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Complex number

Index Complex number

A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation. [1]

295 relations: Abraham de Moivre, Absolute convergence, Absolute value, Addison-Wesley, Addition, Additive inverse, Algebra over a field, Algebraic closure, Algebraic equation, Algebraic extension, Algebraic number, Algebraic number field, Algebraic number theory, Algebraic solution, Algebraic surface, Algebraically closed field, American Mathematical Monthly, Amplitude, Amplitude modulation, Analytic continuation, Analytic signal, Angle notation, Angular frequency, Anno Domini, Applied mathematics, Argument (complex analysis), Atan2, Augustin-Louis Cauchy, Automorphism, Axiom of choice, Base (topology), Bernhard Riemann, BIBO stability, Blackboard bold, Branch point, C. V. Mourey, Capacitor, Cardinality of the continuum, Carl Friedrich Gauss, Carl Gustav Jacob Jacobi, Cartesian coordinate system, Cartography, Caspar Wessel, Casus irreducibilis, Cauchy–Riemann equations, Cayley–Dickson construction, Characteristic (algebra), Circle group, Cis (mathematics), Coefficient, ..., Commutative property, Compass-and-straightedge construction, Complex analysis, Complex conjugate, Complex geometry, Complex logarithm, Complex number, Complex plane, Complex-base system, Complex-valued function, Congruence (geometry), Connected space, Continuous function, Contour integration, Control theory, Convergent series, Coset, Cubic function, Data compression, De Moivre's formula, Determinant, Differential equation, Digital data, Digital image processing, Digital signal processing, Dimension, Distributive property, Dover Publications, E (mathematical constant), Eigenvalues and eigenvectors, Eisenstein integer, Electric current, Electrical engineering, Electrical impedance, Electrical network, Electromagnetism, Electronics, Elementary function, Elements of Algebra, Ellipse, Essential singularity, Euclidean vector, Euler's formula, Euler's identity, Exponential function, Exponentiation, Felix Klein, Fermat's theorem on sums of two squares, Field (mathematics), Field extension, Fluid dynamics, Focus (geometry), Fourier analysis, Fourier transform, Fractal, Fraction (mathematics), Frequency, Frequency domain, Frustum, Fundamental theorem of algebra, G. H. Hardy, Galois theory, Gaussian integer, General relativity, Gerolamo Cardano, Giusto Bellavitis, Graph of a function, Greek mathematics, Henri Poincaré, Hermann Schwarz, Hero of Alexandria, Hilbert space, Holomorphic function, Hurwitz's theorem (composition algebras), Hyperbolic function, Hypercomplex number, Ideal (ring theory), Identity matrix, If and only if, Imaginary number, Imaginary unit, Improper integral, Indeterminate (variable), Inductor, Infinite set, Instability, Interval (mathematics), Inverse trigonometric functions, Inversive geometry, Involution (mathematics), Irreducible polynomial, Isomorphism, Jean le Rond d'Alembert, Jean-Robert Argand, John Wallis, John Wiley & Sons, Julia set, Karl Weierstrass, Laplace transform, Leonhard Euler, Lie algebra, Limit (mathematics), Limit of a sequence, Linear combination, Linear complex structure, Linear differential equation, Liouville's theorem (complex analysis), Local field, Locally compact space, Machin-like formula, Mandelbrot set, Marden's theorem, Marginal stability, Mathematical analysis, Mathematical formulation of quantum mechanics, Matrix (mathematics), Matrix mechanics, Matrix multiplication, Maximal ideal, Maximum power transfer theorem, McGraw-Hill Education, Meromorphic function, Metric (mathematics), Metric space, Minimum phase, Multiplicative inverse, Multivalued function, Natural logarithm, Negative number, Neighbourhood (mathematics), Network analysis (electrical circuits), Niccolò Fontana Tartaglia, Nichols plot, Nicolas Bourbaki, Niels Henrik Abel, Nonagon, Nth root, Number, Number line, Number theory, Nyquist stability criterion, Octonion, Open set, Ordered field, Ordered pair, Orientation (geometry), Origin (mathematics), Ostrowski's theorem, Otto Hölder, Oxford University Press, P-adic number, Parallelogram, Phase (waves), Phasor, Pi, Polynomial, Polynomial ring, Positive real numbers, Potential flow, Power series, Prime ideal, Prime number, Prime number theorem, Princeton University Press, Principal ideal domain, Principal value, Puiseux series, Pyramid, Pythagorean theorem, Quantum field theory, Quantum mechanics, Quartic function, Quaternion, Quotient ring, Radian, Radius, Rafael Bombelli, Rational number, Rational root theorem, Rationalisation (mathematics), Real analysis, Real number, Recurrence relation, Reflection symmetry, Regular representation, René Descartes, Representation theory, Resistor, Richard Dedekind, Riemann sphere, Riemann zeta function, Right angle, Ring (mathematics), Roger Penrose, Root locus, Root of unity, Rotation matrix, Royal Danish Academy of Fine Arts, Schrödinger equation, Sedenion, Series (mathematics), Set (mathematics), Sign function, Signal processing, Sine, Sine wave, Sound, Spacetime, Special relativity, Spinor, Split-complex number, Square (algebra), Square matrix, Square root, Square root of 2, Steiner inellipse, Stereographic projection, Subtraction, Taylor series, Tensor, Time domain, Tom M. Apostol, Topological ring, Topological space, Topology, Total order, Transcendence degree, Transpose, Triangle, Triangle inequality, Trigonometric functions, Tristan Needham, Turn (geometry), Two-dimensional space, Vector space, Video, Voltage, Wavelet, Wick rotation, William Rowan Hamilton, Winding number, Zero of a function, Zeros and poles, (ε, δ)-definition of limit, 2 × 2 real matrices. Expand index (245 more) »

