Mathematics: Mathematics is a field encompassing numbers, formulas, shapes, quantities, and their changes. It comprises subdisciplines like number theory, algebra, geometry, and analysis. A precise definition for the discipline is still debated among mathematicians.

Scientific notation: Scientific notation is a method used by scientists, mathematicians, and engineers to express extremely large or small numbers. It avoids long decimal strings by using a base ten format. This notation simplifies calculations and is referred to as scientific form or standard index form. It's also known as "SCI" display mode on scientific calculators.

Pure mathematics: Pure mathematics is the examination of mathematical concepts without consideration of their practical applications. The study involves investigating logical outcomes of fundamental principles, with the focus centered on intellectual challenge and aesthetic appeal rather than real-world utility.

Arithmetic: Arithmetic is a branch of mathematics that studies numerical operations like addition, subtraction, multiplication, and division. It also includes exponentiation, roots, and logarithms. It operates on different types of numbers, such as integers, rational numbers, and real numbers. The numeral system used can be decimal or binary. Some arithmetic systems work with mathematical objects other than numbers.

Addition: Addition is a basic arithmetic operation that combines two numbers to give a total or sum. It is one of the four fundamental operations, alongside subtraction, multiplication, and division. An example with apples demonstrates adding two columns of three and two apples to get a total of five apples, which is represented by the equation 3 + 2 = 5.

Subtraction: Subtraction is an arithmetic operation that involves removing objects from a collection. It is one of the four basic arithmetic operations – addition, subtraction, multiplication, and division. Subtraction is commonly associated with natural numbers, but it can also involve negative numbers, fractions, decimals, and other mathematical concepts. Its purpose is to find the difference or decrease between quantities.

Multiplication: Multiplication is a basic arithmetic operation that combines two or more numbers to give a product. It is one of the essential mathematical operations, along with addition, subtraction, and division.

Division (mathematics): Division is a fundamental arithmetic operation alongside addition, subtraction, and multiplication. It involves dividing a dividend by a divisor, leading to a quotient as the result.

Euclidean algorithm: The Euclidean algorithm is a mathematical method for finding the greatest common divisor of two numbers. It was described by the ancient Greek mathematician Euclid and is one of the oldest and most widely used algorithms. It simplifies fractions and is used in various number-theoretic and cryptographic calculations.

Fraction: A fraction represents part of a whole or a number of equal parts. It consists of a numerator (above a line) and a denominator (below the line). The numerator indicates the number of equal parts, while the denominator represents how many of those parts make up a whole. A common example is 3/4, where 3 represents 3 equal parts and 4 indicates that 4 parts make up a whole.

Percentage: A percentage is a number or ratio out of 100, commonly represented by the % sign. It is a dimensionless unit used to express proportions.

Modular arithmetic: Modular arithmetic is a mathematical system that deals with integers. It involves numbers "wrapping around" when they reach a certain value known as the modulus. Carl Friedrich Gauss developed the modern approach to modular arithmetic, which was introduced in his book Disquisitiones Arithmeticae in 1801.

Constant (mathematics): A constant in mathematics is a non-changing quantity, which can be either a fixed number or a function that does not vary with the main variable(s). It can be referred to as a mathematical or physical constant.

Pi: Pi is a mathematical constant (approximately 3.14159) representing the ratio of a circle's circumference to its diameter. It is irrational, meaning it is not expressible as a ratio of two integers. Pi is transcendental, making it impossible to solve the ancient challenge of squaring the circle. Its decimal digits seem random, though no proof of this has been found.

e(mathematical constant): The constant e, approximately equal to 2.71828, serves as the base for natural logarithms. It can be derived as the limit of (1 + 1/n)n as n approaches infinity and is used in compound interest calculations. Additionally, e can be obtained as the sum of an infinite series.

Imaginary unit: The imaginary unit, denoted as 'i', is a solution to the equation x^2 + 1 = 0. It allows for the extension of real numbers to complex numbers, which involve both real and imaginary components. An example of a complex number is 2 + 3i.

Golden ratio: The golden ratio is a mathematical concept where two quantities are in a specific ratio that is equal to the ratio of their sum to the larger quantity. It can be expressed algebraically as a/b = (a+b)/a.

−1: The term '-1' in mathematics refers to the additive inverse of 1, denoting the number that, when added to 1, results in the additive identity, 0. It is the negative integer between -2 and 0.

0: Zero, denoted as '0', is a number that represents an empty quantity. It is the additive identity for various number systems, such as integers, rational numbers, real numbers, and complex numbers. When added to any number, it does not change the value of that number. Multiplying any number by zero results in zero, and division by zero is undefined in arithmetic.

1: '1' is a number that represents a single or the only entity. It is the first positive integer, a numerical digit, and a unit of counting or measurement. In some conventions, it is the first natural number while in others it is considered the second, following '0'.

Equation: An equation is a mathematical formula that connects two expressions with an equals sign. Different languages may have varying definitions, such as French considering equations to have variables. In English, any well-formed formula with two related expressions is considered an equation.

Equation solving: Equation solving involves finding values that satisfy the given equation. These values are called solutions or roots of the equation. A solution is obtained by assigning values to the unknown variables that make the equation true. The solution set refers to the collection of all solutions for an equation.

Polynomial: A polynomial is a mathematical expression made up of variables and coefficients, using addition, subtraction, multiplication, and non-negative integer powers of variables. It can have multiple indeterminates and is written in the form of terms. Examples include x^2 - 4x + 7 and x^3 + 2xyz^2 - yz + 1.

Infinity: Infinity refers to something without limits or restrictions, surpassing all natural numbers. Its symbol is the infinity symbol.

Hilbert's problems: Hilbert's problems, 23 unsolved mathematical problems published by David Hilbert in 1900, had a significant impact on 20th-century mathematics. Hilbert presented ten of the problems at a conference in Paris, and the full list was later published in English translation in the Bulletin of the American Mathematical Society. These problems were influential and widely discussed in the field.

Mathematical proof: A mathematical proof is a deductive argument that logically guarantees a mathematical statement. It uses established statements and axioms with rules of inference. Proofs establish logical certainty and must hold true in all cases. Conjectures or hypotheses are statements believed to be true but not proven.

Formula: A formula is a symbolic representation used in science to convey information concisely, such as in mathematics or chemistry. It is a way of expressing the relationship between given quantities.

Theorem: A theorem is a proven statement in mathematics, backed by a logical argument that uses inference rules and established axioms and theorems.

Conjecture: A conjecture in mathematics is a tentative proposition without proof. Iconic conjectures like the Riemann hypothesis have driven the development of new mathematical areas.

Number: A number is a mathematical concept used for counting, measuring, and labeling. It can be represented by number words or symbols called numerals. Numerals are organized in numeral systems, such as the Hindu-Arabic system, which allows for the representation of any non-negative integer using ten fundamental digits. Numerals are used not only for counting but also for labeling, ordering, and coding. It is important to note that a numeral and the number it represents are often used interchangeably in everyday language.

Natural number: Natural numbers are positive integers used for counting, such as 1, 2, 3, and so on. They can sometimes include zero, and are distinct from negative numbers and fractions. The term 'counting numbers' may be used to distinguish them from other types of numbers.

Integer: An integer is a number that can be zero, positive, or negative. Negative integers are the opposites of positive integers. The entire group of integers is often represented with the symbol Z.

Negative number: A negative number in mathematics represents an opposite and is less than zero in the real number system. They are used to indicate losses or deficiencies, and can also represent debts or negative assets. Negative numbers describe values below zero on scales like Celsius and Fahrenheit. The laws of arithmetic for negative numbers reflect the concept of opposites, such as -(-3) being equal to 3.

Parity (mathematics): Parity in mathematics refers to the classification of an integer as even or odd. An even number is divisible by 2, while an odd number is not divisible. Some examples of even numbers are −4, 0, and 82, while odd numbers include −3, 5, 7, and 21.

Real number: Real numbers are mathematical entities used for measuring continuous quantities like distance, time, and temperature. They can have infinitely precise decimal representations, allowing for arbitrarily small differences between values.

Complex number: A complex number is an extension of the real numbers that includes an imaginary unit denoted as i. It can be expressed as a sum of a real part and an imaginary part. Complex numbers are considered as fundamental in mathematics and science, despite being termed "imaginary" historically. They play a crucial role in describing the natural world.

Rational number: A rational number is a number that can be expressed as a fraction of two integers. It includes integers and is denoted by Q.

Irrational number: Irrational numbers are real numbers that cannot be expressed as a fraction. Line segments with irrational ratios have no common measure, meaning their lengths cannot be expressed as integer multiples of a single length.

