Glossary of mathematical terms — Queen of the Sciences

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

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

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

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

bit

A binary place: a single digit 0 or 1; the smallest unit of information. Chapter 1 · Counting: from notches to bits

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

coefficient

The numerical factor in front of the letter part of a term: in 3x² the coefficient is 3. Chapter 7 · Letters for numbers

common logarithm

The logarithm to base 10: lg b = log₁₀ b. Chapter 12 · Powers and logarithms

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

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

conjecture

A statement that has been neither proved nor disproved yet. Chapter 0 · What mathematics is

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

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

coprime

Numbers with no common divisor other than 1, that is, gcd(a, b) = 1. Chapter 4 · Divisibility and Euclid's algorithm

counterexample

A case in which a general statement fails; a single counterexample is enough to disprove the statement. Chapter 0 · What mathematics is

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

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

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

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

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

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

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

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

equivalent

Equations that have the same set of roots. Chapter 8 · Equations: the art of balance

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

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

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

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

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

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

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

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

function

A rule that assigns to each number x in some set exactly one number y = f(x). Chapter 9 · Functions and graphs

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

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

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

greatest common divisor

The largest natural number that divides both given numbers; written gcd(a, b). Chapter 4 · Divisibility and Euclid's algorithm

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

identity

An equation with variables that holds for all values of the variables at which both sides make sense. Chapter 7 · Letters for numbers

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

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

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

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

least common multiple

The smallest natural number divisible by both given numbers; written lcm(a, b). Chapter 4 · Divisibility and Euclid's algorithm

like terms

Terms that differ only in their numerical coefficient, such as 3x and −5x. Chapter 7 · Letters for numbers

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 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

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 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

lowest terms

A fraction a/b whose numerator and denominator are coprime: gcd(a, b) = 1. Chapter 5 · Fractions

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

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

mixed number

A notation like 2⅓: a whole part and a proper fraction side by side; it means their sum. Chapter 5 · Fractions

monic

A quadratic equation with leading coefficient 1: x² + px + q = 0. Chapter 10 · Quadratic equations

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

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

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

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

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

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

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

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

range

The set of all values a function takes on its domain. Chapter 9 · Functions and graphs

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

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

root

A value of the unknown that turns the equation into a true numerical equality. Chapter 8 · Equations: the art of balance

scientific notation

Writing a number as a·10ⁿ, where 1 ≤ a < 10 and n is an integer. Chapter 12 · Powers and logarithms

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

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

slope

The number k in the line y = kx + b: how much y changes when x increases by one. Chapter 9 · Functions and 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

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

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

theorem

A statement for which a proof has been found. Chapter 0 · What mathematics is

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

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

Twin primes

A pair of primes that differ by 2, such as 11 and 13. Chapter 3 · Prime numbers

variable

A letter in place of which different numbers can be substituted. Chapter 7 · Letters for numbers

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

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