Mathematics RU

Mathematics

Glossary

The terms of the course with short definitions, each linked to the chapter where it first appears.

$\ell^2$
The space of sequences (x₁, x₂, …) with finite sum of squares Σ xₙ²; the inner product is ⟨x, y⟩ = Σ xₙyₙ. Chapter 55 · Functional analysis
$L^2$
The space of measurable functions with finite Lebesgue integral ∫|f|²; functions equal almost everywhere are regarded as one point; ⟨f, g⟩ = ∫ f g. Chapter 55 · Functional analysis
$n$th roots of unity
A solution of zⁿ = 1; there are n such numbers, and they are the vertices of a regular n-gon on the unit circle. Chapter 15 · Complex numbers
A proof assistant
A program that accepts a proof written in a formal language only if each of its steps follows by the rules of logic from the axioms and from statements already checked (Lean, Coq, Isabelle). Chapter 60 · The frontier
abelian
A group in which the operation is commutative: ab = ba for any elements a and b. Chapter 40 · Groups
Absolute geometry
The geometry based on all of Euclid’s axioms (in rigorous form, Hilbert’s) except the axiom of parallels; its theorems hold both for Euclid and for Lobachevsky. Chapter 58 · Non-Euclidean geometries
absolute value
The distance from a number to zero on the number line: |a| = a for a ≥ 0 and |a| = −a for a < 0. Chapter 2 · Zero and minus
absolutely convergent
A series Σaₙ converges absolutely if the series of absolute values Σ|aₙ| converges; such a series converges, and rearranging its terms doesn’t change its sum. Chapter 30 · Series
algebraic multiplicity
The multiplicity of an eigenvalue λ as a root of the characteristic polynomial. Chapter 38 · Eigenvectors
algebraic number
A root of an equation aₙxⁿ + … + a₁x + a₀ = 0 with integer coefficients, not all zero. Chapter 6 · Root two and the real numbers
almost everywhere
At all points except a set of measure zero. For example, the Dirichlet function is zero almost everywhere. Chapter 54 · Measure and the Lebesgue integral
alternating group
The group of all even permutations of n objects; written A_n, for n ≥ 2 it has n!/2 elements. Chapter 40 · Groups
altitude
The perpendicular dropped from a vertex of a triangle to the line containing the opposite side. Chapter 17 · The triangle
analytic
A function that in a neighbourhood of each point of its region equals the sum of its Taylor series; for functions of a complex variable this is the same as holomorphic. Chapter 34 · Complex analysis
analytic continuation
A holomorphic function on a larger region that coincides with the given function on a smaller one; by the uniqueness theorem, if a continuation exists, it is unique. Chapter 34 · Complex analysis
angle bisector of a triangle
The segment of the bisector of an angle of a triangle from the vertex to the opposite side. Chapter 17 · The triangle
angle of parallelism
The angle Π(d) between the perpendicular dropped from a point to a line and the limiting parallel through that point; it depends only on the length d of the perpendicular: tan(Π/2) = e^(−d). Chapter 58 · Non-Euclidean geometries
antiderivative
A function F whose derivative on a given interval equals f: F′(x) = f(x) at every point of the interval. Chapter 28 · The integral
area
A number assigned to a figure so that it is non-negative, the same for congruent figures, additive when the figure is cut into parts, and equal to 1 for the square of side 1. Chapter 20 · Area and volume
argument
The angle φ between the positive direction of the real axis and the ray from zero to the point z ≠ 0; defined up to a whole number of turns. Chapter 15 · Complex numbers
arithmetic progression
A sequence in which each term from the second on equals the previous one plus the same number d, the common difference. Chapter 13 · Sequences and induction
arrangement
An ordered choice of k different objects out of n; their number, A from n by k, is n!/(n − k)!. Chapter 45 · Combinatorics
attractor
A set in the space of states that all paths starting nearby approach over time; a path that reaches it doesn’t leave it. Chapter 59 · Chaos and fractals
augmented matrix
The matrix of coefficients of a system of linear equations with the column of right-hand sides appended on the right; written (A | b). Chapter 36 · Gaussian elimination
average value
The number (1/(b − a))·∫ₐᵇ f(x)dx: the height of the rectangle on [a, b] whose area equals the area under the graph of f. Chapter 28 · The integral
axiom
A statement accepted without proof in order to derive all the other statements of a theory from it. Chapter 16 · The Elements: axioms and constructions
ball
The solid consisting of all points of space at distance at most R from the centre; its boundary is a sphere. Chapter 20 · Area and volume
Banach space
A complete normed space. Chapter 55 · Functional analysis
base of the numeral system
An integer b ≥ 2 in a positional numeral system: each place is worth b times the place to its right, and the system has exactly b digits, from 0 to b − 1. Chapter 1 · Counting: from notches to bits
basic reproduction number
R₀ = β/γ: how many people one infected person infects on average in a fully susceptible population. Chapter 31 · Differential equations
basis
A linearly independent set of vectors whose span is the whole space; every vector can be expanded in a basis in exactly one way. Chapter 37 · Vector spaces
Bernoulli trials
A series of independent trials with two outcomes (‘success’ with probability p, ‘failure’ with probability 1 − p), the same for all trials. Chapter 48 · Random variables
bifurcation
A qualitative change in a system’s behaviour as a parameter changes smoothly: for example, a fixed point loses stability and a cycle of double the period is born. Chapter 59 · Chaos and fractals
bijection
A map that’s both an injection and a surjection: a one-to-one correspondence in which each element of B comes from exactly one element of A. Chapter 51 · Logic and sets
binomial coefficient
The number of ways to choose k objects out of n without regard to order: n!/(k!(n − k)!); written (n over k), or C from n by k in some books. Chapter 45 · Combinatorics
binomial distribution
The distribution of the number of successes in n Bernoulli trials: P(X = k) = C(n, k)·p^k·(1 − p)^(n − k), k = 0, …, n. Chapter 48 · Random variables
bipartite
A graph whose vertices split into two groups so that edges join only vertices from different groups. Chapter 46 · Graphs
bisector of an angle
The ray from the vertex of an angle that divides it into two equal angles. Chapter 16 · The Elements: axioms and constructions
bit
A binary place: a single digit 0 or 1; the smallest unit of information. Chapter 1 · Counting: from notches to bits
bounded
A linear operator T on a normed space with ‖Tx‖ ≤ C‖x‖ for one constant C; the smallest such C is the norm of the operator ‖T‖. Chapter 55 · Functional analysis
box-counting dimension
D = lim ln N(ε) / ln(1/ε), where N(ε) is the number of boxes of a grid of side ε that the set touches; 1 for a segment, 2 for a square, ln 4/ln 3 for the Koch curve. Chapter 59 · Chaos and fractals
busy beaver
BB(n): the largest number of steps made before halting by a Turing machine with n states and the symbols 0, 1, started on a blank tape; the function was introduced by Tibor Radó (1962). Chapter 56 · Gödel, Turing and the limits of proof
byte
A group of eight bits; a byte holds one of 256 values, from 0 to 255. Chapter 1 · Counting: from notches to bits
Bézout’s identity
The equation ax + by = gcd(a, b) with integers x and y; such x and y always exist and are found by the extended Euclidean algorithm. Chapter 4 · Divisibility and Euclid's algorithm
Cantor set
The set of points of the segment [0, 1] that remain when the middle third is removed from each segment infinitely many times; the points whose ternary expansion can avoid the digit 1. Chapter 52 · Infinities
cardinality
The cardinality of a set is what all sets equinumerous with it have in common; for a finite set it’s the number of its elements. Written |A|. Chapter 52 · Infinities
Carmichael numbers
A composite number n for which a^(n−1) ≡ 1 (mod n) for all a coprime to n; the smallest is 561. Chapter 41 · Modular arithmetic and ciphers
Cartesian product
The set A × B of all ordered pairs (a, b) with a ∈ A, b ∈ B. Chapter 51 · Logic and sets
Catalan numbers
The sequence 1, 1, 2, 5, 14, 42, 132, …: the number of ways to bracket a product of n + 1 factors, the number of ways to cut an (n + 2)-gon into triangles, and much else. Chapter 45 · Combinatorics
Cauchy sequence
A sequence whose terms are arbitrarily close to one another for large indices: for every ε > 0 there’s an N such that |aₘ − aₙ| < ε for all m, n > N. Chapter 53 · Rigorous analysis
Cayley table
A table in which the intersection of row a and column b holds the result of the operation a ∘ b; for a finite set it defines the operation completely. Chapter 40 · Groups
central angle
An angle with its vertex at the centre of a circle; its size is taken as the degree measure of the arc it stands on. Chapter 19 · The circle and the number π
centroid
The point where the medians of a triangle meet; it divides each median in the ratio 2 : 1 from the vertex and is the centre of mass of a uniform triangular plate. Chapter 17 · The triangle
change-of-basis matrix
The matrix whose columns are the coordinates of the new basis vectors in the old basis; it converts the new coordinates of a vector into the old ones: x = Px′. Chapter 37 · Vector spaces
characteristic equation
For the equation x″ + px′ + qx = 0, the quadratic equation λ² + pλ + q = 0; its roots give solutions of the form e^(λt). Chapter 31 · Differential equations
characteristic of a field
The smallest natural number p for which 1 + 1 + … + 1 (p ones) = 0 in the field; if there is none, the characteristic is zero. The characteristic of a finite field is a prime. Chapter 42 · Rings, fields and codes
characteristic polynomial
The polynomial det(A − λI) in the variable λ; its roots are the eigenvalues of the matrix A. Chapter 38 · Eigenvectors
Chebyshev function
The function ψ(x): the sum of ln p over all prime powers pᵏ ≤ x; the prime number theorem is equivalent to ψ(x)/x → 1. Chapter 44 · The zeta function and elliptic curves
chromatic number
The smallest number of colours with which the vertices of a graph can be coloured so that the ends of each edge have different colours. Chapter 46 · Graphs
circle
The set of all points of the plane at the same distance (the radius) from a given point (the centre). Chapter 19 · The circle and the number π
circumcircle
The circle through all three vertices of a triangle; its centre is where the perpendicular bisectors of the sides meet. Chapter 17 · The triangle
closed
A set that contains the limits of all its convergent sequences; for example, a segment, a sphere, a closed ball. Chapter 53 · Rigorous analysis
coefficient
The numerical factor in front of the letter part of a term: in 3x² the coefficient is 3. Chapter 7 · Letters for numbers
Collinear vectors
Vectors lying on one line or on parallel lines; the zero vector is collinear with any vector. Chapter 22 · Vectors and coordinates
combination
An unordered choice of k objects out of n (a subset of k elements); their number is the binomial coefficient n choose k. Chapter 45 · Combinatorics
common logarithm
The logarithm to base 10: lg b = log₁₀ b. Chapter 12 · Powers and logarithms
commutator
The element x y x⁻¹ y⁻¹ of a group; it equals e if and only if x and y commute. Chapter 40 · Groups
compact
A set in which every sequence of points has a subsequence converging to a point of the same set; in ℝⁿ these are the bounded closed sets. Chapter 55 · Functional analysis
compass-and-straightedge construction
A construction in which the only moves are drawing a line through two points already constructed and a circle with a constructed centre through a constructed point; new points are the intersections. Chapter 16 · The Elements: axioms and constructions
complement
The complement of a set A in an enclosing set U: the set Aᶜ = U \ A of all elements of U not lying in A. Chapter 51 · Logic and sets
complete
The graph K_n with n vertices in which every two vertices are joined by an edge. Chapter 46 · Graphs
complete
An orthonormal system in which every vector expands in a series Σ cₖeₖ converging to it in norm; equivalently, Parseval’s identity holds for all vectors. Chapter 55 · Functional analysis
complete
A space in which every Cauchy sequence (‖xₘ − xₙ‖ → 0) converges to an element of the same space. Chapter 55 · Functional analysis
complete
A formal system in which, for each statement of its language, either the statement or its negation can be derived. Chapter 56 · Gödel, Turing and the limits of proof
Completing the square
Writing x² + bx as (x + b/2)² − (b/2)²: the expression becomes the square of a sum minus a number. Chapter 10 · Quadratic equations
complex number
A number of the form a + bi, where a and b are real numbers and i is the imaginary unit; a is the real part, b the imaginary part. Chapter 15 · Complex numbers
complex plane
The plane on which the number a + bi is shown as the point with coordinates (a, b); the horizontal axis is the real axis, the vertical one the imaginary axis. Chapter 15 · Complex numbers
composite
A natural number greater than 1 that has a divisor other than 1 and itself. Chapter 3 · Prime numbers
composition
The function f(g(x)): first g is applied to x, then f to the result. Written f ∘ g. Chapter 9 · Functions and graphs
condition number
The number κ(A) = ‖A‖·‖A⁻¹‖; it shows by what factor the relative error of the right-hand side can grow in the relative error of the solution of Ax = b. Chapter 36 · Gaussian elimination
conditional probability
The probability of an event A given that the event B has occurred (P(B) > 0): P(A | B) = P(A ∩ B) / P(B). Chapter 47 · Probability
conditionally convergent
A series converges conditionally if it converges but the series of the absolute values of its terms diverges; by rearranging the terms its sum can be made anything. Chapter 30 · Series
confidence interval
An interval computed from a sample by a rule that covers the unknown parameter with a given probability (for example 95 %); the probability refers to the rule, not to one computed interval. Chapter 49 · Statistics
conformal
A map that preserves the angles between curves together with their orientation; a holomorphic function with non-zero derivative is conformal. Chapter 34 · Complex analysis
confounder