Abraham de Moivre

Abraham de Moivre (26 May 166727 November 1754) was a French mathematician known for de Moivre's formula, a formula that links complex numbers and trigonometry, and for his work on the normal distribution and probability theory.

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Absolute convergence

In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite.

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Absolute value

In mathematics, the absolute value or modulus of a real number is the non-negative value of without regard to its sign.

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Addison-Wesley is a publisher of textbooks and computer literature.

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Addition (often signified by the plus symbol "+") is one of the four basic operations of arithmetic; the others are subtraction, multiplication and division.

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Additive inverse

In mathematics, the additive inverse of a number is the number that, when added to, yields zero.

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Algebra over a field

In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product.

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Algebraic closure

In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed.

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Algebraic equation

In mathematics, an algebraic equation or polynomial equation is an equation of the form where P and Q are polynomials with coefficients in some field, often the field of the rational numbers.

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Algebraic extension

In abstract algebra, a field extension L/K is called algebraic if every element of L is algebraic over K, i.e. if every element of L is a root of some non-zero polynomial with coefficients in K. Field extensions that are not algebraic, i.e. which contain transcendental elements, are called transcendental.

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Algebraic number

An algebraic number is any complex number (including real numbers) that is a root of a non-zero polynomial (that is, a value which causes the polynomial to equal 0) in one variable with rational coefficients (or equivalently – by clearing denominators – with integer coefficients).

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Algebraic number field

In mathematics, an algebraic number field (or simply number field) F is a finite degree (and hence algebraic) field extension of the field of rational numbers Q. Thus F is a field that contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory.

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Algebraic number theory

Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations.

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Algebraic solution

An algebraic solution or solution in radicals is a closed-form expression, and more specifically a closed-form algebraic expression, that is the solution of an algebraic equation in terms of the coefficients, relying only on addition, subtraction, multiplication, division, raising to integer powers, and the extraction of nth roots (square roots, cube roots, and other integer roots).

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Algebraic surface

In mathematics, an algebraic surface is an algebraic variety of dimension two.

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Algebraically closed field

In abstract algebra, an algebraically closed field F contains a root for every non-constant polynomial in F, the ring of polynomials in the variable x with coefficients in F.

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American Mathematical Monthly

The American Mathematical Monthly is a mathematical journal founded by Benjamin Finkel in 1894.

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The amplitude of a periodic variable is a measure of its change over a single period (such as time or spatial period).

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Amplitude modulation

Amplitude modulation (AM) is a modulation technique used in electronic communication, most commonly for transmitting information via a radio carrier wave.

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Analytic continuation

In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function.

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Analytic signal

In mathematics and signal processing, an analytic signal is a complex-valued function that has no negative frequency components.

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Angle notation

Angle notation or phasor notation is a notation used in electronics.