Imaginary number: An imaginary number is obtained by multiplying a real number with the imaginary unit "i" where i² = -1. The square of an imaginary number is always negative. Zero is considered both real and imaginary.

Order of operations: Order of operations refers to the rules followed in mathematics and computer programming to determine the sequence of operations performed when evaluating a mathematical expression. These rules dictate the priority of each operation to ensure accurate calculation.

Positional notation: 'Positional notation' is an extension of the Hindu-Arabic numeral system, where the value of a digit is determined by its position in the number. In contrast to ancient systems like Roman numerals, modern positional systems (such as the decimal system) assign different values to digits based on their position in the number string. For example, in the number 555, each digit represents different quantities of hundreds, tens, and units.

Numerical digit: A numerical digit is a symbol or combination of symbols used to represent numbers in a positional numeral system. The term "digit" originates from the correlation between the ten symbols of the base 10 numeral system and the ten digits of human hands.

Decimal: The decimal system is the standard way of representing both whole and fractional numbers, derived from the Hindu-Arabic numeral system. It is commonly referred to as decimal notation.

Arabic numerals: Arabic numerals (0-9) are widely used for writing numbers in a decimal system. They are also employed in other bases like octal and used for non-numerical purposes such as trademarks or license plates identifiers.

Roman numerals: Roman numerals originated in ancient Rome and were widely used in Europe until the Late Middle Ages. They consist of letters from the Latin alphabet that represent specific numerical values. The modern style is limited to seven letters for writing numbers.

Binary number: A binary number is a base-2 numeral system that uses only two symbols, 0 and 1, to express numbers.

Variable (mathematics): In mathematics, a variable is a symbol representing a mathematical object like a number, function, or set element. It can represent various mathematical entities, such as vectors, matrices, arguments, or sets, for the purpose of mathematical equations and expressions.

Function (mathematics): A mathematical function assigns each element of one set to a unique element of another set. The first set is called the domain, and the second set is called the codomain.

Inverse function: The inverse function is a mathematical function that reverses the operation of another function. It exists only if the original function is bijective, and is denoted as ''.

Implicit function: An implicit function is a mathematical relation written as where R is a function of multiple variables. It is used to describe equations without directly solving for one variable. The unit circle provides an example of an implicit equation, often represented by x^2 + y^2 = 1.

Function composition: Function composition is a mathematical operation that combines two functions to create a new function. It takes functions f and g and produces a function h = g ∘ f. The resulting function applies g to the result of applying f to x. It is denoted g ∘ f : X → Z and defined by (g ∘ f )(x) = g(f(x)) for all x in X. This allows for the relationship between the functions to be expressed in terms of mappings from one domain to another.

Graph of a function: A graph of a function is a set of ordered pairs that represent the relationship between inputs and outputs of the function. Typically represented as points on a plane, these pairs create a curve known as a plot.

Parametric equation: A parametric equation is a mathematical expression that defines quantities in terms of one or more parameters. It is commonly used to describe the coordinates of points on a curve or surface, known as a parametric curve or surface. These equations are referred to as a parameterization or parametric representation of the object.

Limit of a function: The limit of a function is a crucial concept in calculus and analysis that describes the behavior of the function near a specific input, whether or not it belongs to the function's domain.

Exponentiation: Exponentiation is a mathematical operation involving a base and an exponent. It is represented as bn, where b is the base and n is the power. When n is a positive integer, exponentiation is equivalent to repeated multiplication of the base.

Exponential function: The exponential function is a widely used mathematical function denoted by or . It is typically used to represent positive values of a real variable but can also be extended to complex numbers, matrices, and Lie algebras. It originated from the operation of taking powers of a number and can be defined for all real arguments, including irrational numbers. Due to its extensive use in various branches of mathematics, it is regarded as the most important function by mathematician Walter Rudin.

Logarithm: The logarithm is the inverse of exponentiation in mathematics. It calculates the exponent to which a given base must be raised to achieve a certain number. For example, log10 (1000) equals 3, as 1000 can be expressed as 10 raised to the power of 3. The logarithm is denoted as logb (x), and can also be written as logb x or log x when the base is implied or not important. It is commonly used in big O notation and has various applications in mathematical calculations.

nth root: In mathematics, the nth root is an operation that involves two numbers: the radicand (x) and the index (n). It is written as √x, where x is the radicand and n is the index. The nth root of a number x is a number r that, when raised to the power of n, equals x.

Square root: A square root is a mathematical term representing a number that, when multiplied by itself, equals a given number. For instance, 4 and -4 are square roots of 16.

Trigonometric functions: Trigonometric functions are real functions that relate angles of right-angled triangles to ratios of side lengths. They have widespread applications in geometry-based sciences like navigation, mechanics, and astronomy. Trigonometric functions are simple periodic functions and are often used in the study of periodic phenomena using Fourier analysis.

Hyperbolic functions: Hyperbolic functions are mathematical functions that are similar to trigonometric functions, but are defined using the hyperbola instead of the circle. They are represented by the points (cosh t, sinh t), which form the right half of the unit hyperbola. The derivatives of these functions, cosh(t) and sinh(t), are also related to each other, similar to how sin(t) and cos(t) are related.

Gamma function: The gamma function is a generalized version of the factorial function for complex numbers, excluding non-positive integers. It is defined for all complex numbers except for non-positive integers.

Factorial: The factorial is the product of all positive integers up to a given number. It is denoted by the exclamation mark (!). For a non-negative integer n, n! is equal to the product of n and all smaller positive integers.

Geometry: Geometry is a mathematical branch exploring space's properties like distance, shape, size, and relative position of figures. It is one of the oldest branches and encompasses Euclidean geometry, focusing on fundamental concepts like point, line, plane, angle, surface, and curve.

Coordinate system: A coordinate system is a system that uses numbers or coordinates to determine the position of points in geometry. The order of the coordinates matters, and they can be identified by their position or a letter. Coordinates can be real or complex numbers and are used to translate geometric problems into numerical problems. This is the basis of analytic geometry.

Cartesian coordinate system: A Cartesian coordinate system is a geometric system that uniquely identifies points with pairs of real numbers. These numbers represent the distances from two perpendicular lines known as axes. The system includes an origin at (0, 0).

Polar coordinate system: The polar coordinate system is a way to locate points on a plane using a distance from a reference point and an angle from a reference direction. The reference point is called the pole, and the distance is called the radial coordinate or radius. The angle is called the angular coordinate or polar angle. Angles are usually expressed in degrees or radians.

Dimension: In physics and mathematics, dimension refers to the number of coordinates needed to locate a point in a mathematical space. For example, a line is one-dimensional (1D) as it requires only one coordinate, while the surface of a cylinder or sphere is two-dimensional (2D) and requires two coordinates. A three-dimensional (3D) space, like the inside of a cube, cylinder, or sphere, requires three coordinates to locate a point within.

Point (geometry): A point in geometry represents an exact position without size in physical or mathematical spaces. It is the fundamental indivisible element of space, and can be determined by the intersection of curves or surfaces.

Line (geometry): A line in geometry is an idealized straight object that extends infinitely in both directions, with no width, depth, or curvature. It can be thought of as a one-dimensional space and may exist within higher-dimensional spaces. In everyday language, the term line can also refer to a part of a line that is bounded by two points, known as a line segment.

Curve: A curve is a mathematical object that can be likened to a non-straight line.

Euclidean plane: A Euclidean plane is a two-dimensional geometric space where a pair of real numbers determine the position of each point. It includes parallel lines and allows for the measurement of angles and the definition of circles based on distance metrics.

Plane (mathematics): A plane in mathematics is an infinite, flat two-dimensional space that is comparable to a point, line, or three-dimensional space.

Angle: An angle is a figure formed by two rays sharing a common endpoint. It can be formed in a plane or by the intersection of two planes. Angles can also be formed by intersecting curves. They are used to measure the separation or inclination between two lines or surfaces.

Right angle: A right angle is a 90-degree angle or /2 radians, representing a quarter turn. When a ray's endpoint is on a line and its adjacent angles are equal, it forms a right angle. The term originates from the Latin phrase angulus rectus, meaning "upright," referring to the vertical perpendicular to a horizontal base line.

Polygon: A polygon is a closed shape formed by connecting line segments together on a plane.

Triangle: A triangle is a three-sided polygon and one of the fundamental shapes in geometry. It consists of three corners (or vertices) and three sides (or edges). The interior of a triangle forms a two-dimensional region. The base, chosen arbitrarily, has an opposite vertex referred to as the apex.