A hidden variable that affects both the supposed cause and the effect and creates a correlation between them without a direct causal link. Chapter 49 · Statistics
congruent modulo
Numbers a and b are congruent modulo n if n divides the difference a − b, that is, a and b leave the same remainder on division by n; written a ≡ b (mod n). Chapter 41 · Modular arithmetic and ciphers
Congruent triangles
Triangles that can be made to coincide by superposition; all their corresponding sides and angles are equal. Chapter 17 · The triangle
conic section
The curve along which a plane not passing through the apex cuts a double circular cone: an ellipse (including a circle), a parabola or a hyperbola. Chapter 24 · Conic sections
conjecture
A statement that has been neither proved nor disproved yet. Chapter 0 · What mathematics is
conjugate
For z = a + bi, the number z̄ = a − bi: the same real part, the imaginary part with the opposite sign. Chapter 15 · Complex numbers
connected graph
A graph in which you can get from any vertex to any other along edges. Chapter 46 · Graphs
connected sum
The surface M # N: a disc is cut out of each of the surfaces M and N, and the edges of the resulting holes are glued together. Chapter 57 · Topology
consistent
A system of equations that has at least one solution; a system with no solutions is inconsistent. Chapter 36 · Gaussian elimination
consistent
A formal system in which a statement and its negation can’t both be derived. Chapter 56 · Gödel, Turing and the limits of proof
constrained extremum
An extremum of a function f among the points satisfying an extra condition, for example g(x, y) = c. Chapter 32 · Functions of several variables
constructible
A real number a for which a segment of length |a| can be constructed with compass and straightedge, starting from a segment of length 1. Chapter 43 · Galois theory
continued fraction
Writing a number as a₀ + 1/(a₁ + 1/(a₂ + …)) with integer a₀ and natural a₁, a₂, …; for short [a₀; a₁, a₂, …]. Chapter 6 · Root two and the real numbers
continuous
A function defined at a point a and near it whose limit at a equals its value f(a); a function is continuous on an interval if it is continuous at each of its points. Chapter 25 · Limits
continuum
The cardinality of the set of real numbers (and of any set equinumerous with it: an interval, a segment, the plane); written 𝔠 and equal to 2^ℵ₀. Chapter 52 · Infinities
continuum hypothesis
The statement that there’s no set whose cardinality is strictly greater than countable and strictly less than the continuum; it’s neither provable nor refutable in ZFC (Gödel, 1940; Cohen, 1963). Chapter 52 · Infinities
contour integral
The limit of the sums Σ f(zₖ)(zₖ₊₁ − zₖ) over points of a curve γ as the steps tend to zero; written ∫_γ f(z) dz, and ∮ for a closed curve. Chapter 34 · Complex analysis
contraction
A map T of a metric space to itself with d(Tx, Ty) ≤ q·d(x, y) for one number q < 1 and all x, y. Chapter 55 · Functional analysis
convergent
The fraction obtained by cutting off a number’s continued fraction at the k-th term: p_k/q_k. Chapter 6 · Root two and the real numbers
convergent
A sequence that has a finite limit; a sequence without a limit is called divergent. Chapter 25 · Limits
convergent
A series whose partial sums have a finite limit; this limit is called the sum of the series. Chapter 30 · Series
converges pointwise
Convergence of a sequence of functions fₙ to f on a set D in which at each point x ∈ D the numbers fₙ(x) tend to f(x); the index N may depend on the point. Chapter 53 · Rigorous analysis
converges uniformly
Convergence of fₙ to f on a set D in which sup |fₙ(x) − f(x)| over x ∈ D tends to zero: one index N works for all points at once. Chapter 53 · Rigorous analysis
converse
A statement in which the hypothesis and conclusion of a given theorem are swapped; it may be true or false. Chapter 18 · Pythagoras' theorem
convex
A function is convex on an interval if its graph lies on or below each of its chords there (like a cup); concave if it lies on or above them (like a cap). Chapter 27 · Extrema and optimization
coordinates of a vector
The numbers x and y in the decomposition a = x·i + y·j along the unit vectors of the coordinate axes. Chapter 22 · Vectors and coordinates
coprime
Numbers with no common divisor other than 1, that is, gcd(a, b) = 1. Chapter 4 · Divisibility and Euclid's algorithm
correlation coefficient
A number r from −1 to 1 showing how closely the points (xᵢ, yᵢ) hug a straight line: r = Σ(xᵢ − x̄)(yᵢ − ȳ) / √(Σ(xᵢ − x̄)² · Σ(yᵢ − ȳ)²). Chapter 49 · Statistics
coset
The set gH = {gh : h ∈ H}: everything obtained by multiplying the element g by the elements of the subgroup H. Chapter 40 · Groups
cosine
In a right triangle, the ratio of the adjacent leg to the hypotenuse; for any angle, the x-coordinate of the point of the unit circle rotated by that angle. Chapter 21 · Trigonometry
cotangent
The ratio of the cosine of an angle to its sine, that is, 1/tan α. Chapter 21 · Trigonometry
countable
A set equinumerous with the natural numbers: its elements can be numbered 1, 2, 3, … so that each gets exactly one number. An infinite set that can’t be numbered like this is called uncountable. Chapter 52 · Infinities
counterexample
A case in which a general statement fails; a single counterexample is enough to disprove the statement. Chapter 0 · What mathematics is
covariance matrix
For a cloud of n points with its centre of mass at the origin, the symmetric matrix C = XᵀX / n (sometimes divided by n − 1); the diagonal holds the variances of the features, the off-diagonal entries the covariances. Chapter 39 · Orthogonality, least squares and SVD
critical points
An interior point of the domain at which the derivative of a function is zero or does not exist; only at such points can there be a local extremum. Chapter 27 · Extrema and optimization
critical strip
The strip 0 ≤ Re s ≤ 1 of the complex plane, in which all the non-trivial zeros of the zeta function lie. Chapter 44 · The zeta function and elliptic curves
cross product
The vector a × b, perpendicular to a and b, of length |a|·|b|·sin φ (the area of the parallelogram on a and b), pointing by the right-hand rule. Chapter 22 · Vectors and coordinates
cycle
A permutation that takes a₁ to a₂, a₂ to a₃, …, a_k to a₁ and leaves the other elements in place; written (a₁ a₂ … a_k), where k is the length of the cycle. Chapter 40 · Groups
cycle of period
Points x₀, x₁, …, x_{p−1} that the map carries into one another in a circle: f^p(x₀) = x₀, and there’s no smaller such p. Chapter 59 · Chaos and fractals
cyclic
A group all of whose elements are powers of one element, its generator; a finite cyclic group of n elements is built like the clock face Z_n. Chapter 40 · Groups
decimal fraction
A fraction with denominator 10, 100, 1000, …, written in places after the decimal point: 2.375 = 2375/1000. Chapter 5 · Fractions
decreases
A function decreases on an interval if a larger value of the argument in the interval gives a smaller value of the function. Chapter 9 · Functions and graphs
Dedekind cut
A set of fractions A that isn’t empty, isn’t all of ℚ, contains all smaller fractions along with each of its fractions, and has no largest element; this is how Dedekind defined a real number. Chapter 53 · Rigorous analysis
defect
The difference π − (α + β + γ) between the angle sum of a plane triangle and that of a triangle in the Lobachevsky plane; for curvature −1 it equals the triangle’s area. Chapter 58 · Non-Euclidean geometries
definite integral
The limit of the Riemann sums of a function f on an interval [a, b] as the partition gets finer; written ∫ₐᵇ f(x)dx, it equals the signed area under the graph. Chapter 28 · The integral
degree of a vertex
The number of edges leaving a vertex (a loop is counted twice). Chapter 46 · Graphs
degree of the extension
The dimension of the field L as a vector space over its subfield K: the largest number of elements of L linearly independent over K. Written [L : K]. Chapter 43 · Galois theory
degree of the polynomial
The largest exponent of x that has a non-zero coefficient in the polynomial. Chapter 14 · Polynomials
derangement
A permutation in which no object stays in its own place; the number of such permutations of n objects is written D_n. Chapter 45 · Combinatorics
derivative
The limit of the ratio of the increment of a function to the increment of its argument: f′(x) = lim (f(x + h) − f(x))/h as h → 0; the slope of the tangent to the graph. Chapter 26 · The derivative
determinant
A number assigned to a square matrix; for a 2 × 2 matrix it’s ad − bc, the signed area of the parallelogram on its columns and the factor by which the transformation multiplies areas. Chapter 35 · Matrices as transformations
deterministic chaos
The behaviour of a system with an exact, non-random rule in which the motion stays bounded while small differences in initial conditions grow exponentially: the Lyapunov exponent is positive. Chapter 59 · Chaos and fractals
diagonalisable
A square matrix that has a basis of eigenvectors; then A = PDP⁻¹ with P invertible and D diagonal. Chapter 38 · Eigenvectors
differentiable
A function that has a finite derivative at a point (or at every point of an interval). Chapter 26 · The derivative
differentiable
A function f(x, y) is differentiable at (a, b) if f(a + h, b + k) = f(a, b) + Ah + Bk + ε·√(h² + k²), where ε → 0 as (h, k) → (0, 0); then A = fₓ(a, b), B = f_y(a, b). Chapter 32 · Functions of several variables
differential equation
An equation in which the unknown is a function and its derivatives appear; the highest order of derivative is the order of the equation. Chapter 31 · Differential equations
digits
One of the signs from which the notation of a number is made up. The decimal system has ten digits, 0 to 9. Chapter 1 · Counting: from notches to bits
dihedral group
The group of symmetries of a regular n-gon: n rotations and n reflections, 2n elements in all; written D_n. Chapter 40 · Groups
dimension
The number of vectors in any basis of a vector space; if there is no finite basis, the space is infinite-dimensional. Chapter 37 · Vector spaces
Diophantine equations
An equation with integer coefficients whose solutions are sought in integers, such as ax + by = c. Chapter 4 · Divisibility and Euclid's algorithm
direction field
A picture made of short segments: at each point (t, y), a segment with slope f(t, y) from the equation y′ = f(t, y); solutions run along the segments. Chapter 31 · Differential equations
directional derivative
The rate of change of a function when moving from the point P in the direction of a unit vector u: the limit of (f(P + tu) − f(P))/t as t → 0. Chapter 32 · Functions of several variables
directrix
A line associated with a conic section: the ratio of the distances from any point of the curve to the focus and to the directrix is the same and equals the eccentricity. Chapter 24 · Conic sections
Dirichlet function
The function equal to 1 at rational points and 0 at irrational ones; discontinuous at every point and not Riemann integrable. Chapter 53 · Rigorous analysis
Dirichlet kernel
The function D_N(t) = 1/2 + cos t + … + cos Nt = sin((N + 1/2)t) / (2 sin(t/2)); a partial sum of a Fourier series is the integral of the signal with this weight. Chapter 33 · Fourier series
disc
The part of the plane bounded by a circle, together with the circle itself. Chapter 19 · The circle and the number π
discrete Fourier transform
The transformation of N numbers x₀, …, x_{N−1} into N numbers X_k = Σ xₙ e^{−2πikn/N}; the discrete counterpart of the Fourier coefficients. Chapter 33 · Fourier series
discrete logarithm
The exponent x for which g^x ≡ A (mod p); no fast way of finding it is known for large p. Chapter 41 · Modular arithmetic and ciphers
discriminant
The number D = b² − 4ac for the equation ax² + bx + c = 0; its sign shows how many real roots the equation has. Chapter 10 · Quadratic equations
distribution
For a discrete random variable, the list of its values x_i with probabilities p_i = P(X = x_i), p_i ≥ 0, the p_i adding up to 1. Chapter 48 · Random variables
distributive law
The law a(b + c) = ab + ac: multiplying a number by a sum is the same as multiplying by each term and adding. Chapter 2 · Zero and minus
divergent
A series whose sequence of partial sums has no finite limit. Chapter 30 · Series
divide with remainder
Writing an integer a as a = bq + r, where b is a natural number, q an integer and 0 ≤ r < b; r is called the remainder. Chapter 4 · Divisibility and Euclid's algorithm
divisor
An integer d is a divisor of n if n = d · k for some integer k; we write d | n. Chapter 3 · Prime numbers
domain
The set of all values of the argument for which a function is defined. Chapter 9 · Functions and graphs
Domain colouring
A way of picturing a function f: ℂ → ℂ: each point z of the plane is painted the colour of the number f(z), with the hue giving the argument of f(z) and the brightness its modulus. Chapter 34 · Complex analysis
dot product
The number a·b = |a|·|b|·cos φ, where φ is the angle between the vectors; in coordinates a₁b₁ + a₂b₂ (+ a₃b₃). Chapter 22 · Vectors and coordinates
double integral
The limit of the sums Σ f(xᵢ, yⱼ)·ΔxΔy over ever finer partitions of a region into cells; for f ≥ 0, the volume under the surface z = f(x, y). Chapter 32 · Functions of several variables
eccentricity
The number e that shows how far a conic section differs from a circle: e = c/a for an ellipse and a hyperbola; a circle has e = 0, an ellipse 0 < e < 1, a parabola e = 1, a hyperbola e > 1. Chapter 24 · Conic sections
edges
A line of a graph joining two vertices; in problems, a bridge, a road, an acquaintance, a border. Chapter 46 · Graphs
Egyptian fraction
Writing a number as a sum of distinct shares of the form 1/n, such as 3/4 = 1/2 + 1/4. Chapter 5 · Fractions
eigenspace
All the eigenvectors of a matrix A with a given eigenvalue λ, together with the zero vector; it is the kernel of the matrix A − λI. Chapter 38 · Eigenvectors
eigenvalue
A number λ for which Av = λv for some non-zero vector v; a root of the characteristic polynomial. Chapter 38 · Eigenvectors
eigenvector
A non-zero vector v that a matrix takes to a vector collinear with it: Av = λv for some number λ. Chapter 38 · Eigenvectors
elementary row operations
Three moves on the rows of a matrix: swap two rows, multiply a row by a non-zero number, add to a row another row multiplied by a number. Chapter 36 · Gaussian elimination
ellipse