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Angular frequency

In physics, angular frequency ω (also referred to by the terms angular speed, radial frequency, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate.

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Anno Domini

The terms anno Domini (AD) and before Christ (BC) are used to label or number years in the Julian and Gregorian calendars.

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Applied mathematics

Applied mathematics is the application of mathematical methods by different fields such as science, engineering, business, computer science, and industry.

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Argument (complex analysis)

In mathematics, the argument is a multi-valued function operating on the nonzero complex numbers.

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The function \operatorname (y,x) or \operatorname (y,x) is defined as the angle in the Euclidean plane, given in rad, between the positive x-axis and the ray to the Points in the upper half-plane deliver values in points with.

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Augustin-Louis Cauchy

Baron Augustin-Louis Cauchy FRS FRSE (21 August 178923 May 1857) was a French mathematician, engineer and physicist who made pioneering contributions to several branches of mathematics, including: mathematical analysis and continuum mechanics.

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In mathematics, an automorphism is an isomorphism from a mathematical object to itself.

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Axiom of choice

In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that the Cartesian product of a collection of non-empty sets is non-empty.

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Base (topology)

In mathematics, a base (or basis) B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B.We are using a convention that the union of empty collection of sets is the empty set.

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Bernhard Riemann

Georg Friedrich Bernhard Riemann (17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry.

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BIBO stability

In signal processing, specifically control theory, bounded-input, bounded-output (BIBO) stability is a form of stability for linear signals and systems that take inputs.

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Blackboard bold

Blackboard bold is a typeface style that is often used for certain symbols in mathematical texts, in which certain lines of the symbol (usually vertical or near-vertical lines) are doubled.

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Branch point

In the mathematical field of complex analysis, a branch point of a multi-valued function (usually referred to as a "multifunction" in the context of complex analysis) is a point such that the function is discontinuous when going around an arbitrarily small circuit around this point.

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C. V. Mourey


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A capacitor is a passive two-terminal electrical component that stores potential energy in an electric field.

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Cardinality of the continuum

In set theory, the cardinality of the continuum is the cardinality or “size” of the set of real numbers \mathbb R, sometimes called the continuum.

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Carl Friedrich Gauss

Johann Carl Friedrich Gauss (Gauß; Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields, including algebra, analysis, astronomy, differential geometry, electrostatics, geodesy, geophysics, magnetic fields, matrix theory, mechanics, number theory, optics and statistics.

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Carl Gustav Jacob Jacobi

Carl Gustav Jacob Jacobi (10 December 1804 – 18 February 1851) was a German mathematician, who made fundamental contributions to elliptic functions, dynamics, differential equations, and number theory.

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Cartesian coordinate system

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length.

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Cartography (from Greek χάρτης chartēs, "papyrus, sheet of paper, map"; and γράφειν graphein, "write") is the study and practice of making maps.

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Caspar Wessel

Caspar Wessel (June 8, 1745, Vestby – March 25, 1818, Copenhagen) was a Danish–Norwegian mathematician and cartographer.

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Casus irreducibilis

In algebra, casus irreducibilis (Latin for "the irreducible case") is one of the cases that may arise in attempting to solve a cubic equation with integer coefficients with roots that are expressed with radicals.

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Cauchy–Riemann equations

In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which, together with certain continuity and differentiability criteria, form a necessary and sufficient condition for a complex function to be complex differentiable, that is, holomorphic.

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Cayley–Dickson construction

In mathematics, the Cayley–Dickson construction, named after Arthur Cayley and Leonard Eugene Dickson, produces a sequence of algebras over the field of real numbers, each with twice the dimension of the previous one.

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Characteristic (algebra)

In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0) if the sum does indeed eventually attain 0.

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Circle group

In mathematics, the circle group, denoted by T, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers The circle group forms a subgroup of C×, the multiplicative group of all nonzero complex numbers.

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Cis (mathematics)

is a less commonly used mathematical notation defined by, where is the cosine function, is the imaginary unit and is the sine.

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In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series or any expression; it is usually a number, but may be any expression.

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Commutative property

In mathematics, a binary operation is commutative if changing the order of the operands does not change the result.

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Compass-and-straightedge construction

Compass-and-straightedge construction, also known as ruler-and-compass construction or classical construction, is the construction of lengths, angles, and other geometric figures using only an idealized ruler and compass.

