Mathematics for Quantum — Track 2

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Complex Numbers

Why quantum needs 2D numbers. Phase, rotation, Euler's formula, interference as multiplication. The engine behind every quantum speedup.

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Vectors + Dirac Notation

Quantum states as column vectors. Inner products. The ket |ψ⟩ is just a name for [α,β]ᵀ. Born rule as inner product squared. The language of quantum mechanics.

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Matrices & Operators

Gates are matrices. Unitary means reversible. Pauli, Hadamard, eigenvectors, spectral theorem. Everything in Track 3 is built from this section.

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Geometry & Quantum Info

Tensor products, entanglement as non-factorizability, Bloch sphere math, fidelity, trace distance. Plus probability, entropy, and Big-O notation.

Full Curriculum

Full curriculum

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§0 complete — Bridge Builder badge earned.

You've mapped every Track 1 concept to its Track 2 symbol.

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§0 · 3 lessons · ~28 min

Mindset & Bridge

T1→T2 translation, learning contract, growth mindset

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M00

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Welcome — Math as Your New Superpower

You already understand quantum — this track gives you the symbols. An animated before/after reveal maps every Track 1 concept to its Track 2 mathematical form. 8 pairs, click by click.

How to Learn Track 2 — Prediction Battles

Predict first, calculate second — this is how physicists actually think. A live Prediction Battle demo shows you the format: commit to an answer before the reveal, every time.

Track 1 → Track 2 Bridge — 8 Translations

Every Track 1 concept has a mathematical symbol. Eight flip-cards — T1 concept on front, T2 symbol on back, confidence 1–5 slider per card. Completion unlocks §1.

8 min✦ Interactive

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§1 complete — Phase Master badge earned.

You can compute quantum probabilities from complex amplitudes. That's a bigger deal than it sounds.

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§1 · 10 lessons · ~148 min

Complex Numbers — Quantum's Phase Engine

i as rotation, complex plane, polar form, Euler's formula, interference, probability rule

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M03

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Why Quantum Needs Complex Numbers

Real amplitude sliders can't cancel to zero — interference is impossible. Unlock the imaginary dimension and cancellation appears. This lesson shows the gap before introducing i.

i is not imaginary — it rotates the number line 90° into a second dimension. Click ×i four times and watch the point return to its origin. Discover i²=−1 geometrically.

The Complex Plane — Every z as a 2D Point

z = a+bi is a point (a,b) with a length and a direction. Drag a point on the complex plane and watch Re, Im projections and |z| update live. Prediction Battle: if z=3+4i, what is |z|?

Polar Form — Length and Angle

Every complex number is r·e^(iθ): a length r and a rotation angle θ. Radius + angle sliders update the Cartesian form live. Convert z=1+i to polar: r=√2, θ=45°.

Euler's Formula — The Equation That Unifies Everything

e^(iπ)+1=0. Five constants walk into a bar. A θ slider traces the unit circle, Taylor series terms add up live. Drag θ to π and discover the identity yourself. Full derivation inside.

18 min✦ Interactive⚠ T3 gates

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M08

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Multiplication = Rotate and Scale — the Engine of Interference

Multiplying complex numbers adds angles and multiplies lengths. Two phasor sliders — drag them together for constructive (green) or opposite for destructive (red) interference.

14 min✦ Interactive⚠ T4 algorithms

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M09

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Complex Conjugate and |z|² — the Probability Rule

Probability = z·z* = |z|². The conjugate is not optional. Drag amplitude z → z* mirrors across the real axis → probability bar updates live. Algebraic proof: (a+bi)(a−bi)=a²+b².

Global vs Relative Phase — the First Surprise

Multiply the whole state by e^(iφ) — probabilities don't change. Relative phase does change everything. Two sliders show the asymmetry. Most counterintuitive result in §1.

14 min✦ Interactive⚠ T3 gate design

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Modular Arithmetic & Roots of Unity — Shor's Secret Weapon

Clock-face shows N equally spaced unit circle points. Compute 7 mod 3 interactively. The N-th roots of unity ω_N=e^(2πi/N) are the QFT phase factors. Not in any other beginner course.

16 min✦ Interactive⚠ T4 QFT + Shor

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M12

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§1 Mastery Challenge — Complex Numbers in Quantum

5 problems: |z|², polar convert, probability from amplitude, global vs relative phase, normalization from scratch (no scaffolding on the last one). Badge on completion.

18 min✦ Milestone

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§2 complete — State Vector badge earned.

You can represent any qubit state as a vector and compute measurement probabilities.

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§2 · 8 lessons · ~110 min

Vectors & Early Dirac Exposure

State representation, inner product, normalization, Born rule, basis independence

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M13

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Vectors as Quantum States — the Column Picture

A quantum state |ψ⟩ is a column of complex numbers — the vector IS the state. Sliders for α,β sync live to a Bloch sphere. |ψ⟩=[α,β]ᵀ notation updates in real time.

