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Interactive Explainers

Play with the theorems

Interactive explainers that let you move the mathematics: drag ε and watch δ answer, walk the topologist's sine curve, trap a subsequence by bisection, watch Newton's method fall into a cycle. 15 are free to play right here — the rest come with Trap Analysis.

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Video explainers

Narrated walk-throughs of the trickiest ideas, made for this site — no ads, no recommendations pulling you elsewhere. Free to watch whether or not the interactive below is unlocked.

Unit 2Zeros and Mappings › Argument principle, Rouché's theorem, open mapping#

Rouché in action: counting the zeros of z⁵ + 3z + 1interactivefree

Step 1 / 5The question

How many zeros does p(z)=z5+3z+1p(z) = z^5 + 3z + 1 have inside ∣z∣<1|z| < 1? And inside ∣z∣<2|z| < 2?

No formula finds these roots — but Rouché's theorem counts them without finding them.

Unit 2Analytic Functions › Cauchy–Riemann equations, harmonic functions#

Cauchy–Riemann: satisfied at a point, holomorphic nowhereinteractivefree

Step 1 / 4The equations

Writing f=u+ivf = u+iv, complex differentiability at a point requires

ux=vy,uy=−vx.u_x = v_y, \qquad u_y = -v_x.

These are necessary. They are not, on their own, sufficient.

Unit 2Cauchy Theory › Cauchy's theorem and integral formula#

Cauchy's integral formula, and what it gives you for freeinteractivefree

Step 1 / 4The theorem

If ff is holomorphic on and inside a simple closed contour CC, then ∮Cf=0\oint_C f = 0. If aa is inside CC,

f(a)=12πi∮Cf(z)z−a dz.f(a) = \frac{1}{2\pi i}\oint_C \frac{f(z)}{z-a}\,dz.

The value at an interior point is determined entirely by the boundary values.

Unit 2Topology › Connectedness and components#

The topologist's sine curve, walkedinteractive

This visual is part of Trap Analysis

Walk along sin(1/x) toward the segment and watch the distance meter diverge — connected, but no path gets there.

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Unit 2Zeros and Mappings › Conformal maps, Möbius transformations, Schwarz lemma#

Möbius maps bending the gridinteractive

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Slide the map on and watch straight lines bow into circles — every one still a circle or a line, angles preserved throughout.

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Unit 2Groups › Subgroups, cosets, Lagrange, cyclic groups#

Lagrange's theorem, tiledinteractive

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Cosets colouring a cyclic group: same size, never overlapping, always tiling it exactly. The picture will not let you build a counterexample.

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Unit 2Singularities and Residues › Laurent series, classification of singularities, Casorati–Weierstrass#

Classifying an isolated singularity in one lookinteractive

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Removable, pole or essential — decided by the Laurent tail, with the test for each.

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Unit 2Singularities and Residues › Residue theorem and standard contour integrals#

Computing residues without expanding the seriesinteractive

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Simple poles, higher-order poles and the check that catches arithmetic slips.

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Unit 2Cauchy Theory › Liouville, Morera, maximum modulus principle#

Liouville, and the growth bounds that do not force constancyinteractive

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Four hypotheses — which collapse the function to a constant and which only make it a polynomial.

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Unit 2Analytic Functions › Power series and analyticity#

Why the radius of convergence is set by the nearest singularityinteractive

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A function smooth on all of ℝ whose series stops at |x| = 1 — explained by a pole you cannot see on the real line.

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Unit 2Groups › Subgroups, cosets, Lagrange, cyclic groups#

Lagrange's theorem and its false converseinteractive

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Order divides — but a divisor need not give a subgroup, and A₄ is the reason.

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Unit 2Groups › Normal subgroups, quotients, isomorphism theorems#

The first isomorphism theorem, used as a toolinteractive

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Three quotients identified in one line each — the fastest way to answer 'what is G/N?'

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Unit 2Groups › Permutation groups: cycles, sign, conjugacy in S_n and A_n#

Cycle type decides everything in Sₙinteractive

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Order, sign and conjugacy all read off one decomposition — plus where the rule changes in Aₙ.

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Unit 2Groups › Group actions, class equation, p-groups#

The class equation, and what it proves about p-groupsinteractive

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Counting orbits mod p — the argument behind almost every small-group structure result.

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Unit 2Groups › Sylow theorems and groups of small order#

Sylow counting: proving a group is not simpleinteractive

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The standard argument, run on order 12 and order 15 — including the element count that finishes it.

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Unit 2Groups › Finite abelian groups#

Classifying finite abelian groups, and telling two apartinteractive

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Counting the groups of a given order, and the invariant that distinguishes them in one step.

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Unit 2Rings and Fields › Ideals, quotient rings, prime & maximal ideals, CRT#

Prime versus maximal, decided by the quotientinteractive

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One test settles both — worked in ℤ[x], where an ideal can be prime and still not maximal.

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Unit 2Rings and Fields › Euclidean, PID, UFD hierarchy#

Euclidean ⊂ PID ⊂ UFD, with a separating example at each stepinteractive

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The chain is strict — here is the ring that sits in each gap.

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Unit 2Rings and Fields › Polynomial rings and irreducibility tests#

Irreducibility tests, in the order worth trying theminteractive

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Rational roots, Eisenstein, reduction mod p — and the quartic that factors despite having no real roots.

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Unit 2Rings and Fields › Field extensions, splitting fields, finite fields#

Degrees multiply — computing [K:ℚ] and spotting the primitive elementinteractive

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The tower law in practice, plus the subfield rule for finite fields that decides many options.

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Unit 2Rings and Fields › Galois theory essentials#

The Galois correspondence, used on one exampleinteractive

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A degree-4 extension, its group, and the subfield lattice read straight off the subgroup lattice.

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Unit 2Topology › Topological spaces, bases, subspace/product/quotient#

Which properties survive subspaces, products and quotientsinteractive

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A table you can reconstruct — and the operation that destroys Hausdorff.

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Unit 2Topology › Compactness and Tychonoff#

The compactness facts that need Hausdorff, and the ones that do notinteractive

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Compact implies closed is false without it — here is the space where it fails.

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Unit 2Topology › Continuity, homeomorphism, separation axioms#

The separation axioms, in the order they matterinteractive

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T₁ versus T₂ decided by one example, and the finite-space collapse that is a favourite question.

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Unit 2Topology › Standard spaces: cofinite, cocountable, Sorgenfrey, Cantor set#

The standard counterexample spaces, and what each is forinteractive

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Cofinite, Sorgenfrey and the Cantor set — the three that answer most topology options.

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Unit 2Zeros and Mappings › Conformal maps, Möbius transformations, Schwarz lemma#

A grid, deformed by a complex functioninteractive

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z², 1/z, eᶻ, sin z — watch which ones preserve angles everywhere, and where each one doesn't.

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Every interactive explainer comes with Trap Analysis — along with the worked solutions, the per-option trap diagnosis and the counterexample objects. The papers, the book and the video map are free without it.

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