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Complex analysis

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers.

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Complex conjugate

In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign.

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Complex geometry

In mathematics, complex geometry is the study of complex manifolds and functions of several complex variables.

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Complex logarithm

In complex analysis, a complex logarithm of the non-zero complex number, denoted by, is defined to be any complex number for which.

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Complex number

A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.

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Complex plane

In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the perpendicular imaginary axis.

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Complex-base system

In arithmetic, a complex-base system is a positional numeral system whose radix is an imaginary (proposed by Donald Knuth in 1955) or complex number (proposed by S. Khmelnik in 1964 and Walter F. Penney in 1965W. Penney, A "binary" system for complex numbers, JACM 12 (1965) 247-248.).

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Complex-valued function

In mathematics, a complex-valued function (not to be confused with complex variable function) is a function whose values are complex numbers.

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Congruence (geometry)

In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other.

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Connected space

In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets.

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Continuous function

In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.

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Contour integration

In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane.

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Control theory

Control theory in control systems engineering deals with the control of continuously operating dynamical systems in engineered processes and machines.

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Convergent series

In mathematics, a series is the sum of the terms of an infinite sequence of numbers.

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In mathematics, if G is a group, and H is a subgroup of G, and g is an element of G, then Only when H is normal will the set of right cosets and the set of left cosets of H coincide, which is one definition of normality of a subgroup.

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Cubic function

In algebra, a cubic function is a function of the form in which is nonzero.

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Data compression

In signal processing, data compression, source coding, or bit-rate reduction involves encoding information using fewer bits than the original representation.

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De Moivre's formula

In mathematics, de Moivre's formula (also known as de Moivre's theorem and de Moivre's identity), named after Abraham de Moivre, states that for any complex number (and, in particular, for any real number) and integer it holds that where is the imaginary unit.

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In linear algebra, the determinant is a value that can be computed from the elements of a square matrix.

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Differential equation

A differential equation is a mathematical equation that relates some function with its derivatives.

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Digital data

Digital data, in information theory and information systems, is the discrete, discontinuous representation of information or works.

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Digital image processing

In computer science, Digital image processing is the use of computer algorithms to perform image processing on digital images.

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Digital signal processing

Digital signal processing (DSP) is the use of digital processing, such as by computers or more specialized digital signal processors, to perform a wide variety of signal processing operations.

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In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it.

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Distributive property

In abstract algebra and formal logic, the distributive property of binary operations generalizes the distributive law from boolean algebra and elementary algebra.

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Dover Publications

Dover Publications, also known as Dover Books, is an American book publisher founded in 1941 by Hayward Cirker and his wife, Blanche.

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E (mathematical constant)

The number is a mathematical constant, approximately equal to 2.71828, which appears in many different settings throughout mathematics.

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Eigenvalues and eigenvectors

In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that changes by only a scalar factor when that linear transformation is applied to it.

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Eisenstein integer

In mathematics, Eisenstein integers (named after Gotthold Eisenstein), occasionally also known as Eulerian integers (after Leonhard Euler), are complex numbers of the form where and are integers and is a primitive (hence non-real) cube root of unity.

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Electric current

An electric current is a flow of electric charge.

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Electrical engineering

Electrical engineering is a professional engineering discipline that generally deals with the study and application of electricity, electronics, and electromagnetism.

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Electrical impedance

Electrical impedance is the measure of the opposition that a circuit presents to a current when a voltage is applied.

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Electrical network

An electrical network is an interconnection of electrical components (e.g. batteries, resistors, inductors, capacitors, switches) or a model of such an interconnection, consisting of electrical elements (e.g. voltage sources, current sources, resistances, inductances, capacitances).

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Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electrically charged particles.

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Electronics is the discipline dealing with the development and application of devices and systems involving the flow of electrons in a vacuum, in gaseous media, and in semiconductors.

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Elementary function

In mathematics, an elementary function is a function of one variable which is the composition of a finite number of arithmetic operations, exponentials, logarithms, constants, and solutions of algebraic equations (a generalization of ''n''th roots).

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Elements of Algebra

Elements of Algebra is an elementary mathematics textbook written by mathematician Leonhard Euler and originally published in 1770 in German.

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In mathematics, an ellipse is a curve in a plane surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve.

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Essential singularity

In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits odd behavior.