Quadrilateral: A quadrilateral is a four-sided polygon with four edges and four corners. Its name comes from the Latin words quadri, meaning "four," and latus, meaning "side." It is also known as a tetragon, derived from Greek words meaning "four" and "corner," and as a quadrangle or 4-angle.

Square: A square is a regular quadrilateral with four equal sides and four equal angles. It can be seen as a rectangle with two equal-length adjacent sides. It has internal, central, and external angles of 90°, and its diagonals are all equal in length.

Conic section: A conic section is a curve formed when a plane intersects a cone's surface. The three main types are the hyperbola, parabola, and ellipse, with the circle considered a special case of the ellipse. Ancient Greek mathematicians extensively studied conic sections, with Apollonius of Perga being a significant contributor in understanding their properties.

Circle: A circle is a geometric shape formed by points in a plane equidistant from a center point called the radius.

Ellipse: An ellipse is a curved shape with two focal points, where the sum of the distances from any point on the curve to the focal points is constant. It is a generalized form of a circle, where the focal points are the same. The elongation of an ellipse is determined by its eccentricity, a numerical value.

Parabola: A parabola is a U-shaped plane curve that has mirror symmetry. It is defined by various mathematical descriptions which all represent the same curve.

Hyperbola: A hyperbola is a smooth curve in a plane, consisting of two mirror-image branches resembling infinite bows. It is formed by the intersection of a plane and a double cone, representing one of the three types of conic sections.

Three-dimensional space: A three-dimensional space refers to a mathematical space that requires three values to determine the position of a point. It is commonly known as the three-dimensional Euclidean space, which models physical space. It can also refer to a subset of space, a three-dimensional region, or a solid figure.

Shape: A shape is a visual representation of an object's form or external boundary, separate from qualities like color or texture. It pertains solely to the object's structure and does not consider its location, size, orientation, or reflection. A figure, on the other hand, encompasses both shape and size.

Sphere: A sphere is a three-dimensional object, similar to a circle in two dimensions. It consists of all points that are equidistant from a central point, called the center, with a radius denoting the distance. Spheres have been mentioned by ancient Greek mathematicians.

Polyhedron: A polyhedron is a 3D shape made up of flat faces, straight edges, and sharp corners.

Cube: A cube is a 3D shape with six square faces. It has three squares meeting at each corner and its net resembles a cross.

Trigonometry: Trigonometry is a branch of math that studies the relationships between angles and side lengths of triangles. It emerged during the 3rd century BC from geometry's application in astronomy. Trigonometric functions, like sine, use ratios of side lengths to calculate angles in right triangles. Ancient Greeks calculated chords, while Indian mathematicians developed early trigonometric tables.

Topology: Topology is a branch of mathematics that studies the properties of objects that remain unchanged under continuous deformations like stretching, bending, and twisting. It focuses on preserving shape without tearing, gluing, or passing through itself.

General topology: General topology is a fundamental branch of mathematics that explores set theory and the foundational aspects used in topology. It serves as the basis for various other branches like differential topology, geometric topology, and algebraic topology.

Topological space: A topological space is a geometrical space where closeness is defined without numeric distances. It consists of a set of points and a topology that defines neighborhoods satisfying axioms of closeness. The most common definition is through open sets, which is more easily manipulated.

Compact space: Compact space is a mathematical concept that generalizes closed and bounded subsets. It includes all limiting values of points and has no "punctures" or "missing endpoints". For instance, the open interval (0,1) is not compact because it excludes the limiting values of 0 and 1. On the other hand, the closed interval [0,1] is compact. The concept can be made precise in various ways, typically agreeing in metric spaces but not always in other topological spaces.

Manifold: A manifold is a mathematical concept representing a space that locally resembles Euclidean space near each point. It is a topological space where every point has a neighborhood that is similar to an open subset of Euclidean space.

Open set: An open set is a mathematical concept that extends the idea of an open interval in the real line. It encompasses sets where every point has a neighborhood entirely contained within the set.

Homology (mathematics): Homology in mathematics associates algebraic objects with other mathematical objects, like topological spaces. Originally defined in algebraic topology, it is now used in diverse contexts such as abstract algebra, groups, Lie algebras, Galois theory, and algebraic geometry.

Homotopy: Homotopy is a concept in topology, a branch of mathematics. It involves the continuous deformation of one function into another within a topological space. This deformation is known as a homotopy. Homotopy is used to define important invariants called homotopy groups and cohomotopy groups in algebraic topology.

Euler characteristic: The Euler characteristic is a mathematical concept that measures the shape or structure of a topological space, regardless of how it is distorted. It is a topological invariant and is often denoted by .

Knot theory: Knot theory is the mathematical study of knots, which are formed by joining the ends of a closed loop. It differs from everyday knots by being unable to be undone. Knots are represented as circles embedded in 3-dimensional space. Equivalent knots can be transformed into each other without cutting or passing the string through itself.

Poincaré conjecture: The Poincaré conjecture is a mathematical theorem regarding the description of a 3-dimensional hypersphere known as the 3-sphere. It deals with the boundaries of a unit ball in four-dimensional space.

Area: Area is a measure of size on a surface. It can refer to the size of a shape or the boundary of a three-dimensional object. It represents the amount of material or paint needed to cover a surface. Area is the two-dimensional equivalent of length or volume.

Volume: Volume is a numerical measure of three-dimensional space using SI or imperial units. It is related to length cubed and represents the capacity of a container. Additionally, the term "volume" can also refer to the corresponding region.

Length: Length is a quantifiable distance, with the base unit in the SI system being the meter. It serves as the foundation for other units of measurement.

Distance: Distance is a measure of the separation between objects or points, either in terms of physical length or based on other criteria. It can also represent the difference or degree of separation between similar objects. Mathematics formalizes distance using the concept of a metric space.

Metric space: A metric space is a mathematical concept that defines distance between points in a set. It is used to study mathematical analysis and geometry.

Euclidean geometry: Euclidean geometry, attributed to Euclid, is a mathematical system described in his textbook Elements. It is based on intuitively appealing axioms, from which numerous theorems are deduced. Euclid is recognized as the first to organize these propositions into a logical system with rigorous proofs.

Convex set: A convex set is a subset of a space that contains the line segment between any two points within the set. It is characterized by intersecting any line into a single line segment.

Pythagorean theorem: The Pythagorean theorem is a key concept in math that relates the sides of a right triangle. It states that the sum of the areas of the squares on the two shorter sides is equal to the area of the square on the longest side.

Non-Euclidean geometry: Non-Euclidean geometry is a branch of mathematics that encompasses two geometries related to Euclidean geometry. It involves either replacing the parallel postulate or relaxing the metric requirement. Hyperbolic and elliptic geometries are traditional non-Euclidean geometries obtained through replacing the parallel postulate. Relaxing the metric requirement leads to kinematic geometries associated with affine planes.

Symmetry: Symmetry refers to harmonious and balanced proportion. In mathematics, it specifically describes objects that remain unchanged under transformations like translation, reflection, rotation, or scaling. The term's everyday and mathematical meanings are interconnected and explored in this article.

Symmetry in mathematics: Symmetry in mathematics refers to the property of a mathematical object remaining unchanged under operations or transformations. It is not limited to geometry but is found in various branches of mathematics.

Analytic geometry: Analytic geometry, also called coordinate or Cartesian geometry, is a mathematical study that uses a coordinate system to analyze geometry. It differs from synthetic geometry.

Differential geometry: Differential geometry is a branch of mathematics that explores the geometry of smooth shapes and spaces called smooth manifolds. It draws techniques from differential calculus, integral calculus, linear algebra, and multilinear algebra. Its origins lie in the study of spherical geometry and later expanded to include astronomy, geodesy, and hyperbolic geometry. The field developed during the 18th and 19th centuries, primarily by studying plane and space curves and surfaces in three-dimensional Euclidean space.

Riemannian geometry: Riemannian geometry studies smooth manifolds with a Riemannian metric. It provides local measures of angles, length, surface area, and volume. It also allows for deriving global quantities by integrating local contributions.

Fractal: A fractal is a mathematical shape that shows intricate patterns at all levels of magnification, with a dimension surpassing its topological dimension. These self-similar shapes exhibit similar details as they are zoomed in, and if the replication is identical at every scale, it is known as affine self-similar. Fractal geometry is a part of measure theory in mathematics.

Tessellation: Tessellation, or tiling, is the process of covering a surface (often a plane) using geometric tiles without any overlaps or gaps. This mathematical concept can also be applied to higher dimensions and various geometries.

Algebra: Algebra is the math study of variables and their manipulation in formulas, originating from ancient Babylonian methods. It has become an integral part of various mathematical disciplines.

