The Complete Mathematics Reference 2026: Algebra, Geometry, Trigonometry, Calculus, Statistics, and Beyond

The non-negative value of a number without regard to its sign, denoted |x|. Geometrically, the distance from zero on the number line. |5| = 5, |-5| = 5. In complex numbers, |a+bi| = sqrt(a^2 + b^2) gives the modulus. Properties: |xy| = |x||y|, |x+y| <= |x|+|y| (triangle inequality), |x| >= 0 always.

A line that a curve approaches but never reaches. Three types: horizontal (y = L as x -> +/-infinity), vertical (x = a where f(x) -> +/-infinity), and oblique/slant (y = mx + b for rational functions where degree numerator = degree denominator + 1). Example: y = 1/x has vertical asymptote x = 0 and horizontal asymptote y = 0.

A statement accepted as true without proof, serving as a starting point for deductive reasoning. Examples: Euclid axioms for geometry (through any two points there is exactly one line), Peano axioms for natural numbers, ZFC axioms for set theory. Axioms are chosen to be consistent (non-contradictory) and independent (not derivable from each other).

A function that is both injective (one-to-one) and surjective (onto). Every element in the domain maps to a unique element in the codomain, and every element in the codomain is mapped to. Bijections establish a one-to-one correspondence between sets and prove they have the same cardinality. Example: f(x) = 2x+1 from R to R is a bijection.

C(n,k) = n! / (k!(n-k)!), read as "n choose k." Counts the number of ways to choose k items from n items without regard to order. Properties: C(n,0) = C(n,n) = 1, C(n,k) = C(n, n-k), sum of row n in Pascal triangle = 2^n. Central to the binomial theorem, probability, and combinatorics.

A number that indicates quantity (how many). For finite sets, it is a non-negative integer. For infinite sets, Cantor defined: aleph-null (|N| = |Z| = |Q|), the cardinality of the continuum c = |R| = 2^aleph-null. The continuum hypothesis asks whether there is a cardinality between aleph-null and c.

A sequence where terms become arbitrarily close to each other as the sequence progresses. Formally: for every epsilon > 0, there exists N such that |a_m - a_n| < epsilon for all m, n > N. In complete metric spaces (like R), every Cauchy sequence converges. Cauchy sequences are fundamental to real analysis and the construction of the real numbers.

The branch of mathematics studying counting, arrangement, and combination of objects. Key topics: permutations (ordered arrangements: n!), combinations (unordered selections: C(n,k)), pigeonhole principle, inclusion-exclusion, generating functions, graph theory, Ramsey theory. Applications: probability, cryptography, algorithm analysis, optimization.

A number of the form a + bi where a and b are real and i = sqrt(-1). a is the real part, b is the imaginary part. Complex numbers extend the reals and guarantee every polynomial equation has a solution (Fundamental Theorem of Algebra). Operations: addition component-wise, multiplication uses i^2 = -1, division by conjugate. Euler formula: e^(ix) = cos(x) + i*sin(x).

In number theory: a is congruent to b modulo n (a = b mod n) if n divides (a-b). Example: 17 = 2 (mod 5) because 5 divides 15. In geometry: two figures are congruent if they have the same shape and size (can be mapped by rigid motions: translation, rotation, reflection). Triangle congruence: SSS, SAS, ASA, AAS.

A function where small changes in input produce small changes in output. Formally: f is continuous at x=a if lim(x->a) f(x) = f(a). Informally: the graph has no breaks, jumps, or holes. Key theorems: Intermediate Value Theorem (continuous function on [a,b] takes every value between f(a) and f(b)), Extreme Value Theorem (continuous function on closed interval attains its max and min).

A sequence or series approaches a definite value (limit) as terms increase. Sequence convergence: lim(n->inf) a_n = L. Series convergence: sum converges if partial sums converge. Tests: comparison, ratio, root, integral, alternating series. Absolute convergence implies convergence. Rate of convergence: how quickly a sequence approaches its limit (linear, quadratic, exponential).

