Math






Study Planet · Mathematics


Foundations

  • Sets & Logic: elements, subsets, unions ∪, intersections ∩, complements; statements, truth tables, implication, equivalence.
  • Proofs: direct, contrapositive, contradiction, induction; counterexamples.
  • Order of operations & properties: PEMDAS; commutative, associative, distributive; identities and inverses.
  • Notations: interval, function, sigma Σ, product Π, factorial n!, absolute value |x|.
Mindset: define precisely, draw a picture, try small cases, look for invariants or symmetry.

Arithmetic

  • Integers & operations; divisibility; GCD/LCM; prime factorization.
  • Fractions, decimals, percents; ratios & rates; unit conversions; estimation.
  • Powers & roots; scientific notation; exponential growth/decay basics.
Mental Math Tips
  • Break numbers (distributive property), use complements to 10/100, and friendly benchmarks.
  • Check reasonableness with magnitude and last‑digit tests.

Algebra

Core tools
  • Expressions, equations, inequalities; factoring; expanding; completing the square.
  • Polynomials: degree, roots/zeros, Remainder & Factor theorems; synthetic division.
  • Systems: substitution, elimination, matrices (intro).
Exponential & Logarithmic
  • Exponential a^x; logarithm log_a(x); change‑of‑base; properties (product, quotient, power).
  • Applications: compound interest, growth/decay, half‑life.
Quadratics
  • Forms: standard, factored, vertex. Vertex by completing the square.
  • Quadratic formula; discriminant; parabolas & transformations.

Functions & Graphing

  • Definition; domain & range; composition; inverses (swap x/y + horizontal line test).
  • Families: linear, quadratic, polynomial, rational, radical, exponential, logarithmic, piecewise.
  • Transformations: shifts, stretches, reflections; asymptotes and end behavior.

Geometry

  • Euclidean basics: points/lines/angles; triangle congruence (SSS, SAS, ASA, AAS, HL).
  • Similarity & right triangles; Pythagorean theorem; special triangles (30‑60‑90, 45‑45‑90).
  • Circles: chords, tangents, arcs; central/inscribed angles.
  • Coordinate geometry: slope, distance, midpoint; conic sections (parabola, circle, ellipse, hyperbola).
  • Area & volume: polygons, solids (prisms, cylinders, pyramids/cones, spheres); surface area.
  • Transformations: translations, rotations, reflections, dilations; symmetry.

Trigonometry

  • Unit circle; radian measure; values for special angles.
  • Right‑triangle trig; Law of Sines & Cosines.
  • Identities: Pythagorean, angle sum/difference, double/half‑angle; trig equations and graphs.
  • Inverse trig; polar coordinates; complex numbers in polar form (De Moivre’s idea).

Calculus

Limits & Derivatives
  • Limit definitions; continuity; derivative as instantaneous rate/slope.
  • Rules: power, product, quotient, chain; derivatives of e^x, ln x, trig/inverse trig.
  • Applications: tangents, velocity/acceleration, optimization, related rates, L’Hôpital’s rule (indeterminate forms).
Integrals
  • Antiderivatives; definite integral as signed area; Fundamental Theorem of Calculus.
  • Techniques: substitution, parts, partial fractions (overview), numerical (trapezoid/Simpson ideas).
  • Applications: area/volume, accumulation, average value, probability density.
Series & Multivariable (brief)
  • Sequences & series; tests for convergence (integral, comparison, ratio); power series & Taylor ideas.
  • Multivariable: partial derivatives, gradients; multiple integrals (concepts), vectors & parameterization.

Probability & Statistics

  • Counting: permutations, combinations, binomial coefficients; inclusion–exclusion (idea).
  • Probability rules: sample spaces, independence, conditional probability, Bayes’ rule.
  • Random variables: discrete vs continuous; expectation E[X], variance Var(X).
  • Distributions: binomial, geometric, Poisson (discrete); uniform, normal, exponential (continuous); CLT idea.
  • Statistics: sampling, bias, confidence intervals, hypothesis tests (conceptual), correlation vs causation.

Discrete Math & Computer Science

  • Logic & proof; recursion and recurrence relations; induction & strong induction.
  • Graphs: vertices/edges, paths, cycles, trees; Eulerian/Hamiltonian ideas; connectivity; shortest paths (concepts).
  • Algorithms & complexity (big‑O intuition); greedy vs dynamic programming (conceptual).
  • Modular arithmetic & cryptography (very high level).