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Euclidean vector

In mathematics, physics, and engineering, a Euclidean vector (sometimes called a geometric or spatial vector, or—as here—simply a vector) is a geometric object that has magnitude (or length) and direction.

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Euler's formula

Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function.

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Euler's identity

In mathematics, Euler's identity (also known as Euler's equation) is the equality where Euler's identity is named after the Swiss mathematician Leonhard Euler.

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Exponential function

In mathematics, an exponential function is a function of the form in which the argument occurs as an exponent.

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Exponentiation is a mathematical operation, written as, involving two numbers, the base and the exponent.

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Felix Klein

Christian Felix Klein (25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group theory.

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Fermat's theorem on sums of two squares

In additive number theory, Fermat's theorem on sums of two squares states that an odd prime p can be expressed as: p.

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Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.

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Field extension

In mathematics, and in particular, algebra, a field E is an extension field of a field F if E contains F and the operations of F are those of E restricted to F. Equivalently, F is a subfield of E. For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers.

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Fluid dynamics

In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids - liquids and gases.

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Focus (geometry)

In geometry, focuses or foci, singular focus, are special points with reference to which any of a variety of curves is constructed.

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Fourier analysis

In mathematics, Fourier analysis is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions.

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Fourier transform

The Fourier transform (FT) decomposes a function of time (a signal) into the frequencies that make it up, in a way similar to how a musical chord can be expressed as the frequencies (or pitches) of its constituent notes.

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In mathematics, a fractal is an abstract object used to describe and simulate naturally occurring objects.

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Fraction (mathematics)

A fraction (from Latin fractus, "broken") represents a part of a whole or, more generally, any number of equal parts.

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Frequency is the number of occurrences of a repeating event per unit of time.

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Frequency domain

In electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time.

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In geometry, a frustum (plural: frusta or frustums) is the portion of a solid (normally a cone or pyramid) that lies between one or two parallel planes cutting it.

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Fundamental theorem of algebra

The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root.

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G. H. Hardy

Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis.

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Galois theory

In the field of algebra within mathematics, Galois theory, provides a connection between field theory and group theory.

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Gaussian integer

In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers.

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General relativity

General relativity (GR, also known as the general theory of relativity or GTR) is the geometric theory of gravitation published by Albert Einstein in 1915 and the current description of gravitation in modern physics.

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Gerolamo Cardano

Gerolamo (or Girolamo, or Geronimo) Cardano (Jérôme Cardan; Hieronymus Cardanus; 24 September 1501 – 21 September 1576) was an Italian polymath, whose interests and proficiencies ranged from being a mathematician, physician, biologist, physicist, chemist, astrologer, astronomer, philosopher, writer, and gambler.

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Giusto Bellavitis

Giusto Bellavitis (22 November 1803 – 6 November 1880) was an Italian mathematician, senator, and municipal councilor.

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Graph of a function

In mathematics, the graph of a function f is, formally, the set of all ordered pairs, and, in practice, the graphical representation of this set.

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Greek mathematics

Greek mathematics refers to mathematics texts and advances written in Greek, developed from the 7th century BC to the 4th century AD around the shores of the Eastern Mediterranean.

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Henri Poincaré

Jules Henri Poincaré (29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science.

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Hermann Schwarz

Karl Hermann Amandus Schwarz (25 January 1843 – 30 November 1921) was a German mathematician, known for his work in complex analysis.

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Hero of Alexandria

Hero of Alexandria (ἭρωνGenitive: Ἥρωνος., Heron ho Alexandreus; also known as Heron of Alexandria; c. 10 AD – c. 70 AD) was a mathematician and engineer who was active in his native city of Alexandria, Roman Egypt.

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Hilbert space

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space.

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Holomorphic function

In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain.

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Hurwitz's theorem (composition algebras)

In mathematics, Hurwitz's theorem is a theorem of Adolf Hurwitz (1859–1919), published posthumously in 1923, solving the Hurwitz problem for finite-dimensional unital real non-associative algebras endowed with a positive-definite quadratic form.

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Hyperbolic function

In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions.

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Hypercomplex number

In mathematics, a hypercomplex number is a traditional term for an element of a unital algebra over the field of real numbers.

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Ideal (ring theory)

In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring.

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Identity matrix

In linear algebra, the identity matrix, or sometimes ambiguously called a unit matrix, of size n is the n × n square matrix with ones on the main diagonal and zeros elsewhere.