A statement that follows readily from a theorem that has already been proven. It is a direct consequence requiring little or no additional proof. Example: Corollary of Pythagorean Theorem: the diagonal of a unit square is sqrt(2). Distinguished from a lemma (helper result before a theorem) and a proposition (a proven statement of moderate importance).

A binary operation on two vectors in R^3 producing a vector perpendicular to both: a x b = |a||b|sin(theta) n-hat. Result: a vector (unlike dot product which gives a scalar). Properties: anti-commutative (a x b = -(b x a)), |a x b| = area of parallelogram. Formula: (a2*b3 - a3*b2, a3*b1 - a1*b3, a1*b2 - a2*b1). Used in physics (torque, angular momentum) and computer graphics.

The instantaneous rate of change of a function. f'(x) = lim(h->0) [f(x+h) - f(x)] / h. Geometrically: slope of the tangent line. Physically: velocity is the derivative of position, acceleration is the derivative of velocity. Higher-order derivatives: f'' (second derivative, concavity), f''' (third, jerk). Notation: f'(x), dy/dx, Df, dot notation.

A scalar value computed from a square matrix that encodes important properties. For 2x2: det = ad-bc. Properties: det = 0 means singular (no inverse), det of product = product of dets, det changes sign when rows swap, det scales when row is multiplied. Geometric meaning: signed volume of the parallelepiped spanned by columns/rows. Used in: solving linear systems (Cramer rule), eigenvalues, change of variables in integrals.

An equation involving a function and its derivatives. ODE (ordinary): involves one independent variable. PDE (partial): involves multiple independent variables. Order: highest derivative that appears. Linear vs nonlinear. Solution methods: separation of variables, integrating factors, Laplace transforms, power series, numerical methods (Euler, Runge-Kutta). Applications: physics (Newton law, heat equation, wave equation), biology, economics.

A binary operation on two vectors producing a scalar: a . b = |a||b|cos(theta) = sum(a_i * b_i). Result: a scalar (unlike cross product). Properties: commutative, distributive. a . b = 0 means perpendicular (orthogonal). a . a = |a|^2. Used for: projections, angles between vectors, work (force dot displacement), checking orthogonality.

A scalar lambda such that Av = lambda*v for a nonzero vector v (eigenvector). Found by solving det(A - lambda*I) = 0 (characteristic polynomial). Properties: sum of eigenvalues = trace(A), product = det(A). Diagonalization: A = PDP^(-1) where D is diagonal of eigenvalues. Applications: PCA in data science, Google PageRank, quantum mechanics, vibration analysis, stability analysis.

e^(ix) = cos(x) + i*sin(x). Called the most beautiful equation in mathematics. Special case: e^(i*pi) + 1 = 0 (Euler identity) connects five fundamental constants (e, i, pi, 1, 0). Applications: simplifies trigonometric calculations, signal processing (Fourier transforms), electrical engineering (phasors), quantum mechanics, complex analysis.

n! = 1 * 2 * 3 * ... * n for positive integers. 0! = 1 by convention. Counts the number of permutations of n distinct objects. Stirling approximation: n! ~ sqrt(2*pi*n) * (n/e)^n for large n. The gamma function extends factorials to non-integers: Gamma(n) = (n-1)! for positive integers. Growth: n! grows faster than any exponential (super-exponential growth).

The sequence where each term is the sum of the two preceding terms: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, .... F(n) = F(n-1) + F(n-2) with F(0)=0, F(1)=1. Closed form: Binet formula using the golden ratio phi. Ratio F(n+1)/F(n) -> phi = (1+sqrt(5))/2 ~ 1.618. Appears in: nature (phyllotaxis, pinecones, sunflowers), art, algorithms (Fibonacci heap), number theory.

A relation that assigns each element in the domain exactly one element in the codomain. Notation: f: A -> B, y = f(x). Types: injective (one-to-one), surjective (onto), bijective (both), linear (f(ax+by) = af(x)+bf(y)), continuous, differentiable. Composition: (f o g)(x) = f(g(x)). Inverse: f^(-1) exists iff f is bijective. Fundamental concept in all mathematics.