Linear Algebra

  • Vectors & dot product; projections; equations of lines/planes.
  • Matrices: addition, multiplication, inverse (when it exists); determinants.
  • Linear systems: row‑reduction, pivoting, rank; least squares (conceptual).
  • Vector spaces & subspaces; linear independence; basis & dimension.
  • Eigenvalues/eigenvectors; diagonalization (idea); applications (transformations, PCA concept).

Number Theory

  • Divisibility, prime numbers, Euclidean algorithm; GCD as linear combination.
  • Congruences: a ≡ b (mod n); modular inverses; Euler’s φ(n); Fermat’s little theorem.
  • Diophantine equations (linear, Pythagorean triples idea); Chinese Remainder Theorem (concept).

Problem‑Solving Playbook

  • Understand → Plan → Execute → Check. Draw, organize data, try small cases, generalize.
  • Heuristics: invariants, parity, extremal principle, symmetry, Pigeonhole, bounding, construction, reverse thinking.
  • Communication: define variables, show steps, label diagrams, sanity‑check units and order of magnitude.

Formula Sheet

Topic Key Formulas
Algebra Quadratic: x = [-b ± √(b²−4ac)]/(2a);
Binomial: (x+y)^n = Σ C(n,k) x^{n−k} y^k;
Log rules: log_a(xy)=log_a x + log_a y; log_a(x^r)=r log_a x.
Geometry Triangle area: A = ½bh = ½ab sin C; Heron: A=√[s(s−a)(s−b)(s−c)];
Circle: C=2πr, A=πr²;
Distance: d = √[(x₂−x₁)²+(y₂−y₁)²].
Trigonometry Identities: sin²x+cos²x=1; 1+tan²x=sec²x;
Addition: sin(a±b)=sin a cos b ± cos a sin b;
Law of Cosines: c²=a²+b²−2ab cos C.
Calculus d(x^n)/dx = n x^{n−1}; d(e^x)/dx = e^x; d(ln x)/dx = 1/x;
∫x^n dx = x^{n+1}/(n+1)+C (n≠−1);
∫e^x dx = e^x + C;
FTC: \(\int_a^b f'(x)\,dx = f(b)−f(a)\) (conceptual statement).
Probability/Stats Permutations: nP r = n!/(n−r)!; Combinations: nC r = n!/[r!(n−r)!];
Bayes: P(A|B)=P(B|A)P(A)/P(B);
Mean: \(\bar x=\frac{1}{n}\sum x_i\); Var: \(\frac{1}{n}\sum (x_i-\bar x)^2\).
Linear Algebra Matrix multiplication (AB)_{ij} = Σ_k A_{ik} B_{kj};
Det 2×2: |a b; c d| = ad − bc.
Number Theory Euclid: gcd(a,b)=gcd(b, a mod b);
Fermat: a^{p−1}≡1 (mod p) for prime p ∤ a.

Tip: memorize structures, not just formulas — know when and why to use each.

Timeline of Mathematics (Select Milestones)

  • Prehistory: counting, tally marks, early geometry in surveying and building.
  • Ancient: Egyptian/ Babylonian arithmetic & geometry; Greek axiomatic geometry; Indian place‑value & zero ideas; Chinese algorithms; Mayan calendars.
  • Medieval: algebra named and advanced; trigonometry systematized; transmission through Islamic, Indian, Chinese, and European scholars.
  • Early Modern: analytic geometry; calculus (Newton/Leibniz); probability foundations.
  • 19th Century: rigorous analysis; non‑Euclidean geometries; group theory; linear algebra formalization.
  • 20th–21st: topology, abstract algebra expansion, measure probability, computing & algorithms, statistics, optimization, modern data science.

Glossary

  • Function: Rule pairing each input with exactly one output.
  • Derivative: Instantaneous rate of change; slope of a curve at a point.
  • Integral: Accumulated quantity; signed area under a curve.
  • Vector: Quantity with magnitude and direction.
  • Matrix: Rectangular array representing linear maps or systems.
  • Eigenvalue/Eigenvector: Scalar λ and vector v with Av=λv for matrix A.
  • Algorithm: Step‑by‑step procedure for computation.
  • Combinatorics: Counting structures and arrangements.
  • Congruence (mod n): a and b leave the same remainder on division by n.
  • Random variable: Numerical outcome of a random process.