every basis has the same size; coordinates.
6. 6Linear MapsStructure-preserving functions between spaces; kernel and image.
7. 7Rank–Nullitydim(ker) + dim(im) = dim(domain): the conservation law of linear algebra.
8. 8Change of BasisSame vector, different coordinates; similarity A ↦ P⁻¹AP.
9. 9Eigenvalues & EigenvectorsAv = λv: the directions a transformation merely stretches.
10. 10DiagonalizationA = PDP⁻¹: rewriting a transformation in its own favorite basis.
11. 11Orthogonality & Gram–SchmidtOrthonormal bases, orthogonal projection onto subspaces, and the Gram–Schmidt machine.
12. 12The Spectral TheoremSymmetric matrices diagonalize orthogonally: A = QDQᵀ. The summit.