The set of points of the plane whose distances to two given points (the foci) have a constant sum greater than the distance between the foci. Chapter 24 · Conic sections
elliptic curve
The curve y² = x³ + ax + b with 4a³ + 27b² ≠ 0, together with a point at infinity O; its points form a commutative group under the chord-and-tangent addition. Chapter 44 · The zeta function and elliptic curves
empty
The set with no elements at all; written ∅. It’s unique and is a subset of every set. Chapter 51 · Logic and sets
entire
A function of a complex variable that is holomorphic on the whole plane: polynomials, eᶻ, sin z, cos z. Chapter 34 · Complex analysis
entropy
The average amount of information in one outcome: H = −Σ pᵢ log₂ pᵢ bits; a measure of the uncertainty of a distribution and the limit of compression. Chapter 50 · Markov chains and information
equal in area
Figures whose areas are equal. Chapter 20 · Area and volume
equation
An equality with an unknown, about which we ask for which values of the unknown it is true. Chapter 8 · Equations: the art of balance
equidecomposable
Two polygons such that one can be cut into finitely many polygons that can be reassembled into the other. Chapter 20 · Area and volume
equilibrium point
A state in which the right-hand sides of all the equations of a system are zero: a solution that starts there stays there forever. Chapter 31 · Differential equations
equinumerous
Sets A and B are equinumerous if there’s a bijection, a one-to-one correspondence, between them. Chapter 52 · Infinities
equivalence class
The set [a] of all elements equivalent to a given element a. Chapter 51 · Logic and sets
equivalence relation
A relation on a set that’s reflexive (a ∼ a), symmetric (a ∼ b implies b ∼ a) and transitive (a ∼ b and b ∼ c imply a ∼ c). Chapter 51 · Logic and sets
equivalent
Equations that have the same set of roots. Chapter 8 · Equations: the art of balance
equivalent
Two logical formulas are equivalent if their truth tables coincide: for any values of the variables they’re true or false together. Written F ≡ G. Chapter 51 · Logic and sets
error-correcting code
A rule for adding check symbols to a message so that the original can be recovered from a message corrupted in transit. Chapter 42 · Rings, fields and codes
Euclid’s algorithm
A way to find the gcd of two numbers: replace the pair (a, b) by (b, r), where r is the remainder of a divided by b, until the remainder is zero; the last non-zero remainder is the gcd. Chapter 4 · Divisibility and Euclid's algorithm
Euler characteristic
The number χ = V − E + F for a division of a surface into polygons; it doesn’t depend on the division: 2 for the sphere, 0 for the torus, 2 − 2g for a sphere with g handles. Chapter 57 · Topology
Euler line
The line containing the circumcentre, the centroid and the orthocentre of a triangle (Euler, 1765). Chapter 17 · The triangle
Eulerian path
A route in a graph that traverses each edge exactly once; if it returns to the start, it’s called an Eulerian circuit. Chapter 46 · Graphs
Euler’s method
An approximate solution of y′ = f(t, y) in steps of length h, each along the tangent: yₙ₊₁ = yₙ + h·f(tₙ, yₙ). Chapter 31 · Differential equations
Euler’s totient function
φ(n) is the number of integers from 1 to n that are coprime to n; for example, φ(10) = 4, and φ(p) = p − 1 for a prime p. Chapter 41 · Modular arithmetic and ciphers
even
A function with f(−x) = f(x) for all x in the domain; its graph is symmetric about the y-axis. Chapter 9 · Functions and graphs
even
A permutation is even if it has an even number of inversions and odd if it has an odd number; its sign sgn σ is +1 or −1 accordingly. Chapter 40 · Groups
event
Any set of outcomes of an experiment (a subset of the sample space Ω); the event has occurred if one of its outcomes came up. Chapter 47 · Probability
expectation
For a discrete variable E X = Σ x_i p_i (if the series converges absolutely): the mean value weighted by probabilities, the centre of gravity of the distribution. Chapter 48 · Random variables
exponential function
The function y = aˣ, where a > 0 and a ≠ 1; defined for all x, takes only positive values, increasing when a > 1 and decreasing when 0 < a < 1. Chapter 12 · Powers and logarithms
exponential function
The function y = eˣ (also written exp x): the exponential function with base e; the only function that equals its own derivative and equals 1 at zero. Chapter 29 · The number e and the exponential
expression
A combination of numbers, variables, operation signs and brackets built by the rules of arithmetic, such as 2x + 10. Chapter 7 · Letters for numbers
extraneous root
A number obtained during solving because of a non-equivalent transformation, which does not satisfy the original equation. Chapter 8 · Equations: the art of balance
face
One of the pieces into which the edges of a plane drawing of a graph cut the plane; the outer infinite piece is a face too. Chapter 46 · Graphs
factor an expression
Writing an expression as a product of several factors, such as x² − 9 = (x − 3)(x + 3). Chapter 7 · Letters for numbers
factorial
The product of the natural numbers from 1 to n; written n!, with 0! = 1 by convention. Chapter 45 · Combinatorics
Fast exponentiation
Computing a^e mod n by a chain of squarings and multiplying together the squares that correspond to the ones in the binary expansion of e; it needs at most 2·log₂ e multiplications. Chapter 41 · Modular arithmetic and ciphers
fast Fourier transform
A way to compute a DFT of length N = 2^m in about N log₂ N operations by splitting the sum into even and odd samples (Cooley and Tukey, 1965). Chapter 33 · Fourier series
fat Cantor set
The set left of the segment [0, 1] when, at step n, an interval of length 1/4ⁿ is removed from the middle of each of the 2ⁿ⁻¹ remaining segments; it contains no segment, and its measure is 1/2. Chapter 54 · Measure and the Lebesgue integral
feasible region
The set of all solutions of the constraints of an optimization problem; for linear constraints in the plane it is usually a polygon or an unbounded region (and it may be empty). Chapter 11 · Systems of equations and inequalities
Fermat primes
A prime of the form 2^(2^m) + 1; five are known: 3, 5, 17, 257, 65,537. Chapter 43 · Galois theory
Fibonacci numbers
The sequence F₁ = F₂ = 1, F(n+1) = F(n) + F(n−1): 1, 1, 2, 3, 5, 8, 13, 21, …, each number the sum of the two before it. Chapter 13 · Sequences and induction
field
A commutative ring with a one different from zero in which every non-zero element has an inverse; examples are Q, R, C, Zₚ. Chapter 42 · Rings, fields and codes
field extension
A field K containing a field F with the same operations; for example, C is an extension of R, and Q(√2) of Q. Chapter 42 · Rings, fields and codes
fifth postulate
Euclid’s postulate about parallels; equivalent to the statement that through a point not on a line there passes at most one line parallel to it. Chapter 16 · The Elements: axioms and constructions
finite field
A field with finitely many elements; their number is always a prime power, q = pⁿ, and for each such q the field exists; written GF(q) or F_q. Chapter 42 · Rings, fields and codes
fixed point
A point x* that a map doesn’t move: f(x*) = x*. Chapter 55 · Functional analysis
foci
A special point of a conic section: an ellipse and a hyperbola have two, a parabola one; the curve is defined through the distances to the foci, and the rays it reflects gather at them. Chapter 24 · Conic sections
formal system
A set of symbols, axioms (initial strings) and mechanical rules for deriving new strings from existing ones; the strings that can be obtained from the axioms are called the theorems of the system. Chapter 56 · Gödel, Turing and the limits of proof
Fourier coefficients
The numbers aₙ = (1/π)∫ f(x) cos nx dx and bₙ = (1/π)∫ f(x) sin nx dx over the period [−π, π]: the projections of the function onto the harmonics. Chapter 33 · Fourier series
Fourier series
The series a₀/2 + Σ (aₙ cos nx + bₙ sin nx) whose coefficients are the Fourier coefficients of a function f; one writes f ~ this series until convergence is proved. Chapter 33 · Fourier series
fractal
A set in which new detail appears at every scale and whose dimension exceeds its topological dimension, often being fractional (the Koch curve, the Sierpiński triangle, a coastline). Chapter 59 · Chaos and fractals
fraction
A number of the form a/b with integer a and natural b: a shares of size 1/b of the whole. It is also the quotient of a divided by b. Chapter 5 · Fractions
free
An unknown whose column contains no pivot of the echelon form; it can be given any value, and the leading unknowns are expressed in terms of the free ones. Chapter 36 · Gaussian elimination
free group
The group of reduced words in the letters a, a⁻¹, b, b⁻¹ (no letter stands next to its inverse), including the empty word; multiplication means appending one word to another and cancelling neighbouring pairs like a·a⁻¹. Chapter 54 · Measure and the Lebesgue integral
frieze
A pattern on an infinite strip that goes to itself under translations along the strip; all these translations are multiples of the shortest one. Chapter 23 · Symmetry and transformations
function
A rule that assigns to each number x in some set exactly one number y = f(x). Chapter 9 · Functions and graphs
function of several variables
A rule that assigns to each point (x₁, …, xₙ) of some region a single number f(x₁, …, xₙ). Chapter 32 · Functions of several variables
Galois group
The set of permutations of the roots of an equation with rational coefficients that preserve all true relations between the roots with rational coefficients. Chapter 43 · Galois theory
Gaussian curvature
A measure of the curvedness of a surface at a point that can be measured from inside: K = lim 3(2πr − C(r))/(πr³), where C(r) is the length of the circle of radius r; for a sphere K = 1/R², for the plane 0, for the Lobachevsky plane −1/R². Chapter 58 · Non-Euclidean geometries
generators
Any of the lines that make up the surface of a cone (or a cylinder); on a cone all of them pass through the apex. Chapter 24 · Conic sections
genus
The number of handles of a closed orientable surface (through-holes, as in a doughnut): 0 for the sphere, 1 for the torus, 2 for the pretzel. Chapter 57 · Topology
geodesics
A line on a surface each short piece of which is a shortest path between its ends; in the plane these are straight lines, on a sphere great circles. Chapter 58 · Non-Euclidean geometries
geometric distribution
The distribution of the number of the first success in Bernoulli trials: P(T = k) = (1 − p)^(k − 1)·p, k = 1, 2, …; E T = 1/p. Chapter 48 · Random variables
geometric multiplicity
The dimension of the eigenspace for λ, that is, the largest number of linearly independent eigenvectors with this λ. Chapter 38 · Eigenvectors
geometric progression
A sequence of non-zero numbers in which each term from the second on equals the previous one multiplied by the same number q, the common ratio. Chapter 13 · Sequences and induction
Gibbs phenomenon
The overshoot of the partial sums of a Fourier series at a jump of the function: about 9 % of the height of the jump; it doesn’t shrink as the number of harmonics grows, it only gets narrower. Chapter 33 · Fourier series
glide reflection
A reflection in a line followed by a translation along the same line by a non-zero vector. Chapter 23 · Symmetry and transformations
golden ratio
The number φ = (1 + √5)/2 ≈ 1.618; φ² = φ + 1, continued fraction [1; 1, 1, …]. Chapter 6 · Root two and the real numbers
gradient
The vector ∇f = (∂f/∂x, ∂f/∂y) of the partial derivatives of a function; it points in the direction of fastest growth of the function, and its length is the greatest rate of growth. Chapter 32 · Functions of several variables
Gradient descent
An iterative method for finding a minimum: from the current point take a step against the gradient, x ← x − η∇f(x). Chapter 32 · Functions of several variables
graph
The set of all points (x, f(x)) of the coordinate plane, where x runs through the domain of f. Chapter 9 · Functions and graphs
graph
A set of vertices and edges, where each edge joins two vertices; all that matters is what is joined to what. Chapter 46 · Graphs
great circle
The section of a sphere by a plane through its centre; the shortest path between two points of a sphere is an arc of a great circle. Chapter 58 · Non-Euclidean geometries
greatest common divisor
The largest natural number that divides both given numbers; written gcd(a, b). Chapter 4 · Divisibility and Euclid's algorithm
greatest lower bound
The greatest lower bound of a set A, inf A. Chapter 53 · Rigorous analysis
group
A set with an operation that assigns to any two elements an element of the same set, is associative, has an identity element and an inverse for each element. Chapter 40 · Groups
Gödel numbering
A way of assigning a natural number to each string of symbols (formula, proof) so that the string can be recovered from the number uniquely and mechanically; for example, 2^{a₁}·3^{a₂}·5^{a₃}·… Chapter 56 · Gödel, Turing and the limits of proof
half-life
The time in which half the atoms of a radioactive sample decay; it doesn’t depend on the size of the sample: T½ = ln 2 / λ. Chapter 29 · The number e and the exponential
half-planes
The part of the plane on one side of a line (together with the line itself if the boundary is included). Chapter 11 · Systems of equations and inequalities
half-turn
A rotation by 180° about a point: X goes to X′, and the centre is the midpoint of XX′. Chapter 23 · Symmetry and transformations
halting problem
The question: given the description of a program P and an input x, decide whether P halts on x in finitely many steps. Turing (1936) proved that no algorithm solves it for all P and x. Chapter 56 · Gödel, Turing and the limits of proof
Hamiltonian cycle
A cycle in a graph that passes through each vertex exactly once. Chapter 46 · Graphs
Hamming code
A binary code that adds 3 check bits in positions 1, 2, 4 to 4 message bits and corrects any single error in a block of 7 bits. Chapter 42 · Rings, fields and codes
Hamming distance
The number of positions in which two words of the same length differ; for example, for 10110 and 11100 it is 2. Chapter 42 · Rings, fields and codes
harmonic
A sine wave whose frequency is a whole number of times the fundamental frequency of a signal; with period 2π it is aₙ cos nx + bₙ sin nx. Chapter 33 · Fourier series
harmonic series