Elementary algebra: Elementary algebra, also called college algebra, covers fundamental algebraic concepts. It differs from arithmetic by introducing variables instead of working with specific numbers.

Quadratic equation: A quadratic equation is an algebraic equation that can be rearranged into standard form. It is characterized by having a variable raised to the power of 2 and includes terms of lower powers.

Linear algebra: Linear algebra is a mathematical branch studying linear equations. It deals with matrices, vectors, and linear transformations.

Linear equation: A linear equation is a mathematical equation that consists of variables and coefficients. The variables are typically represented by letters, while the coefficients are usually real numbers. The equation must have at least one nonzero coefficient to be meaningful. The coefficients can be considered as parameters and can be any arbitrary expressions, as long as they do not contain the variables.

Gaussian elimination: Gaussian elimination, also known as row reduction, is a mathematical algorithm used to solve systems of linear equations. It involves performing row-wise operations on a coefficient matrix to reach a matrix with zeros in the lower left-hand corner. This method can also be used to find the rank, determinant, and inverse of a matrix. The algorithm is named after Carl Friedrich Gauss and uses three types of elementary row operations: swapping rows, multiplying a row by a nonzero number, and adding a multiple of one row to another row.

Matrix (mathematics): In math, a matrix is a rectangular arrangement of numbers or symbols organized in rows and columns. It represents mathematical objects or their properties.

Determinant: The determinant is a scalar value that characterizes a square matrix and the linear map it represents. It is denoted as det(A) or |A|. The determinant is nonzero if the matrix is invertible and the linear map is an isomorphism. The determinant of a product of matrices is the product of their determinants.

Eigenvalues and eigenvectors: Eigenvalues and eigenvectors refer to special vectors that remain unchanged, except for scaling, when a linear transformation is applied to them. The eigenvector gets multiplied by a constant factor known as the eigenvalue, which represents the scaling factor.

Vector space: A vector space is a set of vectors that can be added and scaled by numbers. Scalars can be real numbers, complex numbers, or elements of any field. Vector addition and scalar multiplication must satisfy certain requirements. There are different types of vector spaces based on the kind of scalars used.

Euclidean vector: A Euclidean vector is a geometric object representing magnitude and direction. It can be added to other vectors using vector algebra. Typically represented by a directed line segment or arrow, it is denoted as 'A→B'.

Cross product: The cross product is a vector operation between two vectors in a three-dimensional space. It results in a vector perpendicular to the two input vectors and finds applications in various fields such as mathematics, physics, engineering, and computer programming. It is different from the dot product.

Dot product: The dot product is a mathematical operation that combines two sequences of numbers into a single number. It is commonly used in Euclidean geometry to calculate the inner product of vectors. Note that there are other inner products possible in Euclidean space.

Tensor: A tensor is an algebraic object describing relationships between sets of algebraic objects in a vector space. It can map between different objects such as vectors and scalars. Tensors include scalars, vectors, dual vectors, and multilinear maps. They are defined independently of any basis but are often represented by components in a coordinate system. Tensors can be thought of as high-dimensional matrices.

Algebraic topology: Algebraic topology applies abstract algebra to study topological spaces, aiming to find algebraic invariants that classify these spaces up to similarity. It focuses on determining classifications up to both homeomorphism and homotopy equivalence.

Sheaf (mathematics): A sheaf is a mathematical concept used to track data associated with open sets in a topological space. It allows for local definitions of data, which can be restricted to smaller open sets. The data assigned to an open set is equivalent to compatible data assigned to collections of smaller open sets covering the original open set.

Associative property: The associative property in mathematics states that changing the grouping of terms in an expression does not alter its result. In propositional logic, associativity is a valid rule for replacing expressions in logical proofs.

Commutative algebra: Commutative algebra is a branch of algebra that explores commutative rings, their ideals, and modules over such rings. It serves as a foundation for algebraic geometry and algebraic number theory. Notable examples of commutative rings include polynomial rings, rings of algebraic integers (including ordinary integers), and p-adic integers.

Commutative property: The commutative property in mathematics states that changing the order of operands does not change the result of a binary operation. It is a fundamental property used in arithmetic and other advanced settings. Operations like division and subtraction do not have this property and are called noncommutative. The commutative property was named in the 19th century when math started getting formalized. It also exists in binary relations as symmetry, where the order of operands doesn't affect the relation's validity.

Distributive property: The distributive property is a mathematical concept that extends the distributive law. It states that when operations are performed on two values, the result will be the same whether the operations are performed individually or together.

Field (mathematics): In mathematics, a field is a set where addition, subtraction, multiplication, and division are defined, similar to rational and real numbers. It is a fundamental algebraic structure used extensively in various mathematical disciplines like algebra and number theory.

Fundamental theorem of algebra: The fundamental theorem of algebra states that every non-constant polynomial with complex coefficients has at least one complex root. This applies to both polynomials with real coefficients and those with imaginary parts equal to zero.

Galois theory: Galois theory is a mathematical theory by Évariste Galois that links field theory and group theory. It simplifies and enhances our understanding of field theory problems by reducing them to group theory using the fundamental theorem of Galois theory.

Group (mathematics): A group in mathematics is a set with an operation that is associative and has an identity element. It also requires that every element in the set has an inverse element.

Abelian group: An Abelian group is a commutative group, where the order of applying the group operation doesn't matter. Examples include the integers and real numbers under addition. It is named after mathematician Niels Henrik Abel.

Classification of finite simple groups: The classification of finite simple groups is a fundamental result in group theory. It states that every finite simple group can be categorized as cyclic, alternating, a group of Lie type, or one of a few sporadic exceptions. The proof for this classification spans tens of thousands of pages in hundreds of articles written by about 100 authors over the course of almost 50 years.

Fundamental group: The fundamental group of a topological space is a group that captures information about the shape and holes of the space. It consists of equivalence classes of loops under homotopy. The fundamental group is a homotopy invariant and is denoted by .

Group representation: Group representation in mathematics refers to the way abstract groups are described using linear transformations of a vector space. This representation allows group elements to be represented by invertible matrices, enabling the group operation to be represented through matrix multiplication.

Group theory: Group theory is a branch of abstract algebra that studies the algebraic structures called groups. It is central to abstract algebra and influences other algebraic structures like rings, fields, and vector spaces. The methods of group theory have influenced various parts of algebra, including linear algebraic groups and Lie groups.

Homological algebra: Homological algebra is a branch of mathematics that studies homology in a broader algebraic context. It originated from the research of Poincaré and Hilbert in the late 19th century, combining combinatorial topology and abstract algebra.

Isomorphism: An isomorphism is a structure-preserving mapping between two mathematical structures of the same type. It can be reversed by an inverse mapping. If an isomorphism exists, the two structures are considered isomorphic. The term is derived from Ancient Greek, meaning "equal form".

Module (mathematics): A module is a mathematical generalization of a vector space, where a ring replaces the field of scalars. It also extends the concept of an abelian group, as abelian groups are modules over the ring of integers.

Representation theory: Representation theory is a mathematical branch that represents elements of abstract algebraic structures as linear transformations. It also analyzes modules over these structures. By using matrices and linear algebra operations, representations make abstract objects more understandable, allowing for easier property analysis and sometimes simplifying abstract theories calculations.

Ring theory: Ring theory is the study of algebraic structures called rings, which have properties similar to those of integers. It explores the structure, representations, and properties of rings, including modules and special ring classes. This theory is useful for understanding homological properties and polynomial identities.

Commutative ring: A commutative ring is a mathematical structure where multiplication is commutative. It is studied in commutative algebra, which focuses on properties unique to commutative rings. In contrast, noncommutative algebra explores properties that do not extend to noncommutative rings.

Ring (mathematics): A ring is a mathematical structure that extends the concept of field, allowing for non-commutative multiplication and the absence of multiplicative inverses. It consists of a set with two operations resembling addition and multiplication. Rings can include numeric objects like integers and complex numbers, as well as non-numeric entities like polynomials, matrices, functions, and power series.

Algebraic geometry: Algebraic geometry uses abstract algebra to solve geometric problems by studying zeros of multivariate polynomials. The modern approach generalizes this by incorporating aspects from commutative algebra.

Algebraic variety: An algebraic variety is the main focus in algebraic geometry. It represents the set of solutions to a system of polynomial equations. Modern definitions aim to extend this concept while retaining its geometric intuition.

Inequality (mathematics): In mathematics, inequality is a relation that compares two numbers or expressions in terms of their size. It is commonly used to compare numbers on the number line. Notations include "<" for less than and ">" for greater than.