An algorithm for solving systems of linear equations by transforming the augmented matrix to row echelon form using elementary row operations: swap rows, multiply row by nonzero scalar, add multiple of one row to another. Reduced row echelon form (RREF) gives the solution directly. Complexity: O(n^3). Used for: solving linear systems, finding matrix inverse, computing rank and determinant.

phi = (1 + sqrt(5))/2 ~ 1.6180339887... The unique positive number satisfying phi = 1 + 1/phi, equivalently phi^2 = phi + 1. Geometric property: a rectangle with ratio phi can be divided into a square and a similar rectangle. Appears in: Fibonacci sequence (limit of ratios), regular pentagons, the Parthenon (debated), phyllotaxis in plants, financial markets (Fibonacci retracement).

The study of graphs: mathematical structures consisting of vertices (nodes) and edges (connections). Types: directed/undirected, weighted, bipartite, planar, complete. Key concepts: degree, path, cycle, tree, connectivity, Euler path/circuit, Hamiltonian path/cycle, coloring, planarity. Applications: social networks, navigation (shortest path), scheduling, circuit design, internet routing.

A set G with a binary operation * satisfying: closure (a*b in G), associativity ((a*b)*c = a*(b*c)), identity (exists e: e*a = a*e = a), inverse (for each a exists a^(-1): a*a^(-1) = e). Examples: integers under addition, nonzero rationals under multiplication, symmetries of geometric objects. Abelian group: additionally commutative (a*b = b*a). Foundation of abstract algebra.

A statistical procedure to determine whether sample data provide sufficient evidence to reject a null hypothesis. Steps: (1) state H0 and H1, (2) choose significance level alpha (often 0.05), (3) compute test statistic, (4) find p-value, (5) reject H0 if p < alpha. Types: one-tailed, two-tailed. Common tests: z-test, t-test, chi-squared, F-test, ANOVA. Type I error: false positive (reject true H0). Type II error: false negative (fail to reject false H0).

A number of the form bi where b is a nonzero real number and i = sqrt(-1). i^2 = -1, i^3 = -i, i^4 = 1 (cycles with period 4). Combined with real numbers to form complex numbers a + bi. Originally considered fictional (hence "imaginary"), now indispensable in: AC circuit analysis, quantum mechanics, signal processing, fluid dynamics, control theory.

A mathematical object representing area under a curve, accumulation, or the antiderivative. Definite integral: integral_a^b f(x) dx = signed area between f and x-axis from a to b. Indefinite integral: integral f(x) dx = F(x) + C where F is an antiderivative. Riemann integral: limit of Riemann sums. Techniques: substitution, integration by parts, partial fractions, trigonometric substitution. Applications: area, volume, work, probability, average value.

A real number that cannot be expressed as a ratio of two integers. Decimal expansion is non-terminating and non-repeating. Examples: sqrt(2) (proved irrational by Pythagoreans), pi, e, phi (golden ratio), ln(2). The set of irrationals is uncountable (much larger than the rationals). Algebraic irrationals (roots of polynomials with integer coefficients, like sqrt(2)) vs transcendental (pi, e, which are not roots of any such polynomial).

A bijective mapping between mathematical structures that preserves the structure (operations, relations). If an isomorphism exists between two structures, they are structurally identical (differ only in labeling). Examples: (Z, +) is isomorphic to (2Z, +) via f(n) = 2n; two graphs are isomorphic if there is a bijection between vertices preserving edges. Isomorphism is an equivalence relation.

The matrix of all first-order partial derivatives of a vector-valued function. For f: R^n -> R^m, the Jacobian is the m x n matrix [df_i/dx_j]. The Jacobian determinant measures local scaling of volume under the transformation. Used in: change of variables in multiple integrals, Newton method for systems, inverse function theorem, machine learning (backpropagation).

A method for finding the extrema of a function subject to equality constraints. To optimize f(x,y) subject to g(x,y) = c: solve grad(f) = lambda * grad(g) along with the constraint. lambda is the Lagrange multiplier and represents the rate of change of the optimal value with respect to the constraint. Generalizes to multiple constraints and higher dimensions.