The series 1 + 1/2 + 1/3 + 1/4 + …; it diverges although its terms tend to zero. Chapter 30 · Series
Hausdorff distance
For bounded closed sets A and B in the plane, the smallest r such that each point of A is within r of B and each point of B is within r of A. Chapter 55 · Functional analysis
Hilbert space
A complete inner product space, for example ℓ² or L². Chapter 55 · Functional analysis
holomorphic
A function of a complex variable that has a complex derivative f′(z) = lim (f(z + h) − f(z))/h as h → 0 at every point of a region. Chapter 34 · Complex analysis
homeomorphism
A one-to-one map of one shape onto another that’s continuous together with its inverse; shapes between which there is one are called homeomorphic. Chapter 57 · Topology
homogeneous system
A system of linear equations whose right-hand sides are all zero; it always has the zero solution. Chapter 36 · Gaussian elimination
homothety
A transformation with centre O and ratio k ≠ 0 that takes a point X to the point X′ with OX′ = k·OX (as vectors). Chapter 23 · Symmetry and transformations
Horner’s method
A way to compute P(a) and at the same time divide P(x) by x − a: b₀ = aₙ, b_k = a_{n−k} + a·b_{k−1}; the last number is the remainder P(a). Chapter 14 · Polynomials
Huffman code
The prefix code of smallest average length for a given distribution; built by repeatedly merging the two least likely symbols into one. Chapter 50 · Markov chains and information
hyperbola
The set of points of the plane whose distances to two given points (the foci) have a constant absolute difference smaller than the distance between the foci. Chapter 24 · Conic sections
hypotenuse
The side of a right triangle opposite the right angle; the longest of the three. Chapter 18 · Pythagoras' theorem
ideal
A triangle in the Lobachevsky plane all three of whose vertices are at infinity; all its angles are zero, and for curvature −1 its area is π. Chapter 58 · Non-Euclidean geometries
identity
An equation with variables that holds for all values of the variables at which both sides make sense. Chapter 7 · Letters for numbers
identity matrix
The square matrix I with ones on the main diagonal and zeros elsewhere; the matrix of the identity transformation: Iv = v. Chapter 35 · Matrices as transformations
image
The set of all vectors of the form Ax: where the linear map takes the space; it coincides with the span of the columns of the matrix. Chapter 37 · Vector spaces
imaginary unit
The number i whose square is −1. Chapter 15 · Complex numbers
implication
The statement ‘if A then B’ (A → B); it’s false only when A is true and B is false. Chapter 51 · Logic and sets
implicit differentiation
Finding the derivative of a function y(x) given by an equation F(x, y) = 0: both sides are differentiated with respect to x, treating y as a function of x, and y′ is solved for. Chapter 26 · The derivative
improper integral
An integral over an infinite interval (or of an unbounded function), understood as a limit: ∫ₐ^∞ f(x)dx = lim ∫ₐᴮ f(x)dx as B → ∞. If the limit is finite, the integral converges; otherwise it diverges. Chapter 28 · The integral
incircle
The circle touching all three sides of a triangle; its centre is where the bisectors meet. Chapter 17 · The triangle
inclusion–exclusion principle
A way to count the union of several groups: add the sizes of the groups, subtract the pairwise overlaps, add the triple ones and so on, alternating signs. Chapter 45 · Combinatorics
incommensurable
Segments with no common measure, that is, no segment that fits a whole number of times into each of them; the ratio of their lengths is irrational. Chapter 6 · Root two and the real numbers
increases
A function increases on an interval if a larger value of the argument in the interval gives a larger value of the function. Chapter 9 · Functions and graphs
indefinite integral
The set of all antiderivatives of a function f on an interval; written ∫f(x)dx = F(x) + C, where F is one of the antiderivatives and C any constant. Chapter 28 · The integral
independent
Events A and B with P(A ∩ B) = P(A)·P(B); when P(B) > 0 this means P(A | B) = P(A): knowing about B doesn’t change the chances of A. Chapter 47 · Probability
independent
Variables X and Y with P(X = x, Y = y) = P(X = x)·P(Y = y) for all x, y (in general P(X ≤ s, Y ≤ t) = P(X ≤ s)·P(Y ≤ t)). Chapter 48 · Random variables
indicator
A random variable equal to 1 if the event A occurred and 0 if it didn’t; its expectation equals P(A). Chapter 48 · Random variables
infinitesimal
A sequence or function whose limit is zero. Chapter 25 · Limits
initial value problem
A differential equation together with initial conditions: the value of the unknown function (and of its derivatives up to one less than the order of the equation) at one point. Also called a Cauchy problem. Chapter 31 · Differential equations
injection
A map that sends different elements to different ones: a ≠ a′ implies f(a) ≠ f(a′). Chapter 51 · Logic and sets
inner product space
A vector space with an operation ⟨x, y⟩ giving a number that’s symmetric, linear in each argument, and has ⟨x, x⟩ > 0 for x ≠ 0; length in it is ‖x‖ = √⟨x, x⟩. Chapter 55 · Functional analysis
inscribed angle
An angle whose vertex lies on a circle and whose arms cross the circle. Chapter 19 · The circle and the number π
Instantaneous speed
The limit of the average speed over the stretch from t to t + h as h → 0: v(t) = lim (s(t + h) − s(t))/h. Chapter 26 · The derivative
integers
The natural numbers, their opposite negative numbers and zero: …, −2, −1, 0, 1, 2, …; the set is written ℤ. Chapter 2 · Zero and minus
integrable
A function whose Riemann sums approach one number as the partition gets finer, however the points in the pieces are chosen. Chapter 28 · The integral
integral curve
The graph of a solution of a differential equation; at each of its points it touches the segment of the direction field. Chapter 31 · Differential equations
interpolation polynomial
The polynomial of degree at most n that takes given values at n + 1 given points; such a polynomial exists and is unique. Chapter 14 · Polynomials
intersection
The set A ∩ B of all elements that belong to both A and B. Chapter 51 · Logic and sets
inverse
A function f⁻¹ that undoes f: f⁻¹(f(x)) = x for all x in the domain of f, and f(f⁻¹(y)) = y for all y in the range of f. Chapter 9 · Functions and graphs
inverse
The matrix A⁻¹ for which A⁻¹A = AA⁻¹ = I; its transformation undoes the transformation A. Chapter 35 · Matrices as transformations
inverse of $a$ modulo
A number a′ with a · a′ ≡ 1 (mod n); it exists if and only if gcd(a, n) = 1, and is found by the extended Euclidean algorithm. Chapter 41 · Modular arithmetic and ciphers
Inversion
A transformation with respect to a circle with centre O and radius R: a point X ≠ O goes to the point X′ on the ray OX with OX · OX′ = R². Chapter 23 · Symmetry and transformations
inversion
A pair of places i < j in which a permutation puts the elements in reverse order: σ(i) > σ(j). Chapter 40 · Groups
irrational number
A real number (a point of the number line) that can’t be written as a fraction p/q with integers p and q; for example √2 and π. Chapter 6 · Root two and the real numbers
irreducible
A polynomial of non-zero degree that cannot be written as a product of two polynomials of smaller degree with coefficients from the same field. Chapter 42 · Rings, fields and codes
irreducible case
A cubic equation with three distinct real roots: in this case Cardano’s formula needs the square root of a negative number. Chapter 14 · Polynomials
isomorphism
A one-to-one linear map of one vector space onto another; such spaces are indistinguishable from the point of view of linear algebra. Chapter 37 · Vector spaces
isomorphism
A one-to-one correspondence φ between groups with φ(ab) = φ(a)φ(b); the Cayley tables of isomorphic groups coincide up to renaming the elements. Chapter 40 · Groups
iterated function system
A collection of contractions of the plane f₁, …, f_k; it assigns to a shape A the union f₁(A) ∪ … ∪ f_k(A), and its fixed shape is the attractor, often a fractal. Chapter 55 · Functional analysis
Julia set
For the map z ↦ z² + c: the boundary of the set of those z whose orbits stay bounded (that set itself is the filled Julia set). Chapter 59 · Chaos and fractals
kernel
The set of vectors that a linear map (a matrix A) sends to the zero vector: ker A = {x : Ax = 0}. Chapter 37 · Vector spaces
Klein bottle
A closed non-orientable surface: a square with one pair of opposite sides glued straight and the other upside down (the word aba⁻¹b); χ = 0. Chapter 57 · Topology
knot
A closed curve in space without self-intersections; two knots are the same if one can be moved continuously into the other without passing the curve through itself. Chapter 57 · Topology
Lagrange multiplier
The number λ for which ∇f = λ∇g at a point of constrained extremum of f subject to g = c; it gives the rate at which the extreme value changes as c is shifted. Chapter 32 · Functions of several variables
least common multiple
The smallest natural number divisible by both given numbers; written lcm(a, b). Chapter 4 · Divisibility and Euclid's algorithm
least upper bound
The least upper bound of a set A, sup A; it may not belong to A, as with 1 = sup [0, 1). Chapter 53 · Rigorous analysis
Lebesgue integral
For a measurable f ≥ 0, the number ∫₀^∞ μ{f > t} dt; for a function of any sign, the difference of the integrals of its positive and negative parts, if both are finite. Chapter 54 · Measure and the Lebesgue integral
Lebesgue measurable
A set A ⊂ ℝ that, for every ε > 0, can be enclosed in an open set U so that the outer measure of U \ A is less than ε. Chapter 54 · Measure and the Lebesgue integral
Lebesgue measure
Outer measure considered only on measurable sets; written μ; for a segment it equals its length, it’s countably additive and doesn’t change under shifts. Chapter 54 · Measure and the Lebesgue integral
legs
A side of a right triangle adjacent to the right angle. Chapter 18 · Pythagoras' theorem
level curve
The set of points of the plane at which the function f(x, y) takes the same value c: f(x, y) = c. Chapter 32 · Functions of several variables
like terms
Terms that differ only in their numerical coefficient, such as 3x and −5x. Chapter 7 · Letters for numbers
limit of the function
The number L that the values of a function approach as x approaches a: for every ε > 0 there is δ > 0 such that 0 < |x − a| < δ implies |f(x) − L| < ε. Chapter 25 · Limits
limit of the sequence
The number a that the terms of a sequence approach: for every ε > 0 there is an index N after which all terms differ from a by less than ε. Chapter 25 · Limits
linear
An equation of the form ax + b = 0, where a and b are numbers and x is the unknown. Chapter 8 · Equations: the art of balance
linear
A map L of vectors to vectors for which L(u + v) = Lu + Lv and L(cv) = cLv for any vectors u, v and any number c. Chapter 35 · Matrices as transformations
linear combination
A vector of the form s·a + t·b (and so on for more vectors), where s and t are numbers. Chapter 22 · Vectors and coordinates
linear function
A function of the form y = kx + b, where k and b are numbers; its graph is a straight line. Chapter 9 · Functions and graphs
linear programming
The branch of mathematics about finding the largest or smallest value of a linear function subject to linear inequality constraints. Chapter 11 · Systems of equations and inequalities
linearly independent
Vectors are linearly independent if their linear combination is zero only when all the coefficients are zero; otherwise they are linearly dependent. Chapter 37 · Vector spaces
Lipschitz condition
The estimate |f(t, y₁) − f(t, y₂)| ≤ K·|y₁ − y₂| with a single constant K: the function changes no faster than a linear one. Chapter 31 · Differential equations
Lobachevsky plane
A plane in which all the axioms of absolute geometry and Lobachevsky’s axiom hold; also called the hyperbolic plane. Chapter 58 · Non-Euclidean geometries
local maximum
A point c at which the value f(c) is not less (local maximum) or not greater (local minimum) than all values of the function in some neighbourhood of c. Chapter 27 · Extrema and optimization
logarithm
The exponent to which the base a must be raised to get b: log_a b = c means a^c = b (a > 0, a ≠ 1, b > 0). Chapter 12 · Powers and logarithms
logarithmic integral
The function li x: the integral from 0 to x of 1/ln t (as a principal value); it grows like x/ln x but approximates the prime count π(x) more accurately. Chapter 44 · The zeta function and elliptic curves
logarithmic scale
A scale on which equal distances mean equal ratios of quantities: each point is placed at a distance proportional to the logarithm of the quantity. Chapter 12 · Powers and logarithms
logistic equation
The equation y′ = r·y·(1 − y/K): growth that slows down and stops at the limit K; its solutions are S-shaped logistic curves. Chapter 31 · Differential equations
lowest terms
A fraction a/b whose numerator and denominator are coprime: gcd(a, b) = 1. Chapter 5 · Fractions
lune
The part of a sphere between two halves of great circles with common ends at two opposite points; a lune of angle α has area 2αR². Chapter 58 · Non-Euclidean geometries
Mandelbrot set
The set of complex c for which the orbit of zero, 0, c, c² + c, …, under z ↦ z² + c stays bounded. Chapter 59 · Chaos and fractals
map
A map f from A to B is a subset f ⊂ A × B containing, for each a ∈ A, exactly one pair (a, b); written f: A → B and f(a) = b. Chapter 51 · Logic and sets
Markov chain
A random process jumping between states in which the probabilities of the next state depend only on the current one, not on the history. Chapter 38 · Eigenvectors
Mathematical induction
A way to prove a statement P(n) for all natural n at once: check it for n = 1 (the base) and prove that P(k) implies P(k + 1) for every k (the step). Chapter 13 · Sequences and induction
matrix
A rectangular table of numbers; the columns of the matrix of a linear map hold the coordinates of the images of the vectors e₁, e₂, …. Chapter 35 · Matrices as transformations
measurable
A function f for which the set {x : f(x) > t} is measurable for every t; this holds for all continuous functions, their limits and almost all functions met in analysis. Chapter 54 · Measure and the Lebesgue integral
measure
A function μ ≥ 0 on a sigma-algebra of sets (the value +∞ is allowed) with μ(∅) = 0, such that the measure of a union of countably many disjoint sets equals the sum of their measures. Chapter 54 · Measure and the Lebesgue integral