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If and only if

In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.

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Imaginary number

An imaginary number is a complex number that can be written as a real number multiplied by the imaginary unit,j is usually used in Engineering contexts where i has other meanings (such as electrical current) which is defined by its property.

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Imaginary unit

The imaginary unit or unit imaginary number is a solution to the quadratic equation.

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Improper integral

In mathematical analysis, an improper integral is the limit of a definite integral as an endpoint of the interval(s) of integration approaches either a specified real number, \infty, -\infty, or in some instances as both endpoints approach limits.

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Indeterminate (variable)

In mathematics, and particularly in formal algebra, an indeterminate is a symbol that is treated as a variable, but does not stand for anything else but itself and is used as a placeholder in objects such as polynomials and formal power series.

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An inductor, also called a coil, choke or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it.

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Infinite set

In set theory, an infinite set is a set that is not a finite set.

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In numerous fields of study, the component of instability within a system is generally characterized by some of the outputs or internal states growing without bounds.

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Interval (mathematics)

In mathematics, a (real) interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set.

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Inverse trigonometric functions

In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains).

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Inversive geometry

In geometry, inversive geometry is the study of those properties of figures that are preserved by a generalization of a type of transformation of the Euclidean plane, called inversion.

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Involution (mathematics)

In mathematics, an involution, or an involutory function, is a function that is its own inverse, for all in the domain of.

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Irreducible polynomial

In mathematics, an irreducible polynomial is, roughly speaking, a non-constant polynomial that cannot be factored into the product of two non-constant polynomials.

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In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism.

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Jean le Rond d'Alembert

Jean-Baptiste le Rond d'Alembert (16 November 1717 – 29 October 1783) was a French mathematician, mechanician, physicist, philosopher, and music theorist.

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Jean-Robert Argand

Jean-Robert Argand (July 18, 1768 – August 13, 1822) was an amateur mathematician.

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John Wallis

John Wallis (3 December 1616 – 8 November 1703) was an English clergyman and mathematician who is given partial credit for the development of infinitesimal calculus.

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John Wiley & Sons

John Wiley & Sons, Inc., also referred to as Wiley, is a global publishing company that specializes in academic publishing.

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Julia set

In the context of complex dynamics, a topic of mathematics, the Julia set and the Fatou set are two complementary sets (Julia 'laces' and Fatou 'dusts') defined from a function.

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Karl Weierstrass

Karl Theodor Wilhelm Weierstrass (Weierstraß; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis".

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Laplace transform

In mathematics, the Laplace transform is an integral transform named after its discoverer Pierre-Simon Laplace.

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Leonhard Euler

Leonhard Euler (Swiss Standard German:; German Standard German:; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, logician and engineer, who made important and influential discoveries in many branches of mathematics, such as infinitesimal calculus and graph theory, while also making pioneering contributions to several branches such as topology and analytic number theory.

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Lie algebra

In mathematics, a Lie algebra (pronounced "Lee") is a vector space \mathfrak g together with a non-associative, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g; (x, y) \mapsto, called the Lie bracket, satisfying the Jacobi identity.

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Limit (mathematics)

In mathematics, a limit is the value that a function (or sequence) "approaches" as the input (or index) "approaches" some value.

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Limit of a sequence

As the positive integer n becomes larger and larger, the value n\cdot \sin\bigg(\frac1\bigg) becomes arbitrarily close to 1.

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Linear combination

In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants).

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Linear complex structure

In mathematics, a complex structure on a real vector space V is an automorphism of V that squares to the minus identity, −I.

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Linear differential equation

In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form where,..., and are arbitrary differentiable functions that do not need to be linear, and are the successive derivatives of an unknown function of the variable.

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Liouville's theorem (complex analysis)

In complex analysis, Liouville's theorem, named after Joseph Liouville, states that every bounded entire function must be constant.

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Local field

In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.

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Locally compact space

In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space.

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Machin-like formula

In mathematics, Machin-like formulae are a popular technique for computing pi to a large number of digits.

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Mandelbrot set

The Mandelbrot set is the set of complex numbers c for which the function f_c(z).

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Marden's theorem

In mathematics, Marden's theorem, named after Morris Marden but proven much earlier by Jörg Siebeck, gives a geometric relationship between the zeroes of a third-degree polynomial with complex coefficients and the zeroes of its derivative.