Statistics: Statistics is the study of collecting, organizing, analyzing, interpreting, and presenting data. It is used to address scientific, industrial, or social issues, and involves studying populations or models. It encompasses all aspects of data, including survey and experiment design.

Bayesian inference: Bayesian inference is a statistical method that uses Bayes' theorem to update the probability of a hypothesis as more evidence is gathered. It relies on prior knowledge and a prior distribution to estimate posterior probabilities. This technique is widely used in various fields, including science, engineering, medicine, and law. Bayesian inference is closely linked to subjective probability and is crucial for analyzing sequential data.

Correlation: Correlation refers to the statistical relationship between two random variables or data. It measures the degree of linear association between variables. Examples of correlation include the height of parents and their offspring, and the relationship between price and consumer demand.

Regression analysis: Regression analysis is a statistical tool used to estimate the relationship between a dependent variable and independent variables. The most common method is linear regression, which finds the line that best fits the data. Ordinary least squares is a popular technique that minimizes the differences between the data and the line. This allows researchers to estimate the dependent variable's expected value given specific values of the independent variables. Regression can also estimate alternative location parameters and the conditional expectation across non-linear models.

Ordinary least squares: Ordinary least squares (OLS) is a statistical method for estimating unknown parameters in a linear regression model. It minimizes the sum of squared differences between the observed dependent variable and the predicted value based on the linear relationship with the independent variable.

Data and information visualization: Data and information visualization is the creation of visual representations that make complex data easy to understand. These visuals can be static, dynamic, or interactive and are based on expertise in a particular field. They help a broad audience explore and gain insights from the data by highlighting patterns, trends, and relationships. When used to present known information in a clear and engaging way, it is called information graphics.

Design of experiments: Design of experiments refers to the planning and arrangement of tasks that aim to understand and describe the variation of information under specific conditions. It is commonly used in experiments where conditions directly influence this variation, but can also include observational studies of natural conditions that impact the variation.

Randomized controlled trial: A Randomized Controlled Trial (RCT) is a scientific experiment that compares the effects of different treatments. It is commonly used in clinical trials to control variables not directly manipulated by researchers. RCTs assess the effectiveness of drugs, surgical techniques, medical devices, diagnostic procedures, and other medical treatments.

Mean: A mean is a numerical value that represents the center of a set of numbers. It helps summarize data and understand the overall value of a dataset.

Median: The median is a statistical measure that represents the middle value of a dataset or distribution. It is not influenced by extreme values, making it a more accurate measure of center. Median income is a better indicator of the center of income distribution as it is not affected by the highest incomes. Overall, the median is a key concept in statistics, providing robustness in data analysis.

Meta-analysis: Meta-analysis is the statistical combination of multiple studies on a similar topic. It involves calculating an effect size by extracting data from different studies. Meta-analyses contribute to research proposals, treatment guidelines, and health policies. They summarize existing research and guide future studies, making them crucial in metascience. They are often part of systematic reviews, for example, in clinical trials to evaluate the effectiveness of medical treatments.

Mode (statistics): The mode in statistics refers to the most common value in a set of data. For a discrete random variable, it represents the value with the highest probability mass function. Simply put, it is the value that is most likely to be selected or sampled.

Observational study: Observational studies are conducted in fields like epidemiology, social sciences, psychology, and statistics. They involve drawing conclusions from a sample to a larger population. These studies are done when the researcher cannot control the independent variable due to ethical or logistical reasons. One common type of observational study looks at the impact of a treatment on subjects, where the assignment of subjects to treated or control groups is not controlled by the researcher. This is different from experiments, where subjects are randomly assigned to groups. Due to the lack of control over assignment, observational studies pose challenges for inferential analysis.

Sample mean and covariance: The sample mean and covariance are statistical measures calculated from a dataset representing random variables. They provide insights into the average and relationship of the variables.

Standard deviation: Standard deviation measures the amount of variation expected in a random variable around its mean. A low value suggests values closer to the mean, while a high value indicates a wider range of values.

Standard error: The standard error (SE) is the standard deviation of a statistic's sampling distribution or an estimate of that. When the statistic is the sample mean, it is referred to as the standard error of the mean (SEM). The standard error is crucial for creating confidence intervals.

Statistical hypothesis testing: Statistical hypothesis testing is a technique to determine if data supports a hypothesis. It allows for making probabilistic statements about population parameters and involves making decisions under uncertainty. The process involves selecting a tradeoff between false positives and false negatives.

p-value: The p-value is a measure used in statistical testing to determine the probability of obtaining results as extreme as the observed result, assuming the null hypothesis is true. A low p-value indicates that the observed outcome would be unlikely under the null hypothesis. However, the misuse and misinterpretation of p-values are common and have been widely discussed. The American Statistical Association (ASA) stated that p-values do not measure the probability of a hypothesis being true or the importance of a result. Nevertheless, when properly applied and interpreted, p-values and significance tests enhance the rigor of conclusions drawn from data, as emphasized by a 2019 ASA task force.

Chi-squared test: A chi-squared test is a statistical test used to analyze contingency tables with large sample sizes. It determines whether two categorical variables are independent. The test examines if there is a significant difference between expected and observed frequencies in a contingency table. For smaller sample sizes, Fisher's exact test is used.

Student'st-test: The Student's t-test is a statistical hypothesis test used to assess if the difference between the response of two groups is statistically significant. It is applicable when the test statistic follows a Student's t-distribution and is commonly used to compare means of two populations. In certain scenarios, a t-test and a Z-test yield similar results as the dataset size grows.

Survey methodology: 'Survey methodology' is the study of techniques used in gathering data through surveys. It focuses on sampling, questionnaire design, and strategies for increasing response accuracy. This field explores methods to improve the quality and quantity of survey data.

Statistical population: A statistical population is a set of similar items or events that are of interest for a question or experiment. It can be a group of existing objects or a hypothetical, potentially infinite group based on experience. Statistical analysis aims to gather information about a chosen population.

Sampling (statistics): Sampling in statistics is the process of selecting a subset of individuals from a population to estimate characteristics of the entire population. Statisticians aim to collect representative samples to save time and cost compared to surveying the entire population. It provides valuable insights when surveying the whole population is impractical.

Probability: Probability is a mathematical concept that measures the likelihood of events occurring. It is represented by a number between 0 and 1, where higher numbers indicate greater chances of an event happening. For example, when tossing a fair coin, the probability of landing on either heads or tails is equal at 1/2.

Probability theory: Probability theory is a mathematical branch dealing with the concept of probability. It uses a set of strict axioms to express probability and assigns a probability measure to a set of outcomes, known as the sample space. Events are subsets of the sample space in which probability is calculated.

Bayes' theorem: Bayes' theorem is a mathematical tool used in probability theory and statistics to determine the probability of an event based on prior knowledge. It takes into account conditions that may be related to the event, allowing for more accurate assessments. For instance, it helps assess an individual's health risk by considering their age, rather than assuming they represent the entire population.

Central limit theorem: The central limit theorem (CLT) states that the distribution of the sample mean approaches a normal distribution, regardless of the original variables' distribution, under certain conditions. Different versions of the CLT apply depending on the circumstances.

Law of large numbers: The Law of Large Numbers (LLN) is a mathematical theorem stating that if a large number of independent and identical random samples are taken, their average will converge to the true value. Specifically, the LLN states that for a sample of independent and identically distributed values, the sample mean will converge to the true mean.

Expected value: The expected value is a concept in probability theory that calculates the average value of a random variable. It considers all possible outcomes, weighted by their probabilities. The expected value may not be present in the actual data set and can differ from what is expected in reality.

Variance: Variance measures the spread of numbers in a dataset from their average value. It is obtained by calculating the squared deviation from the mean. The standard deviation is the square root of the variance. Variance is a central moment in probability theory, and it can be represented by various symbols.

Probability distribution: A probability distribution is a mathematical function that assigns probabilities to different outcomes in an experiment. It describes a random phenomenon using the sample space and the probabilities of events.

Probability density function: A probability density function (PDF) is a function that measures the relative likelihood of a continuous random variable taking on a specific value. It provides the probability per unit length and allows us to compare the likelihood of the variable being close to different values.

Exponential distribution: The exponential distribution is a probability distribution that represents the distance between events in a continuous and independent process occurring at a constant average rate. It is a special case of the gamma distribution and the continuous analogue of the geometric distribution. Its key feature is being memoryless. It is widely used for analyzing Poisson point processes and found in various other contexts.

Normal distribution: A normal distribution, also known as Gaussian distribution, is a continuous probability distribution for a real-valued random variable. It exhibits a symmetric bell curve shape in which data is centered around the mean. The probability density function of a normal distribution is commonly used in statistics.