measure zero
A set that, for every ε > 0, can be covered by a finite or countable collection of intervals with total length less than ε; for example, any countable set. Chapter 54 · Measure and the Lebesgue integral
median
The segment joining a vertex of a triangle to the midpoint of the opposite side. Chapter 17 · The triangle
median
The number that splits an ordered set of numbers in half: the middle element when there’s an odd number of them, and the average of the two middle ones when there’s an even number. Chapter 49 · Statistics
mediant
For fractions a/b and c/d, the fraction (a + c)/(b + d); if the fractions differ and the denominators are positive, it lies strictly between them. Chapter 5 · Fractions
method of intervals
A way to solve inequalities of the form f(x) > 0 where f is factored: the zeros of the factors divide the line into intervals, and on each of them the sign of f is constant. Chapter 11 · Systems of equations and inequalities
method of least squares
A way of approximately solving an inconsistent system Ax = b: choose the x with the smallest sum of squared residuals |b − Ax|²; such an x satisfies the normal equations AᵀAx = Aᵀb. Chapter 39 · Orthogonality, least squares and SVD
metric space
A set with a distance d(x, y) ≥ 0 that’s zero only when x = y, is symmetric, and satisfies the triangle inequality d(x, z) ≤ d(x, y) + d(y, z). Chapter 55 · Functional analysis
midline
The segment joining the midpoints of two sides of a triangle; it is parallel to the third side and half its length. Chapter 17 · The triangle
minimal polynomial
The polynomial of smallest degree with leading coefficient 1 and coefficients from the field K that has the given number as a root. Chapter 43 · Galois theory
mixed number
A notation like 2⅓: a whole part and a proper fraction side by side; it means their sum. Chapter 5 · Fractions
modulus
For z = a + bi, the number |z| = √(a² + b²), the distance from zero to the point z in the complex plane. Chapter 15 · Complex numbers
monic
A quadratic equation with leading coefficient 1: x² + px + q = 0. Chapter 10 · Quadratic equations
monodromy
The permutation of the roots produced by a loop: a coefficient of the equation goes round a closed path, the roots move continuously and arrive in one another’s places. Chapter 43 · Galois theory
motion
A transformation of the plane (or space) that preserves the distances between points; also called a rigid motion. Chapter 23 · Symmetry and transformations
mutually exclusive
Events with no outcomes in common: they can’t both occur at once. Chapter 47 · Probability
Möbius band
The surface obtained from a rectangular strip by gluing the short sides with half a twist; it has one side and one edge. Chapter 57 · Topology
n-gram
A sequence of n consecutive symbols of a text (letters or words); an n-gram model chooses the next symbol from the previous n − 1 with frequencies taken from a sample. Chapter 50 · Markov chains and information
natural logarithm
The logarithm to base e = 2.71828…: ln b = log_e b. Chapter 12 · Powers and logarithms
natural numbers
The numbers 1, 2, 3, … used to count objects; the set is written ℕ. Whether 0 belongs to it is a convention, and this course starts at 1. Chapter 1 · Counting: from notches to bits
nine-point circle
The circle through the midpoints of the sides of a triangle, the feet of its altitudes and the midpoints of the segments from the orthocentre to the vertices; its radius is half the circumradius. Chapter 17 · The triangle
non-trivial zeros
A zero of the Riemann zeta function in the critical strip 0 ≤ Re s ≤ 1; the first of them is 1/2 + 14.1347…i. Chapter 44 · The zeta function and elliptic curves
norm
A function ‖x‖ on a vector space: ‖x‖ ≥ 0 and equals zero only for the zero vector, ‖cx‖ = |c|·‖x‖, ‖x + y‖ ≤ ‖x‖ + ‖y‖. Chapter 55 · Functional analysis
norm of a matrix
The largest stretch a matrix produces: ‖M‖ = max |Mx| over unit vectors x; it equals the largest singular value. Chapter 39 · Orthogonality, least squares and SVD
normal
A non-zero vector perpendicular to a line (or a plane). Chapter 22 · Vectors and coordinates
normal distribution
The distribution with density e^(−(x − μ)²/(2σ²))/(σ√(2π)); written N(μ, σ²); μ is the expectation, σ the standard deviation. Chapter 48 · Random variables
normal equations
The system AᵀA x̂ = Aᵀb; its solutions are the least-squares solutions of the inconsistent system Ax = b. Chapter 39 · Orthogonality, least squares and SVD
normed space
A vector space with a chosen norm; the distance between vectors x and y in it is ‖x − y‖. Chapter 55 · Functional analysis
NP
Yes-or-no problems in which a yes can be backed up by a hint that is checked in polynomial time. Chapter 60 · The frontier
NP-complete
A problem in NP to which every problem in NP reduces in polynomial time; a fast algorithm for it would give fast algorithms for all of NP. Chapter 60 · The frontier
null hypothesis
The assumption ‘there’s no effect, everything is explained by chance’; it’s tested by trying to refute it with data. Written H₀. Chapter 49 · Statistics
nullity
The dimension of the kernel of a matrix or linear map. Chapter 37 · Vector spaces
number $\pi$
The ratio of a circle’s circumference to its diameter, the same for every circle; π ≈ 3.14159. Chapter 19 · The circle and the number π
number $e$
The limit of (1 + 1/n)ⁿ as n → ∞, equal to 2.718281828…; the base of natural logarithms: the area under the hyperbola y = 1/t from 1 to e equals 1. Chapter 29 · The number e and the exponential
number line
A line with a marked point 0, a chosen direction and a unit length: every number corresponds to a point on it. Chapter 2 · Zero and minus
numeral system
A way of writing numbers: a set of signs and the rules by which the signs make up the notation of a number. Chapter 1 · Counting: from notches to bits
objective function
The function whose largest or smallest value is sought in an optimization problem, such as profit or cost. Chapter 11 · Systems of equations and inequalities
odd
A function with f(−x) = −f(x) for all x in the domain; its graph is symmetric about the origin. Chapter 9 · Functions and graphs
one-sided limit
The limit of a function as x approaches a from one side only: from the right (x > a) or from the left (x < a). Chapter 25 · Limits
opposites
The number −a which gives zero when added to a; on the number line a and −a are mirror images in zero. Chapter 2 · Zero and minus
orbit
The sequence x₀, f(x₀), f(f(x₀)), …: what you get from a starting point by applying the map f repeatedly. Chapter 59 · Chaos and fractals
order
The number of elements of a finite group G; written |G|. Chapter 40 · Groups
order of an element
The smallest natural number k for which g^k = e; if there is no such k, the order of the element is infinite. Chapter 40 · Groups
orientable
A surface on which no closed path brings a small clock back mirror-reversed; in other words, it contains no Möbius band. Chapter 57 · Topology
orientation
The direction of the shortest rotation from the first vector of a pair to the second: anticlockwise is positive, clockwise negative. A transformation with negative determinant reverses orientation. Chapter 35 · Matrices as transformations
orthocentre
The point where the lines containing the altitudes of a triangle meet. Chapter 17 · The triangle
orthogonal
Functions f and g whose dot product ∫ f(x)g(x) dx over a period is zero. Chapter 33 · Fourier series
orthogonal
Vectors with zero dot product: ⟨u, v⟩ = 0. The zero vector is orthogonal to every vector. Chapter 39 · Orthogonality, least squares and SVD
orthogonal
A square matrix Q with orthonormal columns: QᵀQ = I, that is, Q⁻¹ = Qᵀ. It preserves lengths and angles. Chapter 39 · Orthogonality, least squares and SVD
orthogonal projection
The vector p of a subspace W for which the difference b − p is orthogonal to W; it is the point of W nearest to b. Chapter 39 · Orthogonality, least squares and SVD
orthonormal basis
A basis of pairwise orthogonal vectors of length 1: ⟨qᵢ, qⱼ⟩ = 0 for i ≠ j and |qᵢ| = 1. Coordinates in it are the dot products with the basis vectors. Chapter 39 · Orthogonality, least squares and SVD
orthonormal system
A finite or infinite collection of vectors e₁, e₂, … with ⟨eᵢ, eⱼ⟩ = 0 for i ≠ j and ‖eᵢ‖ = 1. Chapter 55 · Functional analysis
outer measure
The outer measure of a set A ⊂ ℝ: the greatest lower bound of the total lengths of intervals over all finite or countable covers of A by intervals. Chapter 54 · Measure and the Lebesgue integral
P
Yes-or-no problems that an algorithm can solve in polynomial time. Chapter 60 · The frontier
p-hacking
Trying out ways of collecting and analysing data until the p-value dips below 0.05; it sharply raises the share of false discoveries. Chapter 49 · Statistics
p-value
The probability of getting a result at least as striking as the one observed, assuming the null hypothesis is true. Chapter 49 · Statistics
parabola
The set of points of the plane equidistant from a given point (the focus) and a given line (the directrix) not passing through the focus. Chapter 24 · Conic sections
Parallel lines
Lines in the plane that never meet, however far they are extended. Chapter 16 · The Elements: axioms and constructions
parity bit
An extra bit that makes the number of ones in a block even; it lets any odd number of errors in the block be noticed, but not located. Chapter 42 · Rings, fields and codes
partial derivative
The derivative of f(x, y) with respect to one variable with the others held fixed: ∂f/∂x at (a, b) is the derivative of x ↦ f(x, b) at a. Chapter 32 · Functions of several variables
Partial pivoting
A rule of Gaussian elimination: at each step the pivot is the entry of the column with the largest absolute value among the remaining rows, brought up by swapping rows. Chapter 36 · Gaussian elimination
partial sum
The sum of the first n terms of a series: Sₙ = a₁ + … + aₙ. Chapter 30 · Series
Pascal’s triangle
The table of binomial coefficients: row n consists of the numbers n choose k, k = 0…n; each number is the sum of the two above it. Chapter 45 · Combinatorics
Peano arithmetic
The formal system with the symbols 0, S, +, ×, = and the axioms: zero follows no number, S is one-to-one, recursive definitions of addition and multiplication, the induction schema. Chapter 56 · Gödel, Turing and the limits of proof
periodic
A function whose values repeat when the argument is shifted by the same number T ≠ 0, the period: f(x + T) = f(x). Chapter 21 · Trigonometry
permutation
An arrangement of n different objects in some order; there are n! of them. Chapter 45 · Combinatorics
perpendicular bisector
The line through the midpoint of a segment perpendicular to it; its points, and only they, are equidistant from the ends of the segment. Chapter 16 · The Elements: axioms and constructions
phase plane
The plane of states of a system of two quantities (x, y): a solution is shown as a moving point, and its track as a phase trajectory. Chapter 31 · Differential equations
piecewise smooth
A function whose period (or interval) splits into finitely many parts on which it is continuously differentiable, with the function and its derivative having finite limits at the ends of the parts. Chapter 33 · Fourier series
pigeonhole principle
If n + 1 objects are put into n boxes, some box will contain at least two objects. Chapter 45 · Combinatorics
pivots
The first non-zero entry of a row in the row echelon form of a matrix; it is used to eliminate an unknown from the other rows. Chapter 36 · Gaussian elimination
places
The position of a digit in positional notation. Place number k (counting from the right, starting at zero) is worth b^k, where b is the base. Chapter 1 · Counting: from notches to bits
planar graph
A graph that can be drawn in the plane without its edges crossing. Chapter 46 · Graphs
Poincaré disk
A model of the Lobachevsky plane: the points are the inside of a disc, the lines are diameters and arcs of circles perpendicular to the edge; angles are shown without distortion. Chapter 58 · Non-Euclidean geometries
point of discontinuity
A point near which a function is defined but at which it is not continuous: there is no limit, or no value, or the value differs from the limit. Chapter 25 · Limits
point of inflection
A point of the graph of a continuous function where the direction of bending changes (from convex to concave or vice versa). Chapter 27 · Extrema and optimization
Poisson distribution
The distribution with P(X = k) = λ^k·e^(−λ)/k!, k = 0, 1, 2, …; describes the number of rare independent events; E X = Var X = λ. Chapter 48 · Random variables
polar coordinates
Coordinates of a point in the plane: the distance r to the origin and the angle φ with the x-axis; x = r cos φ, y = r sin φ. Chapter 32 · Functions of several variables
polar form
Writing a complex number as z = r(cos φ + i sin φ), where r = |z| and φ = arg z. Chapter 15 · Complex numbers
pole
A point a near which f(z) = g(z)/(z − a)ᵐ with g holomorphic and g(a) ≠ 0; the number m is the order of the pole, and for m = 1 the pole is simple. Chapter 34 · Complex analysis
polynomial
An expression a_n x^n + … + a_1 x + a_0, where a_0, …, a_n are numbers (the coefficients) and n is a non-negative integer. Chapter 14 · Polynomials
polynomial time
An algorithm runs in polynomial time if its number of steps on an input of length n is at most C·n^k for some constants C and k. Chapter 60 · The frontier
population
The whole set of objects we draw inferences about: all the voters in a city, all the cartons of milk off the conveyor. Chapter 49 · Statistics
positional numeral system
A numeral system in which the contribution of a digit depends on its place: the digit is multiplied by a power of the base. Chapter 1 · Counting: from notches to bits
power
The product of n equal factors, each equal to a; a is the base and n the exponent. Chapter 12 · Powers and logarithms
power
The probability of rejecting the null hypothesis when it’s false, that is, of detecting an effect that exists; it grows with the sample size and the size of the effect. Chapter 49 · Statistics
power method
An approximate search for the eigenvector with the eigenvalue of largest absolute value: a vector is repeatedly multiplied by the matrix and normalised. Chapter 38 · Eigenvectors
power of a point
The number OP² − r² for a point P and a circle with centre O and radius r; negative inside the disc, zero on the circle, positive outside. Chapter 19 · The circle and the number π