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Marginal stability

In the theory of dynamical systems and control theory, a linear time-invariant system is marginally stable if it is neither asymptotically stable nor unstable.

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Mathematical analysis

Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.

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Mathematical formulation of quantum mechanics

The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics.

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Matrix (mathematics)

In mathematics, a matrix (plural: matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.

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Matrix mechanics

Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925.

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Matrix multiplication

In mathematics, matrix multiplication or matrix product is a binary operation that produces a matrix from two matrices with entries in a field, or, more generally, in a ring or even a semiring.

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Maximal ideal

In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all proper ideals.

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Maximum power transfer theorem

In electrical engineering, the maximum power transfer theorem states that, to obtain maximum external power from a source with a finite internal resistance, the resistance of the load must equal the resistance of the source as viewed from its output terminals.

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McGraw-Hill Education

McGraw-Hill Education (MHE) is a learning science company and one of the "big three" educational publishers that provides customized educational content, software, and services for pre-K through postgraduate education.

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Meromorphic function

In the mathematical field of complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all of D except for a discrete set of isolated points, which are poles of the function.

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Metric (mathematics)

In mathematics, a metric or distance function is a function that defines a distance between each pair of elements of a set.

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Metric space

In mathematics, a metric space is a set for which distances between all members of the set are defined.

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Minimum phase

In control theory and signal processing, a linear, time-invariant system is said to be minimum-phase if the system and its inverse are causal and stable.

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Multiplicative inverse

In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1.

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Multivalued function

In mathematics, a multivalued function from a domain to a codomain is a heterogeneous relation.

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Natural logarithm

The natural logarithm of a number is its logarithm to the base of the mathematical constant ''e'', where e is an irrational and transcendental number approximately equal to.

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Negative number

In mathematics, a negative number is a real number that is less than zero.

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Neighbourhood (mathematics)

In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space.

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Network analysis (electrical circuits)

A network, in the context of electronics, is a collection of interconnected components.

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Niccolò Fontana Tartaglia

Niccolò Fontana Tartaglia (1499/1500, Brescia – 13 December 1557, Venice) was a Venetian mathematician, engineer (designing fortifications), a surveyor (of topography, seeking the best means of defense or offense) and a bookkeeper from the then-Republic of Venice (now part of Italy).

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Nichols plot

The Nichols plot is a plot used in signal processing and control design, named after American engineer Nathaniel B. Nichols.

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Nicolas Bourbaki

Nicolas Bourbaki is the collective pseudonym under which a group of (mainly French) 20th-century mathematicians, with the aim of reformulating mathematics on an extremely abstract and formal but self-contained basis, wrote a series of books beginning in 1935.

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Niels Henrik Abel

Niels Henrik Abel (5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields.

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In geometry, a nonagon or enneagon is a nine-sided polygon or 9-gon.

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Nth root

In mathematics, an nth root of a number x, where n is usually assumed to be a positive integer, is a number r which, when raised to the power n yields x: where n is the degree of the root.

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A number is a mathematical object used to count, measure and also label.

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Number line

In basic mathematics, a number line is a picture of a graduated straight line that serves as abstraction for real numbers, denoted by \mathbb.

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Number theory

Number theory, or in older usage arithmetic, is a branch of pure mathematics devoted primarily to the study of the integers.

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Nyquist stability criterion

In control theory and stability theory, the Nyquist stability criterion, discovered by Swedish-American electrical engineer Harry Nyquist at Bell Telephone Laboratories in 1932, on is a graphical technique for determining the stability of a dynamical system.

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In mathematics, the octonions are a normed division algebra over the real numbers, usually represented by the capital letter O, using boldface O or blackboard bold \mathbb O. There are three lower-dimensional normed division algebras over the reals: the real numbers R themselves, the complex numbers C, and the quaternions H. The octonions have eight dimensions; twice the number of dimensions of the quaternions, of which they are an extension.

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Open set

In topology, an open set is an abstract concept generalizing the idea of an open interval in the real line.

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Ordered field

In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations.

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Ordered pair

In mathematics, an ordered pair (a, b) is a pair of objects.

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Orientation (geometry)

In geometry the orientation, angular position, or attitude of an object such as a line, plane or rigid body is part of the description of how it is placed in the space it occupies.