The value a function or sequence approaches as the input approaches a value. lim(x->a) f(x) = L means f(x) can be made arbitrarily close to L by taking x sufficiently close to a. Epsilon-delta definition: for every epsilon > 0, there exists delta > 0 such that |f(x) - L| < epsilon whenever 0 < |x - a| < delta. Foundation of calculus (continuity, derivatives, integrals all defined via limits).

A function T: V -> W between vector spaces satisfying: T(u + v) = T(u) + T(v) and T(cv) = cT(v). Every linear transformation from R^n to R^m can be represented by an m x n matrix. Properties: preserves lines, parallelism, ratios of distances along lines. Examples: rotation, reflection, scaling, shearing, projection. Kernel (null space): vectors mapped to zero. Image (range): set of output vectors.

The inverse of exponentiation: log_b(x) = y means b^y = x. Common logarithm: log_10 (written log). Natural logarithm: log_e (written ln). Properties: log(xy) = log(x) + log(y), log(x/y) = log(x) - log(y), log(x^n) = n*log(x), log_b(1) = 0, log_b(b) = 1. Applications: pH scale, decibels, Richter scale, information theory (bits = log_2), algorithm complexity.

A rectangular array of numbers arranged in rows and columns. An m x n matrix has m rows and n columns. Operations: addition, scalar multiplication, matrix multiplication (row-by-column), transpose (swap rows/columns), inverse (A^(-1) such that AA^(-1) = I), determinant. Special matrices: identity, diagonal, symmetric, orthogonal, triangular, sparse. Foundation of linear algebra with applications everywhere.

Three measures of central tendency. Mean (arithmetic average): sum of values divided by count. Median: middle value when sorted (or average of two middle values for even count). Mode: most frequently occurring value. For symmetric distributions, mean = median = mode. For skewed distributions, they differ. Mean is sensitive to outliers; median is robust. Weighted mean, geometric mean, and harmonic mean are variants.

Arithmetic where numbers wrap around after reaching a modulus. a mod n gives the remainder when a is divided by n. Clock arithmetic is mod 12. Properties: (a+b) mod n = ((a mod n) + (b mod n)) mod n (similarly for multiplication). Applications: cryptography (RSA, Diffie-Hellman), hash functions, error-detecting codes, random number generators, calendar calculations.

The bell-shaped continuous probability distribution defined by mean mu and standard deviation sigma. 68-95-99.7 rule: approximately 68% of data falls within 1 sigma of mean, 95% within 2 sigma, 99.7% within 3 sigma. The Central Limit Theorem states that the sum of many independent random variables approximates a normal distribution regardless of their individual distributions. Foundation of statistical inference.

Perpendicular. Two vectors are orthogonal if their dot product is zero: u . v = 0. Orthogonal matrix: Q such that Q^T = Q^(-1), preserves lengths and angles (rotation/reflection). Orthogonal basis: a set of mutually perpendicular basis vectors. Gram-Schmidt process converts a linearly independent set into an orthogonal (or orthonormal) set. Orthogonality is central to least squares, Fourier analysis, and quantum mechanics.

In hypothesis testing, the probability of obtaining results at least as extreme as the observed results, assuming the null hypothesis is true. Small p-value (< alpha) suggests the data is inconsistent with H0, leading to rejection. Common alpha: 0.05 (5%), 0.01 (1%), 0.001 (0.1%). P-value is NOT the probability that H0 is true (common misconception). A significant p-value does not necessarily imply practical significance.

An arrangement of objects in a specific order. The number of permutations of n objects is n!. The number of permutations of k objects from n is P(n,k) = n! / (n-k)!. With repetition: n^k (each of k positions has n choices). Permutations with identical objects: n! / (n1! * n2! * ... * nk!) where n_i are counts of each identical type. Different from combination (where order does not matter).

An expression consisting of variables and coefficients using only addition, subtraction, multiplication, and non-negative integer exponents. General form: a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0. Degree: highest exponent with nonzero coefficient. Fundamental Theorem of Algebra: degree-n polynomial has exactly n roots (counting multiplicity, in C). Operations: addition, multiplication, division (long or synthetic), factoring.