power series
A series of the form Σ cₖ(x − a)ᵏ: an ‘infinite polynomial’ centred at the point a. Chapter 30 · Series
power set
The set of all subsets of a set A, including the empty set and A itself; written 𝒫(A). A set of n elements has 2ⁿ subsets. Chapter 52 · Infinities
predicate
A statement with a variable, for example ‘x > 3’: it becomes a proposition when a particular object is substituted for the variable. Chapter 51 · Logic and sets
prefix code
A code in which no codeword is the beginning of another; a message in such a code can be read unambiguously without separators. Chapter 50 · Markov chains and information
prime
A natural number greater than 1 with exactly two divisors: 1 and itself. Chapter 3 · Prime numbers
prime factorization
Writing a natural number as a product of primes (repeated ones are gathered into powers). Chapter 3 · Prime numbers
prime-counting function $\pi(x)$
The function π(x): the number of primes not exceeding x. Chapter 3 · Prime numbers
primitive
A Pythagorean triple whose three numbers have no common divisor greater than 1. Chapter 18 · Pythagoras' theorem
primitive root
A number g whose powers g, g², …, g^(p−1) give all the non-zero remainders modulo a prime p; in other words, the order of g is p − 1. Chapter 41 · Modular arithmetic and ciphers
principal axes
Mutually perpendicular lines along the eigenvectors of a symmetric matrix; in coordinates along them the quadratic form becomes a sum of squares. Chapter 38 · Eigenvectors
Principal component analysis
A way of describing a data cloud by a few directions: the axes are the eigenvectors of the covariance matrix (the right singular vectors of the centred data matrix), ordered by decreasing spread along them. Chapter 39 · Orthogonality, least squares and SVD
prism
A polyhedron two of whose faces are congruent polygons in parallel planes, and the rest parallelograms. Chapter 20 · Area and volume
probability
A number P(A) from 0 to 1 measuring the chances of an event A. With n equally likely outcomes P(A) = m/n, where m is the number of outcomes favourable to A; in general, any rule obeying Kolmogorov’s axioms. Chapter 47 · Probability
probability density
A function f ≥ 0 with integral 1 such that P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx: probabilities are areas under its graph. Chapter 48 · Random variables
product of matrices
The matrix AB of the composition ‘first B, then A’: (AB)v = A(Bv); its jth column is A times the jth column of B. Chapter 35 · Matrices as transformations
projection
The number |b|·cos φ = (a·b)/|a|: the length of the shadow of b on the line of a, with a minus sign if the angle between them is obtuse. Chapter 22 · Vectors and coordinates
projective plane
A closed non-orientable surface: a square with both pairs of opposite sides glued upside down (the word abab); χ = 1. Chapter 57 · Topology
proof
A chain of reasoning in which every step follows by the rules of logic from the previous steps, from axioms or from statements already proved. Chapter 0 · What mathematics is
Proof by contradiction
A method of proof: assume the statement is false and derive a contradiction from that; hence the statement is true. Chapter 6 · Root two and the real numbers
prosecutor’s fallacy
Mistaking the probability of the evidence given innocence, P(evidence | innocent), for the probability of innocence given the evidence, P(innocent | evidence). Chapter 47 · Probability
Public-key cryptography
Encryption in which messages are encrypted with a public key known to everyone and decrypted with a private key that is practically impossible to compute from the public one. Chapter 41 · Modular arithmetic and ciphers
pyramid
A polyhedron one face of which is a polygon (the base) and the rest triangles with a common vertex. Chapter 20 · Area and volume
Pythagorean triple
Three natural numbers a, b, c with a² + b² = c²: the sides of a right triangle with whole-number lengths. Chapter 18 · Pythagoras' theorem
quadratic
The expression ax² + bx + c with a ≠ 0, a polynomial of degree two in x. Chapter 10 · Quadratic equations
quadratic equation
An equation of the form ax² + bx + c = 0, where a, b, c are numbers and a ≠ 0. Chapter 10 · Quadratic equations
quadratic form
A homogeneous polynomial of degree two in the coordinates of a vector; it is written as ⟨x, Sx⟩ with a symmetric matrix S. Chapter 38 · Eigenvectors
Quantifiers
The words ‘for every’ (∀) and ‘there exists’ (∃), which turn a predicate P(x) into a statement: ∀x P(x) says P holds for all x, ∃x P(x) for at least one. Chapter 51 · Logic and sets
radian
The central angle standing on an arc whose length equals the radius; a full turn is 2π radians, 1 radian ≈ 57.3°. Chapter 19 · The circle and the number π
radius of convergence
The number R for which a power series centred at a converges when |x − a| < R and diverges when |x − a| > R. Chapter 30 · Series
Ramsey number
R(m, n): the smallest number of people among whom there are bound to be m who all know one another or n mutual strangers. Chapter 45 · Combinatorics
random
A sample in which every member of the population had the same chance of being included, and choosing some didn’t affect the choice of others; its observations are independent identically distributed variables. Chapter 49 · Statistics
random variable
A numerical function X on the sample space Ω: each outcome of the experiment is assigned a number X(ω). Chapter 48 · Random variables
random walk
A sequence of positions in which each next step is chosen at random, independently of the past ones; the simplest is steps of ±1 along a line with probabilities 1/2. Chapter 47 · Probability
range
The set of all values a function takes on its domain. Chapter 9 · Functions and graphs
rank
The number of pivots (non-zero rows) in the row echelon form of a matrix; it does not depend on how the matrix was brought to this form. Chapter 36 · Gaussian elimination
rate of a code
The ratio of the number of message bits to the number of transmitted bits; 1/3 for triple repetition, 4/7 for the Hamming (7, 4) code. Chapter 42 · Rings, fields and codes
rational number
A number that can be written as a fraction p/q with integer p and natural q. The set of rational numbers is written ℚ. Chapter 5 · Fractions
real number
A number written as an infinite decimal (with a sign); the set of real numbers is ℝ. Chapter 6 · Root two and the real numbers
reciprocal
The number 1/x for x ≠ 0: multiplied by x it gives one. For a fraction a/b the reciprocal is b/a. Chapter 5 · Fractions
recurrence relation
A rule by which a term of a sequence is computed from the previous ones, for example a(n+1) = 2a(n) + 1; together with the first terms it determines the whole sequence. Chapter 13 · Sequences and induction
reduced row echelon form
Row echelon form in which all the pivots equal 1 and all the other entries in their columns are zero. Chapter 36 · Gaussian elimination
redundancy
The share of a message that can be predicted and therefore needn’t be sent: R = 1 − H/log₂ N, where H is the entropy per symbol and N the number of symbols in the alphabet. Chapter 50 · Markov chains and information
reflection
A motion that takes each point to its mirror image in a given line (the axis): the axis is the perpendicular bisector of the segment from the point to its image. Chapter 23 · Symmetry and transformations
Regression to the mean
If two measurements are imperfectly related (|r| < 1), then for objects with an extreme value of the first, the second is on average less extreme, for no reason other than chance. Chapter 49 · Statistics
Reidemeister moves
Three ways of changing a knot diagram inside a small disc: add or remove a curl; slide one strand over another or undo such an overlap; slide a strand over the crossing of two others. Chapter 57 · Topology
relation
A binary relation on a set A is a subset R ⊂ A × A; one writes a R b if the pair (a, b) belongs to R. Chapter 51 · Logic and sets
repeating decimal
An infinite decimal in which, from some point on, the same block of digits (the period) repeats forever: 1/6 = 0.1666…. Chapter 5 · Fractions
residual
The difference e = b − Ax between the right-hand side of a system and what an approximate solution gives; its coordinates are the misses in each equation. Chapter 39 · Orthogonality, least squares and SVD
residue
The coefficient of 1/(z − a) in the expansion of a function near its singular point a in powers of z − a; at a simple pole Res = lim (z − a)f(z). Chapter 34 · Complex analysis
residue class
The set of all integers congruent to a number r modulo n: …, r − n, r, r + n, r + 2n, … There are exactly n residue classes modulo n. Chapter 41 · Modular arithmetic and ciphers
Riemann hypothesis
The statement that all the non-trivial zeros of the Riemann zeta function lie on the line Re s = 1/2; an open problem. Chapter 44 · The zeta function and elliptic curves
Riemann sum
The sum f(ξ₁)Δx₁ + … + f(ξₙ)Δxₙ for a partition of an interval into pieces of length Δxᵢ and points ξᵢ chosen in the pieces; its terms are (signed) areas of rectangles. Chapter 28 · The integral
Riemann zeta function
The function ζ(s) = 1 + 1/2ˢ + 1/3ˢ + … for Re s greater than 1, and its analytic continuation to the whole complex plane except the point s = 1. Chapter 44 · The zeta function and elliptic curves
ring
A set with two operations, addition and multiplication, in which addition satisfies the axioms of a commutative group, multiplication is associative, and the two are linked by the distributive laws. Chapter 42 · Rings, fields and codes
root
A value of the unknown that turns the equation into a true numerical equality. Chapter 8 · Equations: the art of balance
root of a polynomial
A number a at which the value of the polynomial P(a) is zero. Chapter 14 · Polynomials
root of multiplicity
A root a of a polynomial P has multiplicity k if P is divisible by (x − a)^k but not by (x − a)^(k+1). Chapter 14 · Polynomials
rotation
A motion that rotates every point about the same point (the centre) by the same angle. Chapter 23 · Symmetry and transformations
row echelon form
The form of a matrix in which zero rows are at the bottom and the first non-zero entry of each non-zero row stands further right than that of the row above. Chapter 36 · Gaussian elimination
rule of product
If a choice is made in several steps and at each step the number of options doesn’t depend on what was chosen before, the numbers of options multiply. Chapter 45 · Combinatorics
rule of sum
If an object can be chosen in m ways from one group or in n ways from another group that doesn’t overlap it, there are m + n ways in all. Chapter 45 · Combinatorics
Saccheri quadrilateral
A quadrilateral ABCD whose angles at the base AB are right angles and whose sides AD and BC are equal; its upper angles C and D are equal to each other. Chapter 58 · Non-Euclidean geometries
saddle point
A stationary point in every neighbourhood of which the function takes both larger and smaller values than at the point itself; on a map, a pass. Chapter 32 · Functions of several variables
sample
The part of the population that was actually measured; the whole is judged from it. Chapter 49 · Statistics
sample space
The set Ω of all possible outcomes of an experiment, listed so that each time the experiment is performed exactly one of them occurs. Chapter 47 · Probability
sample standard deviation
The square root of the sum of squared deviations from the mean, divided by n − 1: s = √(Σ(xᵢ − x̄)²/(n − 1)); the typical distance of the sample numbers from their mean. Chapter 49 · Statistics
sampling bias
A systematic difference between the sample and the population (because of how it was selected or who agreed to answer); it doesn’t shrink as the sample grows. Chapter 49 · Statistics
scientific notation
Writing a number as a·10ⁿ, where 1 ≤ a < 10 and n is an integer. Chapter 12 · Powers and logarithms
second derivative
The derivative of the derivative: f″(x) = (f′(x))′. It shows how fast the slope of the graph is changing. Chapter 27 · Extrema and optimization
Secret sharing
A way to hand out a secret to n participants so that any k of them can recover it, while fewer participants learn nothing about it. Chapter 14 · Polynomials
self-information
A measure of the surprise of an event of probability p: I = −log₂ p bits; an event of probability 1/2 carries 1 bit, 1/8 carries 3 bits, a certain event 0. Chapter 50 · Markov chains and information
self-similar
A set made up of several scaled-down copies of itself (the Sierpiński triangle of three copies at half size). Chapter 59 · Chaos and fractals
sensitive dependence on initial conditions
A property of a system: initial states that are arbitrarily close eventually move a noticeable distance apart. Chapter 59 · Chaos and fractals
Sensitivity
The probability that a test gives a positive result for someone who has the disease: P(+ | ill). Chapter 47 · Probability
sequence
Numbers indexed by the natural numbers: a₁, a₂, a₃, …; in other words, a function defined on the natural numbers. Chapter 13 · Sequences and induction
series
An infinite sum a₁ + a₂ + a₃ + … made up of the terms of a sequence; written Σaₙ. Chapter 30 · Series
set
A collection of objects (elements) regarded as a single whole; a set is completely determined by which elements belong to it. The notion is taken as primitive and isn’t defined in terms of others. Chapter 51 · Logic and sets
shape of constant width
A convex figure for which the distance between any two parallel supporting lines is the same. Chapter 19 · The circle and the number π
shear
The linear transformation (x, y) ↦ (x + ky, y): each point moves along the x-axis by a distance proportional to its height y. Chapter 35 · Matrices as transformations
sieve of Eratosthenes
A way to list all primes up to N: take the first number not yet crossed out, declare it prime and cross out its multiples, until its square exceeds N. Chapter 3 · Prime numbers
sigma-algebra
A family of subsets of a set X that contains X, contains the complement of each of its sets, and contains the union of each countable collection of its sets. Chapter 54 · Measure and the Lebesgue integral
significance level
The threshold α with which the p-value is compared (most often 0.05); the probability of rejecting the null hypothesis when it’s true. Chapter 49 · Statistics
similar
Square matrices A and P⁻¹AP with invertible P; they are matrices of the same linear map in different bases. Chapter 37 · Vector spaces