Binomial distribution: The binomial distribution is a probability distribution that models the number of successes in a fixed number of independent experiments. Each experiment has a yes or no outcome, and the distribution is used for statistical tests of significance.

Discrete uniform distribution: The discrete uniform distribution is a symmetric probability distribution where a finite number of equally likely values can be observed. Each value has an equal probability of 1/n. In other words, it represents a known, finite number of outcomes that are equally likely to occur.

Poisson distribution: The Poisson distribution is a discrete probability distribution that calculates the likelihood of a specific number of events happening within a fixed time or space, assuming a constant mean rate and independence from previous events. It is named after mathematician Siméon Denis Poisson and is also applicable to distances, areas, and volumes. The distribution is significant in discrete-stable distributions.

Random variable: A random variable is a mathematical representation of a quantity influenced by random events. It is not truly random nor a variable, but a function mapping possible outcomes to measurable spaces, often real numbers.

Stochastic process: A stochastic process is a sequence of random variables in a probability space, often representing variables changing over time. It is widely used as a mathematical model to study systems and phenomena that seem to vary randomly, such as population growth, electrical fluctuations, or gas molecule movement. These processes have applications in various fields including biology, physics, finance, and computer science.

Markov chain: A Markov chain is a stochastic model where the probability of each event only depends on the previous state. It can be discrete-time (moves in discrete steps) or continuous-time. Named after mathematician Andrey Markov.

Mathematical logic: Mathematical logic, a study within mathematics, explores formal logic. It encompasses model theory, proof theory, set theory, and recursion theory. Research focuses on properties of logical systems, their deductive power, and their use in characterizing correct mathematical reasoning and establishing foundations of mathematics.

Axiom: An axiom, postulate, or assumption is a true statement used as a starting point for reasoning. It is derived from the Greek word ἀξίωμα, meaning 'that which is thought worthy or fit'.

Propositional calculus: Propositional calculus, also known as propositional logic or statement logic, is a branch of logic that deals with propositions and their relations. It involves creating arguments based on these propositions and connecting them using logical connectives. Atomic propositions are those that contain no logical connectives.

Boolean algebra: Boolean algebra is a branch of mathematics and logic that uses truth values (true and false) instead of numbers. It employs logical operators like conjunction (and), disjunction (or), and negation (not) to describe logical operations. This formal system is distinct from elementary algebra, which deals with numerical operations.

First-order logic: First-order logic, or predicate logic, is a formal system used in various fields. It allows quantified variables and sentences with variables, enabling more complex expressions. Unlike propositional logic, it uses quantifiers and relations. Propositional logic serves as the basis for first-order logic.

Set theory: Set theory is a field of mathematical logic that examines sets, defined as collections of objects. It focuses on relevant sets in mathematics as a whole.

Naive set theory: Naive set theory is an informal approach to understanding mathematical sets. It is described in natural language rather than formal logic. It covers familiar concepts of sets and is sufficient for everyday use in mathematics.

Russell's paradox: Russell's paradox is a set-theoretic paradox discovered by Bertrand Russell in 1901. It demonstrates that all set theories with unrestricted comprehension principles result in contradictions. Ernst Zermelo also independently found the paradox in 1899 but did not publish it. This knowledge was limited to David Hilbert, Edmund Husserl, and others at the University of Göttingen. Georg Cantor, the founder of modern set theory, had foreseen this contradiction in his theory and informed Hilbert and Richard Dedekind.

Venn diagram: A Venn diagram is a diagram style devised by John Venn in the 1880s to illustrate the logical relation between sets. It uses closed curves, often circles or ellipses, to represent sets. Venn diagrams are widely used in various fields like set theory, probability, logic, statistics, linguistics, and computer science. They serve as a teaching tool for elementary set theory and depict simple set relationships.

Set (mathematics): A set in mathematics is a collection of different elements, which can be numbers, symbols, shapes, or even other sets. It can be finite or infinite, and may contain no elements or just one element.

Empty set: The empty set in mathematics is a set with no elements, represented by {}. It has a size of zero and is a fundamental concept in set theory. Some set theories include an axiom to ensure the existence of the empty set, while in others, it can be deduced. The empty set satisfies certain properties by default.

Intersection (set theory): Intersection in set theory, denoted by ∩, is the set that contains elements shared by two sets A and B. It represents the common elements between A and B.

Subset: In mathematics, a subset is a set that contains only elements also found in another set. The term "inclusion" describes the relationship where one set is a subset of another. A proper subset is when two sets are unequal, and one completely contains the other. A k-subset is a subset with k elements.

Union (set theory): The union in set theory combines sets by including all their elements. It is a fundamental operation that allows sets to be related and combined. A nullary union, representing no sets, is equal to the empty set.

Equivalence relation: An equivalence relation is a reflexive, symmetric, and transitive binary relation in mathematics. It can be seen in the equipollence relation between line segments in geometry. Equality is a simpler example, where any number is equal to itself (reflexive), and if a equals b, then b equals a (symmetric) and if a equals b and b equals c, then a equals c (transitive).

Cantor's theorem: Cantor's theorem in set theory states that for any set, the set of all its subsets has a higher cardinality than the original set itself. It establishes the existence of infinite levels of infinity within set theory.

Axiom of choice: The axiom of choice is a fundamental principle in set theory. It states that for any collection of sets, even if infinite, it is possible to select one element from each set and form a new set. This axiom, formulated in 1904 by Ernst Zermelo, is equivalent to the statement that the Cartesian product of non-empty sets is non-empty.

Mathematical induction: Mathematical induction is a method to prove that a statement is true for every natural number. It involves first proving a simple case and then showing that if the claim holds for a given case, it also holds for the next case. It can be compared to falling dominoes or climbing a ladder, showing that we can climb as high as we want by starting from the bottom rung and being able to climb up from each rung to the next.

Computability theory: Computability theory, also called recursion theory, emerged in the 1930s to study computable functions and Turing degrees. It is a branch of mathematics, computer science, and the theory of computation. Today, it encompasses the investigation of generalized computability, definability, and overlaps with proof theory and effective descriptive set theory.

Computable function: Computable functions are the fundamental objects in computability theory, representing algorithms that can process inputs and produce outputs. They allow for the study of computability without relying on specific computational models like Turing machines or register machines. Two notable models, Turing-computable functions and general recursive functions, are used to define the set of computable functions.

Gödel's incompleteness theorems: Gödel's incompleteness theorems, discovered by Kurt Gödel in 1931, reveal the limits of provability in formal axiomatic theories. They have significant implications in both mathematical logic and the philosophy of mathematics. These theorems challenge Hilbert's program of finding a complete and consistent set of axioms for all of mathematics.

Category theory: Category theory is a mathematical theory that studies structures and their relationships. It was introduced by Eilenberg and Mac Lane in the mid-20th century. This theory is widely used in various areas of mathematics to express and unify constructions of new objects, such as quotient spaces and direct products, across different contexts. It provides a powerful tool for understanding and organizing mathematical concepts.

Discrete mathematics: Discrete mathematics is the study of mathematical structures that are separate and distinct. It focuses on integers, graphs, and logic statements while excluding continuous topics like real numbers and calculus. It deals with countable sets but lacks a specific definition.

Combinatorics: Combinatorics is a branch of mathematics that deals with counting and understanding properties of finite structures. It has applications in various fields such as logic, statistical physics, evolutionary biology, and computer science.

Combination: A combination in mathematics is a selection of items from a set where the order doesn't matter. For example, choosing two fruits from a set of apples, oranges, and pears gives three possible combinations. A k-combination of a set is a subset of k distinct elements. The number of k-combinations, denoted by , is equal to the binomial coefficient.

Permutation: In mathematics, a permutation is the arrangement or rearrangement of elements in a set's linear order. It can also refer to the process of changing the order of an already-ordered set.

Graph theory: Graph theory is the mathematical study of graphs, which are structures used to model relationships between objects. Graphs consist of vertices connected by edges, with undirected graphs having symmetrical connections and directed graphs having asymmetrical connections. They are a fundamental subject in discrete mathematics.

Graph (discrete mathematics): A graph, in discrete mathematics, is a set of objects with some connections between them. The objects are known as vertices and the connections as edges. Graphs are depicted as dots connected by lines or curves. They are studied in the field of discrete mathematics.

Calculus: Calculus is the mathematical study of continuous change, similar to how geometry studies shape and algebra studies generalizations of arithmetic operations.

Limit (mathematics): A limit is a value that a function approaches when its input approaches a specific value. It is crucial in calculus and mathematical analysis as it defines continuity, derivatives, and integrals.