A natural number greater than 1 whose only positive divisors are 1 and itself: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, .... The Fundamental Theorem of Arithmetic: every integer > 1 has a unique prime factorization. Euclid proved there are infinitely many primes. The Prime Number Theorem: the number of primes up to n is approximately n/ln(n). Open problems: Goldbach conjecture, twin prime conjecture, Riemann hypothesis. Applications: cryptography (RSA).

A measure of the likelihood of an event, ranging from 0 (impossible) to 1 (certain). P(A) = (favorable outcomes) / (total outcomes) for equally likely outcomes. Axioms (Kolmogorov): P(S) = 1 (sample space), P(A) >= 0, P(A union B) = P(A) + P(B) for disjoint events. Conditional probability: P(A|B) = P(A and B)/P(B). Bayes theorem: P(A|B) = P(B|A)*P(A)/P(B). Independence: P(A and B) = P(A)*P(B).

A logical argument establishing the truth of a mathematical statement beyond any doubt. Types: direct proof, proof by contradiction (assume negation, derive contradiction), proof by induction (base case + inductive step), proof by contrapositive, constructive proof (exhibit an example), non-constructive (prove existence without construction). Mathematical proofs are the gold standard of certainty in human knowledge.

A polynomial of degree 2: ax^2 + bx + c where a != 0. Graph is a parabola opening up (a > 0) or down (a < 0). Vertex at x = -b/(2a). Roots found by: factoring, completing the square, quadratic formula. Discriminant D = b^2 - 4ac: D > 0 (two real roots), D = 0 (one repeated root), D < 0 (two complex conjugate roots). Most-studied polynomial type.

A unit of angle measurement: the angle subtended at the center of a circle by an arc equal in length to the radius. Full circle = 2*pi radians = 360 degrees. Conversions: radians = degrees * pi/180, degrees = radians * 180/pi. Key values: pi/6 = 30 degrees, pi/4 = 45 degrees, pi/3 = 60 degrees, pi/2 = 90 degrees. Calculus formulas for trigonometric functions require radians.

A statistical method for modeling the relationship between a dependent variable and one or more independent variables. Linear regression: y = beta_0 + beta_1*x + epsilon, minimizes sum of squared residuals (OLS). Multiple regression: y = beta_0 + beta_1*x_1 + ... + beta_p*x_p. Polynomial regression: includes higher powers of x. R^2 measures goodness of fit. Assumptions: linearity, independence, homoscedasticity, normality of residuals.

The conjecture that all nontrivial zeros of the Riemann zeta function zeta(s) = sum(1/n^s) have real part 1/2. Formulated in 1859, it remains one of the most important unsolved problems in mathematics (Millennium Prize Problem, $1M reward). If true, it would have deep implications for the distribution of prime numbers. Computationally verified for the first 10 trillion zeros but no proof exists.

An algebraic structure with two binary operations (addition and multiplication) satisfying: (R, +) is an abelian group, multiplication is associative, and multiplication distributes over addition. Examples: integers (Z), polynomials (Z[x]), matrices (M_n), modular arithmetic (Z/nZ). A commutative ring with identity and no zero divisors is an integral domain. A field is a commutative ring where every nonzero element has a multiplicative inverse.

A quantity with magnitude only, no direction (as opposed to a vector). Examples: temperature, mass, speed, energy. In linear algebra: elements of the field over which a vector space is defined (typically real or complex numbers). Scalar multiplication: multiplying a vector by a scalar scales its magnitude and possibly reverses its direction (negative scalar).

The sum of terms of a sequence: S = sum_{n=1}^{infinity} a_n. Partial sums S_N = sum_{n=1}^{N} a_n. The series converges if the sequence of partial sums converges. Convergence tests: nth-term test (diverges if a_n does not approach 0), geometric series, p-series, comparison, limit comparison, ratio, root, integral, alternating series. Power series: sum a_n*x^n, converges in a radius of convergence R. Taylor series: represents functions as power series.