Similar triangles
Triangles whose corresponding angles are equal and whose corresponding sides are proportional. Chapter 17 · The triangle
similarity transformation
A transformation of the plane that multiplies all distances by the same number k > 0; it is a homothety followed by a motion. Chapter 23 · Symmetry and transformations
simple function
A measurable function with finitely many values: c₁ on A₁, …, cₖ on Aₖ; its Lebesgue integral is c₁μ(A₁) + … + cₖμ(Aₖ). Chapter 54 · Measure and the Lebesgue integral
simply connected
A shape in which any closed loop can be shrunk continuously to a point without leaving the shape. Chapter 57 · Topology
Simpson’s paradox
A situation in which a relationship present in each of the groups disappears or reverses when the groups are combined; it arises when the populations being compared are distributed differently among the groups. Chapter 49 · Statistics
sine
In a right triangle, the ratio of the opposite leg to the hypotenuse; for any angle, the y-coordinate of the point of the unit circle rotated by that angle. Chapter 21 · Trigonometry
singular
A square matrix with zero determinant; its transformation squashes space into fewer dimensions and sends some non-zero vector to zero. Chapter 35 · Matrices as transformations
singular value decomposition
Writing a matrix as A = UΣVᵀ, where U and V are orthogonal and Σ is rectangular diagonal with non-negative numbers in decreasing order; the singular value decomposition, SVD. Chapter 39 · Orthogonality, least squares and SVD
singular values
The numbers σ₁ ≥ σ₂ ≥ … ≥ 0 on the diagonal of Σ in the singular value decomposition; the square roots of the eigenvalues of AᵀA, the lengths of the semi-axes of the ellipse that is the image of the unit sphere. Chapter 39 · Orthogonality, least squares and SVD
slope
The number k in the line y = kx + b: how much y changes when x increases by one. Chapter 9 · Functions and graphs
solvable
A group in which the chain ‘the subgroup generated by commutators; the subgroup generated by its commutators; …’ reaches the group consisting of the identity alone in finitely many steps. Chapter 43 · Galois theory
solvable by radicals
An equation whose roots are expressed through its coefficients by addition, subtraction, multiplication, division and taking roots of natural degree. Chapter 43 · Galois theory
sophism
An argument that looks correct but leads to a false conclusion; the mistake in it is hidden. Chapter 16 · The Elements: axioms and constructions
span
The set of all linear combinations of given vectors; the smallest subspace containing them. Chapter 37 · Vector spaces
spanning tree
A tree made of edges of a graph and passing through all its vertices. Chapter 46 · Graphs
special products
Ready-made rules for expanding brackets: the square and cube of a sum or difference, the difference of squares, and the sum and difference of cubes. Chapter 7 · Letters for numbers
Specificity
The probability that a test gives a negative result for someone who doesn’t have the disease: P(− | healthy); 1 − specificity is the share of false alarms. Chapter 47 · Probability
spectrum
The set of amplitudes (and phases) of the harmonics that make up a signal; it shows how much of each frequency the signal contains. Chapter 33 · Fourier series
spectrum
The set of all eigenvalues of a square matrix (including complex ones). Chapter 38 · Eigenvectors
spectrum
For a bounded operator T, the set of numbers λ for which T − λI has no bounded inverse; for matrices it coincides with the set of eigenvalues. Chapter 55 · Functional analysis
sphere
The surface consisting of all points of space at a given distance R from a given point, the centre. Chapter 20 · Area and volume
spherical excess
The difference α + β + γ − π between the angle sum of a spherical triangle and that of a plane one; the area of the triangle equals the excess times R². Chapter 58 · Non-Euclidean geometries
standard deviation
σ = √(Var X): the square root of the variance; measures the typical deviation of a variable from its expectation in the same units as the variable itself. Chapter 48 · Random variables
standard error
The standard deviation of the sample mean, σ/√n; in practice it’s estimated as s/√n. The typical error of the mean of a sample of size n. Chapter 49 · Statistics
statement
A declarative sentence of which it makes sense to say that it’s true or false, and exactly one of the two holds. Chapter 51 · Logic and sets
stationary distribution
A probability distribution π that a Markov chain leaves unchanged: Pπ = π; an eigenvector of the transition matrix with eigenvalue 1. Chapter 38 · Eigenvectors
stationary points
A point at which all the partial derivatives of a function are zero: ∇f = 0. A candidate for an extremum; it may also be a saddle. Chapter 32 · Functions of several variables
subgroup
A subset of a group that is itself a group under the same operation: it contains the product of any two of its elements, the identity, and the inverse of each of its elements. Chapter 40 · Groups
subsequence
A sequence a_{n₁}, a_{n₂}, … with indices n₁ < n₂ < …: some of the terms of the original sequence without rearranging them. Chapter 53 · Rigorous analysis
subset
A set A is a subset of B (A ⊂ B) if every element of A belongs to B. Chapter 51 · Logic and sets
subspace
A non-empty subset of a vector space that is closed under addition and multiplication by numbers; it is a vector space itself. Chapter 37 · Vector spaces
sum of the infinite geometric progression
The number that the sums of the first n terms of a geometric progression come arbitrarily close to; for |q| < 1 it equals b₁/(1 − q). Chapter 13 · Sequences and induction
surface
A shape each point of which has a neighbourhood homeomorphic to a disc. A closed surface has no boundary and splits into finitely many polygons. Chapter 57 · Topology
surjection
A map f: A → B under which every element of B is the image of some element of A. Chapter 51 · Logic and sets
symmetric
A square matrix equal to its transpose: S^T = S. Chapter 38 · Eigenvectors
symmetric group
The group of all permutations of n objects under composition; written S_n, it has n! elements. Chapter 40 · Groups
symmetry of a figure
A motion that takes a figure to itself; the identity is a symmetry of any figure. Chapter 23 · Symmetry and transformations
syndrome
The result of the checks applied to a received word, s = Hr; it depends only on the error, not on the message, and the decoder finds the error from it. Chapter 42 · Rings, fields and codes
system of equations
Several equations that must hold at the same time; a solution of the system is a set of values of the unknowns that makes every equation true. Chapter 11 · Systems of equations and inequalities
tangent
A line that has exactly one point in common with a circle, the point of contact. Chapter 19 · The circle and the number π
tangent
The ratio of the sine of an angle to its cosine; in a right triangle, the ratio of the opposite leg to the adjacent one. Chapter 21 · Trigonometry
tangent plane
The plane z = f(a, b) + fₓ(a, b)(x − a) + f_y(a, b)(y − b), which approximates the surface of a differentiable function near (a, b) with an error small compared with the distance to the point. Chapter 32 · Functions of several variables
tangent to the graph
The line through the point (a, f(a)) of a graph with slope f′(a): y = f(a) + f′(a)(x − a); the limiting position of the secants. Chapter 26 · The derivative
tangent vector field
A rule assigning to each point of a surface a vector tangent to it at that point; a hairstyle on a surface is a continuous tangent field. Chapter 57 · Topology
tautology
A logical formula that’s true for any values of its variables, for example A ∨ ¬A. Chapter 51 · Logic and sets
Taylor polynomial
The polynomial Tₙ(x) = Σ f⁽ᵏ⁾(a)/k! · (x − a)ᵏ over k from 0 to n; at the point a it has the same value and first n derivatives as f. Chapter 30 · Series
Taylor series
The power series Σ f⁽ᵏ⁾(a)/k! · (x − a)ᵏ of a function f centred at the point a; for a = 0 it’s also called the Maclaurin series. Chapter 30 · Series
theorem
A statement for which a proof has been found. Chapter 0 · What mathematics is
Thomae’s function
The function equal to 1/q at the point p/q (a fraction in lowest terms, q ≥ 1) and 0 at irrational points; continuous at every irrational point and discontinuous at every rational one. Chapter 53 · Rigorous analysis
three-dimensional manifold
A shape each point of which has a neighbourhood homeomorphic to a ball of n-dimensional space; surfaces are two-dimensional manifolds. Chapter 57 · Topology
topological invariant
A property or number that’s the same for homeomorphic shapes; if two shapes differ in it, they aren’t homeomorphic. Chapter 57 · Topology
trace
The sum of the diagonal entries of a square matrix, tr A; it equals the sum of its eigenvalues. Chapter 38 · Eigenvectors
transcendental
A number that is not a root of any equation with integer coefficients (not all zero); for example π and e. Chapter 6 · Root two and the real numbers
transition matrix
The matrix of transition probabilities of a Markov chain: the entry p_ij is the probability of moving from state j to state i; the entries are non-negative and each column adds up to 1. Chapter 38 · Eigenvectors
translation
A motion that shifts every point by the same vector. Chapter 23 · Symmetry and transformations
transpose
The matrix Aᵀ obtained from A by turning rows into columns: the entry in row i and column j of Aᵀ equals the entry in row j and column i of A. Chapter 35 · Matrices as transformations
transpositions
A permutation that swaps two elements and leaves the rest in place; a cycle of length 2. Chapter 40 · Groups
tree
A connected graph without cycles; a tree with V vertices has exactly V − 1 edges. Chapter 46 · Graphs
triangle inequality
In any triangle each side is less than the sum of the other two: a < b + c. Chapter 17 · The triangle
triangular numbers
The sum 1 + 2 + … + n = n(n + 1)/2; that many balls fit in a triangle of side n: 1, 3, 6, 10, 15, … Chapter 13 · Sequences and induction
Triangulation
A way to measure large distances: the terrain is covered with a chain of triangles in which all the angles and one side (the base line) are measured, and the other sides are computed by the law of sines. Chapter 21 · Trigonometry
tricolourable
A property of a knot: the arcs of its diagram can be coloured with three colours so that at each crossing the three arcs are all the same colour or all different, and at least two colours are used. Chapter 57 · Topology
trigonometric polynomial
A function of the form a₀/2 + Σ (aₙ cos nx + bₙ sin nx) with finitely many terms; it has period 2π. Chapter 33 · Fourier series
Trisecting an angle
Dividing an angle into three equal parts; impossible with compass and straightedge for an arbitrary angle (Wantzel, 1837). Chapter 16 · The Elements: axioms and constructions
truth table
A table giving the value of a formula for each assignment of values (true or false) to its variables; with n variables it has 2ⁿ rows. Chapter 51 · Logic and sets
Turing machine
An imaginary computer: an infinite tape of cells holding symbols, a head over one cell, and a finite table of rules ‘in state q, reading symbol a, write b, move left or right, go to state r’. Chapter 56 · Gödel, Turing and the limits of proof
Twin primes
A pair of primes that differ by 2, such as 11 and 13. Chapter 3 · Prime numbers
type I error
Rejecting a true null hypothesis, announcing an effect that doesn’t exist (a ‘false alarm’); with threshold α this happens with probability at most α. Chapter 49 · Statistics
type II error
Failing to reject a false null hypothesis, that is, missing an effect that exists. Chapter 49 · Statistics
undecidable
A general problem (a family of yes-or-no questions) for which there’s no algorithm deciding every particular case correctly. Chapter 56 · Gödel, Turing and the limits of proof
uniformly continuous
A property of a function on a set D: for every ε > 0 there’s a δ > 0 such that |f(x) − f(y)| < ε for all x, y in D with |x − y| < δ; one δ for all points. Chapter 53 · Rigorous analysis
union
The set A ∪ B of all elements that belong to A or to B (to at least one of them). Chapter 51 · Logic and sets
unit ball
The set of vectors of a normed space whose norm is at most 1: {x : ‖x‖ ≤ 1}. Chapter 55 · Functional analysis
unit circle
The circle of radius 1 centred at the origin; the sine and cosine of any angle are defined on it. Chapter 21 · Trigonometry
upper bound
A number M not less than any element of a set A; a set that has an upper bound is called bounded above. Chapter 53 · Rigorous analysis
variable
A letter in place of which different numbers can be substituted. Chapter 7 · Letters for numbers
variance
Var X = E(X − E X)²: the mean value of the squared deviation of a variable from its expectation; a measure of spread. Chapter 48 · Random variables
vector
A directed segment; two directed segments of the same length and direction define the same vector. Chapter 22 · Vectors and coordinates
vector space
A set whose elements can be added to one another and multiplied by numbers so that eight axioms hold: commutativity and associativity of addition, a zero vector and opposites, 1·v = v and three laws for numbers. Chapter 37 · Vector spaces
vertex
The point of the parabola y = ax² + bx + c with x-coordinate x₀ = −b/(2a): the lowest point when a > 0, the highest when a < 0. Chapter 10 · Quadratic equations
vertices
A point of a graph; in problems, a crossroads, a person, a town, a country. Chapter 46 · Graphs
volume
A number assigned to a solid so that it is non-negative, the same for congruent solids, additive when the solid is cut into parts, and equal to 1 for the cube of edge 1. Chapter 20 · Area and volume
wallpaper pattern
A pattern on the plane that goes to itself under translations by two non-parallel vectors; all its translations are made up of whole multiples of two shortest ones. Chapter 23 · Symmetry and transformations
weighted graph
A graph each of whose edges carries a number, its weight: a length, a time, a price. Chapter 46 · Graphs
zero divisor
A non-zero element a of a ring for which there is a non-zero b with ab = 0; for example, 2 and 3 in Z₆. Chapter 42 · Rings, fields and codes
zeros of the function
A value of the argument at which the function equals zero; on the graph, the x-coordinate of a common point of the graph and the x-axis. Chapter 9 · Functions and graphs
Mathematics