Fundamental theorem of calculus: The fundamental theorem of calculus connects differentiation and integration. These operations are essentially inverse, except for a constant value determined by the starting point in measuring area.

Continuous function: A continuous function is a function where small changes in the input cause small changes in the output. It does not have abrupt changes in value, called discontinuities. In the past, mathematicians only considered continuous functions, but the definition of continuity was formalized in the 19th century using the epsilon-delta definition of a limit.

Differential calculus: Differential calculus is a branch of mathematics that examines how quantities change over time. It is part of calculus, along with integral calculus—the study of the area under curves.

Derivative: The derivative is a fundamental tool in calculus that measures how a function's output changes with respect to its input. It gives the slope of the tangent line to the function's graph at a specific point. This tangent line provides the best linear approximation of the function near that point. The derivative is known as the instantaneous rate of change, representing the ratio of the instantaneous change in the dependent variable to the independent variable. The process of obtaining a derivative is called differentiation.

Integral: Integral is a mathematical concept used to calculate areas, volumes, and more. It is one of the fundamental operations of calculus, alongside differentiation. Initially used for finding areas and displacements, integration is now widely applied in scientific fields.

Numerical integration: Numerical integration is a collection of algorithms used to calculate the numerical value of a definite integral. It is sometimes referred to as numerical quadrature, particularly for one-dimensional integrals. Cubature is the term used for numerical integration in more than one dimension, and "quadrature" can encompass higher-dimensional integration as well.

Multivariable calculus: Multivariable calculus is the study of calculus applied to functions with multiple variables. It involves differentiating and integrating functions with several variables, rather than just one.

Vector calculus: Vector calculus, also known as vector analysis, is an advanced mathematical subject with practical applications in physics and engineering. It involves differentiating and integrating vector fields in 3-dimensional space. It is closely related to multivariable calculus, including partial differentiation and multiple integration. Vector calculus plays a significant role in differential geometry and the study of partial differential equations. It finds extensive use in describing electromagnetic and gravitational fields, as well as fluid flow.

Multiple integral: A multiple integral is a type of definite integral in multivariable calculus. It involves integrating a function that depends on several real variables. It can be used to calculate various physical quantities such as surface area and volume. The variables in the function can represent time, position, and other factors.

Partial derivative: A partial derivative is the derivative of a function with respect to one variable while keeping the others constant. It is an essential tool in vector calculus and differential geometry.

Del: Del (∇) is a vector differential operator in mathematics, commonly used to represent the gradient, divergence, or curl of a field. It can also denote the standard derivative of a one-dimensional function.

Laplace's equation: Laplace's equation, a second-order partial differential equation, is named after Pierre-Simon Laplace. It describes important properties in mathematics and physics.

Differential equation: A differential equation relates unknown functions and their derivatives. It defines the relationship between physical quantities and their rates of change. These equations are essential in various fields such as engineering, physics, economics, and biology.

Ordinary differential equation: An ordinary differential equation (ODE) is a mathematical equation with a single independent variable, involving the derivatives of one or more functions. It is distinct from partial differential equations, which involve multiple independent variables.

Partial differential equation: A partial differential equation (PDE) is a mathematical equation that relates partial derivatives of a function with respect to multiple variables. It is used to compute the function's behavior in terms of its derivatives.

Dynamical systems theory: Dynamical systems theory is a mathematical framework that studies the behavior of complex systems using differential equations or difference equations. It encompasses continuous dynamical systems, which generalize classical mechanics and do not adhere to the least action principle, and discrete dynamical systems. Time scales and mixed operators may also be considered in some cases.

Dynamical system: A dynamical system is a mathematical model that describes the time evolution of a point in a space. It can represent various phenomena like pendulum swinging, water flow, particle motion, and population growth. This concept combines different mathematical concepts such as differential equations and ergodic theory. Time can be measured in various ways, and the space can be a set or a manifold, with or without a smooth structure.

Chaos theory: Chaos theory is an interdisciplinary scientific study that focuses on patterns and laws of highly sensitive and complex systems. It reveals underlying patterns, interconnections, and self-organization within seemingly random states. The butterfly effect, a key principle, highlights how a small change in one state can lead to significant effects later on. Metaphorically, a butterfly flapping its wings can cause a tornado in a different location.

Sequence: A sequence in mathematics is a collection of objects with order and potential repetitions. It is defined as a function from natural numbers to the elements at each position. Repetition of elements and the order of the sequence matter. The concept can be extended to include an arbitrary index set.

Limit of a sequence: The limit of a sequence is the value that the terms tend to, denoted by a symbol. If the limit exists, the sequence is convergent, otherwise it is divergent. It is a fundamental concept in mathematical analysis.

Series (mathematics): A series in mathematics involves adding an infinite number of quantities to a starting quantity. It is a significant topic in calculus and mathematical analysis. Series are utilized in various areas of mathematics and are also extensively applied in disciplines like physics, computer science, statistics, and finance.

Taylor series: The Taylor series is an infinite sum of terms expressed in terms of a function's derivatives at a point, named after mathematicians Brook Taylor and Colin Maclaurin. It is commonly used to approximate a function near a specific point.

Mathematical analysis: Mathematical analysis refers to the study of continuous functions and their properties. It encompasses concepts like limits, differentiation, integration, measure, infinite sequences, series, and analytic functions.

Approximation theory: Approximation theory is a mathematical field that studies how functions can be approximated by simpler functions. It aims to quantitatively describe the errors that arise from these approximations. The definition of "best" and "simpler" varies depending on the specific application.

Numerical analysis: Numerical analysis is the study of algorithms that approximate solutions to mathematical problems. It has applications across various fields, including engineering, sciences, medicine, business, and the arts. The use of computational power has allowed for more detailed and realistic mathematical models. Examples include solving differential equations in celestial mechanics and using linear algebra in data analysis.

Functional analysis: Functional analysis is a branch of mathematical analysis that focuses on the study of vector spaces with limit-related structures. It explores linear functions on these spaces and their relationships with continuous or unitary operators. Functional analysis is especially valuable in studying differential and integral equations, with its roots in the study of function spaces and transformations like the Fourier transform.

Hilbert space: A Hilbert space is a space where linear algebra and calculus methods can be extended to infinite-dimensional spaces. It is frequently used in mathematics and physics, particularly as function spaces. It is a vector space with an inner product that allows for the definition of a distance function and ensures completeness.

Calculus of variations: The calculus of variations analyzes small changes in functions to find maximum and minimum values. It uses integrals to express functionals and employs the Euler-Lagrange equation to find functions that maximize or minimize these functionals.

Real analysis: Real analysis is a mathematical branch that focuses on the study of real numbers, sequences, and functions. It explores important topics like convergence, limits, continuity, differentiability, and integrability to understand the behavior of these elements.

Complex analysis: Complex analysis is a branch of mathematics that explores functions of complex numbers. It has wide-ranging applications in various mathematical fields, as well as in physics and engineering.

Cauchy's integral formula: Cauchy's integral formula is a fundamental concept in complex analysis named after Augustin-Louis Cauchy. It states that a holomorphic function on a disk can be determined entirely by its values on the boundary of the disk. This formula also provides integral formulas for all derivatives of the function. It demonstrates the equivalence of differentiation and integration in complex analysis, a property not seen in real analysis.

Euler's formula: Euler's formula relates trigonometric and exponential functions in complex analysis. It states that for any real number x, a fundamental relationship exists between these functions. Named after Leonhard Euler, this mathematical formula establishes the connection between complex exponential and trigonometric functions.

Euler's identity: Euler's identity is a fundamental equality in mathematics expressed as a concise formula. It connects five important numbers: 0, 1, pi, e (Euler's number), and the imaginary unit i. This identity holds great significance for its elegant combination of these key constants.

Holomorphic function: A holomorphic function is a complex-valued function that is complex differentiable in a neighborhood of each point in a domain in complex coordinate space. It is infinitely differentiable, locally equal to its Taylor series, and is the main focus of study in complex analysis.

Riemann surface: A Riemann surface is a complex manifold used in mathematics, specifically in complex analysis. It is one-dimensional and serves as a fundamental concept in this field.

Harmonic analysis: Harmonic analysis studies the relationship between a function and its representation in frequency using the Fourier transform for functions on the real line or Fourier series for periodic functions. It is a broad field with applications in number theory, representation theory, signal processing, quantum mechanics, tidal analysis, and neuroscience.

Fourier analysis: Fourier analysis is a mathematical study of representing functions using simpler trigonometric functions. It originated from Fourier series and simplifies the analysis of heat transfer by expressing functions as sums of trigonometric functions.