A well-defined collection of distinct objects (elements/members). Notation: A = {1, 2, 3}, x in A. Operations: union (A union B), intersection (A intersection B), complement (A^c), difference (A \ B), Cartesian product (A x B). Special sets: empty set, natural numbers N, integers Z, rationals Q, reals R, complex C. Set theory (Cantor, Zermelo-Fraenkel) is the foundation of modern mathematics.

A measure of the spread (dispersion) of a dataset: sigma = sqrt(variance) = sqrt((1/N) * sum(x_i - mu)^2) for population, s = sqrt((1/(n-1)) * sum(x_i - x_bar)^2) for sample. Small sigma means data points are close to the mean; large sigma means spread out. For normal distributions: 68% within 1 sigma, 95% within 2 sigma, 99.7% within 3 sigma.

The least upper bound (lub) of a set S: the smallest number that is greater than or equal to every element of S. Notation: sup(S). The supremum may or may not be in S. Example: sup((0,1)) = 1, but 1 is not in (0,1). The Completeness Axiom of the reals: every nonempty subset of R that is bounded above has a supremum in R. This axiom distinguishes R from Q and is the foundation of real analysis.

A transformation that leaves an object unchanged. Types: reflective (mirror symmetry), rotational (rotation by an angle), translational (shift in space), glide reflection. Symmetry group: the set of all symmetries of an object forms a group. Examples: equilateral triangle has 6 symmetries (D_3), square has 8 (D_4), circle has infinitely many. Noether theorem: every continuous symmetry in physics corresponds to a conservation law.

A representation of a function as an infinite sum of terms calculated from its derivatives at a single point: f(x) = sum_{n=0}^{inf} f^(n)(a)/n! * (x-a)^n. Maclaurin series: Taylor series centered at a = 0. Key examples: e^x = sum x^n/n!, sin(x) = sum (-1)^n x^(2n+1)/(2n+1)!, cos(x) = sum (-1)^n x^(2n)/(2n)!, 1/(1-x) = sum x^n for |x| < 1.

A multidimensional generalization of vectors and matrices. A scalar is a rank-0 tensor, a vector is rank-1, a matrix is rank-2, and higher ranks extend the pattern. Tensors transform according to specific rules under coordinate changes. Applications: general relativity (metric tensor, Riemann curvature tensor), continuum mechanics (stress tensor), machine learning (multi-dimensional arrays in neural networks, TensorFlow).

A mathematical statement that has been rigorously proved from axioms and previously established theorems using logical deduction. Famous theorems: Pythagorean theorem, Fundamental Theorem of Calculus, Fundamental Theorem of Algebra, Prime Number Theorem, Central Limit Theorem, Fermat Last Theorem (proved 1995 by Wiles), Four Color Theorem. A proved conjecture becomes a theorem.

The branch of mathematics studying properties preserved under continuous deformations (stretching, bending, but not tearing or gluing). A topological space is a set with a collection of open sets satisfying axioms. Key concepts: homeomorphism (continuous bijection with continuous inverse), compactness, connectedness, genus (number of holes). Famous result: a coffee mug is topologically equivalent to a donut (both have genus 1).

A function that maps a geometric figure to another figure. Types: rigid (preserves distance): translation, rotation, reflection, glide reflection. Non-rigid: scaling (dilation), shearing, projection, inversion. In linear algebra: linear transformations represented by matrices. In calculus: change of variables. Composition: applying one transformation after another. The set of invertible transformations forms a group.

A measure of how spread out a dataset is from its mean: Var(X) = E[(X - mu)^2] = E[X^2] - (E[X])^2. Population variance: sigma^2 = (1/N)*sum(x_i - mu)^2. Sample variance: s^2 = (1/(n-1))*sum(x_i - x_bar)^2 (Bessel correction). Properties: Var(aX + b) = a^2*Var(X), Var(X + Y) = Var(X) + Var(Y) if independent. Standard deviation is the square root of variance.