Chapters

Introduction

  1. 0 What mathematics is

I Numbers

  1. 1 Counting
  2. 2 Zero and minus
  3. 3 Primes
  4. 4 Divisibility
  5. 5 Fractions
  6. 6 Root two

II Algebra

  1. 7 Letters
  2. 8 Equations
  3. 9 Functions and graphs
  4. 10 Quadratic equations
  5. 11 Systems and inequalities
  6. 12 Powers and logarithms
  7. 13 Sequences
  8. 14 Polynomials
  9. 15 Complex numbers

III Geometry

  1. 16 Euclid's Elements
  2. 17 The triangle
  3. 18 Pythagoras' theorem
  4. 19 The circle and π
  5. 20 Area and volume
  6. 21 Trigonometry
  7. 22 Vectors
  8. 23 Symmetry
  9. 24 Conic sections

IV Calculus

  1. 25 Limits
  2. 26 The derivative
  3. 27 Extrema
  4. 28 The integral
  5. 29 The number e
  6. 30 Series
  7. 31 Differential equations
  8. 32 Several variables
  9. 33 Fourier series
  10. 34 Complex analysis

V Linear algebra

  1. 35 Matrices
  2. 36 Gaussian elimination
  3. 37 Vector spaces
  4. 38 Eigenvectors
  5. 39 Least squares and SVD

VI Structures

  1. 40 Groups
  2. 41 Remainders and ciphers
  3. 42 Rings and fields
  4. 43 Galois theory
  5. 44 Zeta and curves

VII Chance & data

  1. 45 Combinatorics
  2. 46 Graphs
  3. 47 Probability
  4. 48 Random variables
  5. 49 Statistics
  6. 50 Markov and information

VIII Foundations

  1. 51 Logic and sets
  2. 52 Infinities
  3. 53 Rigorous analysis
  4. 54 Measure and integral
  5. 55 Functional analysis
  6. 56 Gödel and Turing

IX Horizons

  1. 57 Topology
  2. 58 Non-Euclidean geometries
  3. 59 Chaos and fractals
  4. 60 The frontier