Fourier series: The Fourier series is an expansion of a periodic function into a sum of trigonometric functions. It is useful for analyzing functions as it simplifies problems by expressing them in terms of sines and cosines. The series was first used to find solutions to the heat equation. While Fourier series cannot approximate all functions and may not always converge, they do converge for well-behaved functions. The coefficients of the series are determined by integrals of the function multiplied by trigonometric functions.

Fourier transform: The Fourier transform is a mathematical operation that converts a function into a complex-valued function of frequency. It describes the frequencies present in the original function and is used in physics, engineering, and mathematics. It can be seen as decomposing a sound into its constituent pitches.

Laplace transform: The Laplace transform is a mathematical tool invented by Pierre-Simon Laplace that converts real variable functions into complex variable functions. It is widely used in science and engineering to solve linear differential equations by transforming them into algebraic equations and convolution into multiplication.

Number theory: Number theory is a branch of pure mathematics that focuses on the study of integers and arithmetic functions. It encompasses prime numbers and the properties of mathematical objects built from integers or their generalizations. German mathematician Carl Friedrich Gauss regarded number theory as the epitome of mathematics.

Prime number: A prime number is a natural number greater than 1 that cannot be divided evenly by any other natural number except 1 and itself. This property makes primes important in number theory. All non-prime numbers are called composite numbers. The fundamental theorem of arithmetic states that every natural number greater than 1 can be written as a unique product of prime numbers.

Euclid's theorem: Euclid's theorem states that there are infinitely many prime numbers. First proven by Euclid in his work Elements, it remains a fundamental statement in number theory. Multiple proofs of the theorem exist.

Prime number theorem: The Prime Number Theorem (PNT) is a mathematical theorem that explains how prime numbers are distributed among positive integers. It shows that as the numbers get larger, prime numbers become less frequent. This theorem was independently proved by Jacques Hadamard and Charles Jean de la Vallée Poussin in 1896, using ideas from Bernhard Riemann.

Fundamental theorem of arithmetic: The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be expressed as a unique product of prime numbers.

Fibonacci sequence: The Fibonacci sequence is a mathematical sequence where each number is the sum of the two preceding ones. It commonly starts with 0 and 1. The sequence is denoted as Fn and begins 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ....

Diophantine equation: A Diophantine equation is a polynomial equation with integer coefficients that only requires integer solutions. It can be linear, where degree one monomials are summed to a constant, or exponential, where unknowns can appear in exponents.

Fermat's Last Theorem: Fermat's Last Theorem is a statement in number theory that states there are no positive integer solutions to the equation an + bn = cn when n is greater than 2. It is known since ancient times that the equation has infinitely many solutions for n = 1 and n = 2.

Algebraic number theory: Algebraic number theory is a branch of number theory that uses abstract algebra techniques to study integers, rational numbers, and their extensions. It focuses on properties of algebraic objects like algebraic number fields, rings of integers, finite fields, and function fields. It helps answer important questions in number theory, such as the existence of solutions to Diophantine equations, unique factorization of rings, and the behavior of ideals and Galois groups.

Algebraic number field: An algebraic number field is a field extension of the rational numbers with finite degree. It is a field that includes the rational numbers and has finite dimension as a vector space over the rationals.

Class field theory: Class field theory is a core branch of algebraic number theory that aims to explain abelian Galois extensions of local and global fields. It utilizes elements linked to the original field.

Analytic number theory: Analytic number theory is a branch of mathematics that uses mathematical analysis to solve problems related to integers. It originated with Dirichlet L-functions, which were introduced by Dirichlet in 1837 to prove his theorem on arithmetic progressions. This field is renowned for its contributions to prime numbers and additive number theory.

Riemann hypothesis: The Riemann hypothesis is a major unsolved problem in math. It proposes that the Riemann zeta function's zeros are limited to negative even integers and complex numbers with a real part of 1/2. It has significant implications for prime number distribution and was introduced by Bernhard Riemann in 1859.

Applied mathematics: Applied mathematics is the use of mathematical methods in various fields like physics, engineering, medicine, biology, finance, business, computer science, and industry. It involves combining mathematical science with specialized knowledge to address practical problems through the formulation and study of mathematical models.

Control theory: Control theory is a field that combines control engineering and mathematics to manage dynamical systems in machines and processes. Its goal is to create models or algorithms to guide inputs, achieving desired states while minimizing delays, overshoot, and errors. It emphasizes stability and strives for optimal control.

Game theory: Game theory is the study of strategic interactions among rational agents, using mathematical models. It has diverse applications in social science, economics, logic, systems science, and computer science. Originally focused on two-person zero-sum games, game theory now encompasses a broader range of behavioral relations, exploring logical decision-making in humans, animals, and computers.

Information theory: Information theory is the study of quantifying, storing, and communicating information. It was established by Nyquist, Hartley, and Shannon in the 1920s and 1940s. This field sits at the intersection of various disciplines like probability theory, statistics, computer science, statistical mechanics, information engineering, and electrical engineering.

Coding theory: Coding theory is the study of codes and their applications in data compression, cryptography, error detection and correction, data transmission, and storage. It involves various disciplines like information theory, electrical engineering, mathematics, linguistics, and computer science to design efficient and reliable data transmission methods by removing redundancy and detecting/correcting errors in transmitted data.

Data compression: Data compression is the method of encoding information with fewer bits than the original representation. It can be lossless, where no information is lost, or lossy, where unnecessary information is removed. Compression reduces bits by eliminating statistical redundancy. A compressor is an encoder, and a decompressor is a decoder.

Error detection and correction: Error detection and correction (EDAC) refers to techniques used in computer science and telecommunication to ensure the accurate transmission of digital data. By detecting and correcting errors caused by channel noise during transmission, EDAC enables reliable delivery of information. This approach allows for the identification of errors and, in most cases, the reconstruction of the original data.

Mathematical optimization: Mathematical optimization involves finding the best option from a group of choices based on a specific criteria. It includes discrete and continuous optimization. This field is relevant in various disciplines such as computer science, engineering, operations research, and economics, and has been studied for centuries in mathematics.

Linear programming: Linear programming (LP) or linear optimization is a mathematical approach used to optimize outcomes in models with linear relationships between requirements and objectives. It is a subset of mathematical programming.

Nonlinear programming: Nonlinear programming is a branch of mathematical optimization that deals with solving optimization problems where constraints or the objective function are nonlinear. It involves determining the extrema of an objective function over a set of unknown variables while satisfying a system of equalities and inequalities. Unlike linear programming, it focuses on addressing problems that are not linear.

Operations research: Operations research (OR) is a field that uses analytical methods to enhance decision-making. It is also referred to as management science.

Theoretical computer science: Theoretical computer science (TCS) is a branch of computer science and mathematics emphasizing mathematical aspects of computing. It encompasses subjects like computation theory, formal language theory, lambda calculus, and type theory.

Abstract machine: An abstract machine is a theoretical model used in computer science to analyze how a computer system works. It functions like a mathematical function, taking inputs and producing outputs based on predefined rules. Unlike physical machines, abstract machines are expected to perform correctly regardless of hardware. They enable step-by-step execution of programs by defining inputs, outputs, and allowable operations. Abstract machines are used for theoretical purposes and as models for real-world computers. In the theory of computation, they play a fundamental role in analyzing algorithm complexity and studying computability. Some examples of abstract machines include finite state machines, Mealy machines, push-down automata, and Turing machines.

Theory of computation: Theory of computation is a field in computer science and mathematics that explores the solvability and efficiency of problems using algorithms on different models of computation. It consists of three main branches: automata theory and formal languages, computability theory, and computational complexity theory. The key question it addresses is the fundamental capabilities and limitations of computers.

Algorithm: An algorithm is a set of instructions used in mathematic and computer science to solve specific problems or perform computations. They are used to specify calculations and data processing, and can include conditionals to automate decision making. The term "algorithm" draws on metaphors from human characteristics, as Turing did when using terms like "memory", "search", and "stimulus".

Formal language: A formal language, used in logic, mathematics, computer science, and linguistics, is made up of words that adhere to specific rules defined by a formal grammar. The language is constructed from an alphabet and is characterized by its well-formedness.

Computational complexity theory: Computational complexity theory classifies computational problems based on their resource usage and their relationships to each other. It is concerned with solving tasks by applying algorithms, which involve mechanical application of mathematical steps.

P versus NP problem: The P versus NP problem is an important unsolved problem in theoretical computer science. It questions whether problems that can be easily checked can also be efficiently solved.

Data structure: A data structure is a computer science concept that refers to the organization, storage, and management format chosen for efficient data access. It encompasses a collection of data values, their relationships, and the operations that can be performed on the data. Essentially, it is an algebraic structure concerning data.