A quantity with both magnitude and direction. Representation: as coordinates (a_1, a_2, ..., a_n) in R^n, as arrow from origin. Operations: addition (component-wise), scalar multiplication, dot product (a . b), cross product (a x b, 3D only). Magnitude: ||v|| = sqrt(sum v_i^2). Unit vector: v/||v||. Linear combination: c_1*v_1 + c_2*v_2 + .... Vectors are the fundamental objects of linear algebra.

A set V with two operations (vector addition, scalar multiplication) satisfying 8 axioms: closure under both operations, commutativity and associativity of addition, existence of zero vector and additive inverses, distributive laws, and scalar multiplication identity. Examples: R^n, polynomials of degree at most n, continuous functions on [a,b], matrices. Key concepts: subspace, basis, dimension, span, linear independence.

A standardized score indicating how many standard deviations a data point is from the mean: z = (x - mu) / sigma. z = 0: at the mean. z > 0: above mean. z < 0: below mean. z-scores allow comparison across different scales. In a standard normal distribution (mu=0, sigma=1): z-table gives cumulative probabilities. Example: z = 1.96 corresponds to the 97.5th percentile (used for 95% confidence intervals).

The additive identity: a + 0 = a for any number a. The number that represents nothing, emptiness, or the starting point on the number line. Historical development: originated in India (Brahmagupta, 628 AD), transmitted to Europe via Arabic mathematics. Properties: a * 0 = 0, a/0 is undefined, 0! = 1, 0^0 is conventionally 1 (though debated). Division by zero is the most common mathematical error. Zero is both the smallest non-negative integer and the largest non-positive integer.

A set of philosophical problems posed by Zeno of Elea (5th century BC) about motion and infinity. Most famous: Achilles and the Tortoise (Achilles can never overtake a tortoise with a head start, because he must first reach where the tortoise was, by which time it has moved). Resolved by calculus: the infinite series 1/2 + 1/4 + 1/8 + ... = 1 shows that infinitely many steps can sum to a finite value. Demonstrates the need for rigorous treatment of infinity and limits.

P(A|B) = P(B|A)*P(A) / P(B). Relates conditional probabilities, allowing us to update the probability of a hypothesis given evidence. P(A): prior probability, P(A|B): posterior probability, P(B|A): likelihood, P(B): evidence. Foundation of Bayesian statistics and inference. Applications: spam filters, medical diagnosis, machine learning (Naive Bayes classifier), legal reasoning, A/B testing.

If you take sufficiently large random samples from a population with mean mu and standard deviation sigma, the sampling distribution of the sample mean approaches a normal distribution N(mu, sigma/sqrt(n)) regardless of the population distribution. Formally: (X_bar - mu) / (sigma/sqrt(n)) -> N(0,1) as n -> infinity. Rule of thumb: n >= 30 is usually sufficient. Foundation of statistical inference, confidence intervals, and hypothesis testing.

An interval estimate of a population parameter. A 95% CI means: if we repeated the sampling process many times, 95% of the constructed intervals would contain the true parameter. For the mean: x_bar +/- z*(sigma/sqrt(n)). Common z-values: 1.645 (90% CI), 1.96 (95% CI), 2.576 (99% CI). Wider intervals give more confidence but less precision. NOT the probability that the parameter is in the interval (frequentist interpretation).

A measure of the linear relationship between two variables. Pearson correlation r ranges from -1 (perfect negative) to +1 (perfect positive). r = 0 indicates no linear relationship (but there may be a nonlinear one). r = cov(X,Y) / (sigma_X * sigma_Y). Correlation does NOT imply causation. Spearman rank correlation is a non-parametric alternative. R^2 = r^2 gives the proportion of variance explained.

The long-run average of a random variable. Discrete: E[X] = sum x_i * P(X = x_i). Continuous: E[X] = integral x * f(x) dx. Properties: E[aX + b] = a*E[X] + b, E[X + Y] = E[X] + E[Y] (always, even if not independent), E[XY] = E[X]*E[Y] (only if independent). Also called the mean or expectation. Foundation of decision theory, gambling analysis, insurance pricing.