Dirac Notation Preview — |0⟩ and |1⟩

|0⟩ and |1⟩ are just names for [1,0]ᵀ and [0,1]ᵀ — the notation is shorthand, not magic. Toggle between column vector and Dirac ket. Exercise: write |+⟩ as a column vector.

Vector Addition — Superposition Is a Sum

Quantum superposition IS vector addition. Build |+⟩=(1/√2)|0⟩+(1/√2)|1⟩ geometrically: two vectors, tip-to-tail, column notation updating live. The algebra behind Track 1's intuition.

Inner Product ⟨φ|ψ⟩ — Measuring Overlap

The inner product measures how much two quantum states overlap — it is the engine of probability. Two Bloch sphere states → ⟨φ|ψ⟩ computed step-by-step. Prediction Battle: ⟨0|1⟩=?

Normalization — Why Probabilities Must Sum to 1

⟨ψ|ψ⟩=1 because probabilities sum to 1. Amplitude sliders → live normalization check → valid/invalid indicator. Normalize button shows full calculation step by step.

Orthonormal Bases and Born Rule in Vector Form

P(0)=|⟨0|ψ⟩|² — the full derivation. Rotate the measurement basis slider → probability bars update → histogram builds live. The payoff of §1+§2 combined in one lesson.

Linear Combinations, Span and Basis Vectors

Any quantum state is a linear combination of basis vectors. Switch from Z-basis {|0⟩,|1⟩} to X-basis {|+⟩,|−⟩} — same Bloch sphere point, different coordinates. Choosing a basis is choosing a language.

Mini Milestone — Halfway Celebration

You can now represent any qubit state as a vector and calculate measurement probabilities from scratch. A 10-node knowledge map lights up one concept at a time. Confetti on completion.

10 min✦ Milestone

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§3 complete — Gate Mechanic badge earned.

Gates are matrices. Unitarity is reversibility. You can apply, verify, and derive any single-qubit gate.

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§3 · 9 lessons · ~136 min

Matrices & Operators

Gates as matrices, Pauli family, Hadamard, unitarity, eigenvectors (3-step), SVD

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M21

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Matrices as Operators — Gates Are Matrices

A quantum gate is a matrix. Applying a gate means multiplying a vector by that matrix. 4 entry sliders + input vector → output computed live with every step shown. Build the matrix that maps |0⟩ to |1⟩.

Pauli X, Y, Z — the Fundamental Rotation Family

X flips bits, Z flips phase, Y does both. Axis selector + angle → Bloch sphere rotates live. Y has complex entries (i, −i) — the payoff of §1. Prediction Battle: what does Z do to |+⟩?

16 min✦ Interactive⚠ T6 stabilizers

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M23

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The Hadamard Gate — Superposition as a Rotation Matrix

H=(1/√2)[[1,1],[1,−1]]. Press H → Bloch sphere north pole animates to equator. Full derivation: H|0⟩, H|1⟩, H|+⟩. Surprise: H|+⟩=|0⟩. Verify H²=I step by step.

Unitary Matrices — Reversibility in Matrix Form

U†U=I means every quantum gate is reversible. Enter any matrix → compute U† step by step → check if it equals I. Green=unitary, red=not. Connect to why measurement is irreversible.

Eigenvalues and Eigenvectors — 3-Step Visual Mastery

Step 1: find geometrically on Bloch sphere. Step 2: transformation animation shows eigenvector stays fixed. Step 3: tie to specific gate. Why the Z eigenstates are the measurement outcomes.

20 min✦ Interactive⚠ T4 phase estimation

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M26

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Spectral Theorem — Gates in Their Own Basis

A=Σλᵢ|vᵢ⟩⟨vᵢ|. Angle slider θ → compute e^(iθP) via spectral formula → Bloch sphere rotates. This IS how Rx, Ry, Rz are defined. Hardest derivation in §3 — every step shown.

16 min✦ Interactive⚠ T3 rotation gates

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M27

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Hermitian Operators and Observables

Only Hermitian operators (A=A†) represent physical measurements — they guarantee real outcomes. Enter a matrix → check Hermitian → show real eigenvalues. Proof: Av=λv → λ=λ* in full.

Singular Value Decomposition — Any Matrix as Rotate-Scale-Rotate

Every matrix M=UΣV†. Three animation steps show each transform on a unit circle → ellipse. For unitary gates, all singular values = 1. Preview of T6 noise models where values are < 1.

14 min✦ Interactive⚠ T6 noise models

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§3 Mastery Challenge — Operators in Action

4 problems: H|1⟩ full computation, Y unitarity check, Z eigenvectors, math-only probability from matrix. Final problem: full derivation, no scaffolding, no hints. Badge on completion.

18 min✦ Milestone

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⟨∣⟩

§4 complete — Dirac Fluent badge earned.

You can read and write quantum expressions without looking anything up.

⟨∣⟩

§4 · 5 lessons · ~73 min

Dirac Notation Mastery

Kets, bras, inner + outer products, fluency challenge, Hadamard as Dirac

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M30

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Kets |ψ⟩ — the Natural Language of Quantum States

Full Dirac introduction after 3 sections of exposure. Side-by-side translator: column vector ↔ ket. Gallery: |0⟩, |1⟩, |+⟩, |−⟩, |i⟩, |−i⟩. Should feel like recognition, not learning.

Bras ⟨ψ| — the Conjugate Transpose as a Row

The bra ⟨ψ| is the conjugate transpose of |ψ⟩. Mirror interactive: drag ket → bra appears reflected with conjugated components. Most common error: forgetting the * in complex cases.

Outer Products |ψ⟩⟨φ| — Projectors and Gates as Dirac

|ψ⟩⟨φ| is a matrix. Projector |ψ⟩⟨ψ| projects any state onto |ψ⟩. Completeness relation |0⟩⟨0|+|1⟩⟨1|=I verified step by step. Apply |+⟩⟨+| to any draggable state.

Full Dirac Fluency — Switching Between All Three Languages

Translation challenge: given in one language → convert to both others. 5 timed rounds of increasing complexity. 6-question Instinct Check battery. Score shown for motivation.

§4 Mastery Challenge — Dirac in the Wild

6 problems: ket, bra, inner product, outer product, completeness, and — the finale — derive H=|+⟩⟨0|+|−⟩⟨1| fully from scratch. Students who do this are genuinely Dirac fluent. Badge unlocked.

16 min✦ Milestone

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§5 complete — Entanglement Math badge earned.

Tensor products, density matrices, partial traces — the full multi-qubit toolkit.

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§5 · 9 lessons · ~142 min

Multi-Qubit Math + Density Matrices + GF(2)

Tensor products (expanded), separability, density matrices, partial trace, GF(2)

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M35

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Tensor Products — Intuition and the Kronecker Rule

Two Bloch spheres → combine with ⊗ → 4-component column shown with full Kronecker calculation. Each component a₁b₁, a₁b₂, a₂b₁, a₂b₂ highlighted separately. 2×2=4 states visualized as a grid.

Tensor Products — Worked Examples and Entanglement Preview

5 worked examples with increasing complexity: |0⟩⊗|0⟩ up to a state that cannot be factored — shown to be impossible in the final example. Preview card: "Why can't we factor this? M36 explains."

Separable vs Entangled — When Tensor Products Fail

Factorization failure IS entanglement. Try to factor the Bell state → all attempts show contradiction: αγ=1/√2, αδ=0, βγ=0, βδ=1/√2 → impossible. Full algebraic proof, step by step.

Density Matrices — Mixed States and Decoherence in Math

ρ=|ψ⟩⟨ψ| generalises quantum states to include mixtures. Purity slider: pure state (ρ²=ρ) to mixed (Tr(ρ²)<1) → Bloch sphere point moves from surface to interior. Off-diagonals = coherence.

18 min✦ Interactive⚠ T6 error correction

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Partial Trace — What One Qubit Sees of an Entangled Pair

TrB(ρAB) extracts what system A "knows" when you discard B. Bell state → trace over qubit 2 → qubit 1 becomes I/2. Full derivation. Entanglement makes A maximally mixed — shown algebraically.

16 min✦ Interactive⚠ T6 error correction

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M39

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Multi-Qubit Gates — Tensor Products of Operators

Apply H to qubit 1 → H⊗I as 4×4 matrix → apply to |00⟩. Then CNOT as 4×4 from truth table. Final exercise: (H⊗I)·CNOT·|00⟩ = Bell state. The emotional payoff of §5.

GF(2) Arithmetic — the Language of Stabilizer Codes

1+1=0 in GF(2). XOR=GF(2) addition. Build Hamming code parity check matrix. Inject an error → compute syndrome → identify location. Not taught anywhere else at beginner level.

14 min✦ Interactive⚠ T6 stabilizer codes

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M41

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§5 Mastery Challenge — Multi-Qubit and Density Matrix Problems

5 problems: |+⟩⊗|0⟩ Kronecker, separability check, partial trace, density matrix purity, GF(2) syndrome (math-only, no hints). Badge on completion.

18 min✦ Milestone

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§6 complete — State Geometer badge earned.

You can compute fidelity, trace distance, and quantum angles for any two states.

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§6 · 5 lessons · ~75 min · NEW SECTION

Geometry of Quantum States

Bloch sphere math, fidelity, trace distance, angles between states — not in any other beginner course

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M42

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The Bloch Sphere as a Mathematical Object

Every angle and distance on the Bloch sphere has a precise mathematical meaning. Two draggable states → angle between them computed → shown as |⟨φ|ψ⟩|. The factor of 2 (Bloch vs Hilbert space) explained fully.

15 min✦ Interactive⚠ T5 tomography

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M43

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Fidelity — How Close Are Two Quantum States?

F(|ψ⟩,|φ⟩)=|⟨ψ|φ⟩|² measures indistinguishability: F=1 identical, F=0 orthogonal. Two state sliders → fidelity bar updates live with color coding. Properties proved from first principles.

15 min✦ Interactive⚠ T5 tomography, T6 benchmarking

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M44

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Trace Distance — How Distinguishable Are Two States?

T(ρ,σ)=(1/2)||ρ−σ||₁. For qubits: half the Euclidean distance between Bloch vectors. Red line connects two state points labeled distance/2. Relation to fidelity: 1−√F ≤ T ≤ √(1−F) shown.

15 min✦ Interactive⚠ T6 error thresholds

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M45

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Angles Between States and the Geometry of Hilbert Space

A=arccos(|⟨φ|ψ⟩|) satisfies the triangle inequality. Three states on Bloch sphere → A(1,2)+A(2,3) ≥ A(1,3) shown live. Student tries to "break" it and fails. Satisfying interactive moment.

§6 Mastery — Geometry in Practice

4 problems: fidelity from inner product, trace distance from Bloch vectors, angle satisfying triangle inequality, and — math only — compute Tr(ρ²) and T(ρ, I/2) for a given density matrix.

15 min✦ Milestone

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Track 2 complete — Track 3 Ready badge earned.

You now speak the mathematical language of quantum computing fluently. Track 3 awaits.

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§7 · 4 lessons · ~64 min

Probability, Information & Algorithm Math

Classical probability, Shannon + von Neumann entropy, Big-O notation, Track 2 capstone

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M47

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Classical Probability for Quantum Algorithm Analysis

Conditional probability, expected value, and Bayes are the classical backbone of algorithm analysis. Probability tree builder → joint and marginals live. Exercise: Grover expected iterations. Bayesian measurement update.

14 min✦ Interactive⚠ T4 Grover success prob

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M48

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Shannon and von Neumann Entropy

H(X)=−Σp log p measures classical uncertainty. S(ρ)=−Tr(ρ log ρ) is its quantum analog. Probability slider → Shannon entropy curve live. Purity slider → von Neumann entropy. Bell state subsystem has S=1 ebit.

18 min✦ Interactive⚠ T7 quantum Shannon theory

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M49

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Big-O Notation and Complexity — Why Algorithms Are Fast or Slow

n slider → O(1) through O(2ⁿ) curves on screen. Drag algorithm badges to correct curve: Grover→O(√N), Shor→O(n³), classical factoring→O(e^n^(1/3)). The moment Shor and classical diverge is visceral.

16 min✦ Interactive⚠ T4 algorithm complexity

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M50

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Track 2 Capstone — Full Knowledge Map + Instinct Challenge

27-node clickable knowledge map (all §0–§7 concepts). 8-question mixed retrieval. "Quantum Math Instinct" speed round: 10 questions × 10 seconds. Template cheat sheet. Confetti + Track 3 Ready badge.

16 min✦ Capstone

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Curriculum Roadmap

Eight sections, one mathematical language

Track 2 begins with a bridge from Track 1 intuitions to their mathematical symbols, then builds each piece of formalism from scratch. Every section ends with a mastery challenge before you advance. The physics comes in Track 3.

§0 · 3 lessons · ~28 min

Mindset & Bridge

T1→T2 translation. Growth mindset. 8 flip-cards.

0/3

§1 · 10 lessons · ~148 min

Complex Numbers — Quantum's Phase Engine

i as rotation, Euler's formula, interference, probability rule.

0/10

§2 · 8 lessons · ~110 min

Vectors & Early Dirac Exposure

State representation, inner product, Born rule in vector form.

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§3 · 9 lessons · ~136 min

Matrices & Operators

Pauli family, Hadamard, unitarity, 3-step eigenvectors, SVD.

0/9

§4 · 5 lessons · ~73 min

Dirac Notation Mastery

Kets, bras, outer products, translation challenge, Hadamard as Dirac.

0/5

§5 · 9 lessons · ~142 min

Multi-Qubit Math + Density Matrices + GF(2)

Tensor products, entanglement, density matrices, partial trace.

0/9

§6 · 5 lessons · ~75 min · NEW

Geometry of Quantum States

Bloch sphere math, fidelity, trace distance, quantum angles.

0/5

§7 · 4 lessons · ~64 min

Probability, Information & Algorithm Math

Classical probability, entropy, Big-O notation, Track 2 capstone.

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