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Origin (mathematics)

In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space.

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Ostrowski's theorem

In number theory, Ostrowski's theorem, due to Alexander Ostrowski (1916), states that every non-trivial absolute value on the rational numbers Q is equivalent to either the usual real absolute value or a p-adic absolute value.

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Otto Hölder

Otto Ludwig Hölder (December 22, 1859 – August 29, 1937) was a German mathematician born in Stuttgart.

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Oxford University Press

Oxford University Press (OUP) is the largest university press in the world, and the second oldest after Cambridge University Press.

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P-adic number

In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems.

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In Euclidean geometry, a parallelogram is a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides.

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Phase (waves)

Phase is the position of a point in time (an instant) on a waveform cycle.

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In physics and engineering, a phasor (a portmanteau of phase vector), is a complex number representing a sinusoidal function whose amplitude (A), angular frequency (ω), and initial phase (θ) are time-invariant.

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The number is a mathematical constant.

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In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

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Polynomial ring

In mathematics, especially in the field of abstract algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field.

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Positive real numbers

In mathematics, the set of positive real numbers, \mathbb_.

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Potential flow

In fluid dynamics, potential flow describes the velocity field as the gradient of a scalar function: the velocity potential.

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Power series

In mathematics, a power series (in one variable) is an infinite series of the form where an represents the coefficient of the nth term and c is a constant.

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Prime ideal

In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers.

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Prime number

A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.

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Prime number theorem

In number theory, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers.

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Princeton University Press

Princeton University Press is an independent publisher with close connections to Princeton University.

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Principal ideal domain

In abstract algebra, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element.

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Principal value

In complex analysis, the principal values of a multivalued function are the values along one chosen branch of that function, so that it is single-valued.

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Puiseux series

In mathematics, Puiseux series are a generalization of power series that allow for negative and fractional exponents of the indeterminate T. They were first introduced by Isaac Newton in 1676 and rediscovered by Victor Puiseux in 1850.

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A pyramid (from πυραμίς) is a structure whose outer surfaces are triangular and converge to a single point at the top, making the shape roughly a pyramid in the geometric sense.

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Pythagorean theorem

In mathematics, the Pythagorean theorem, also known as Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle.

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Quantum field theory

In theoretical physics, quantum field theory (QFT) is the theoretical framework for constructing quantum mechanical models of subatomic particles in particle physics and quasiparticles in condensed matter physics.

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Quantum mechanics

Quantum mechanics (QM; also known as quantum physics, quantum theory, the wave mechanical model, or matrix mechanics), including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles.

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Quartic function

In algebra, a quartic function is a function of the form where a is nonzero, which is defined by a polynomial of degree four, called a quartic polynomial.

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In mathematics, the quaternions are a number system that extends the complex numbers.

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Quotient ring

In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient groups of group theory and the quotient spaces of linear algebra.

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The radian (SI symbol rad) is the SI unit for measuring angles, and is the standard unit of angular measure used in many areas of mathematics.

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In classical geometry, a radius of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length.

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Rafael Bombelli

Rafael Bombelli (baptised on 20 January 1526; died 1572) was an Italian mathematician.

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Rational number

In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.

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Rational root theorem

In algebra, the rational root theorem (or rational root test, rational zero theorem, rational zero test or p/q theorem) states a constraint on rational solutions of a polynomial equation with integer coefficients.

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Rationalisation (mathematics)

In elementary algebra, root rationalisation is a process by which radicals in the denominator of an algebraic fraction are eliminated.

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Real analysis

In mathematics, real analysis is the branch of mathematical analysis that studies the behavior of real numbers, sequences and series of real numbers, and real-valued functions.

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Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

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Recurrence relation

In mathematics, a recurrence relation is an equation that recursively defines a sequence or multidimensional array of values, once one or more initial terms are given: each further term of the sequence or array is defined as a function of the preceding terms.

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Reflection symmetry

Reflection symmetry, line symmetry, mirror symmetry, mirror-image symmetry, is symmetry with respect to reflection.

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Regular representation

In mathematics, and in particular the theory of group representations, the regular representation of a group G is the linear representation afforded by the group action of G on itself by translation.

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René Descartes

René Descartes (Latinized: Renatus Cartesius; adjectival form: "Cartesian"; 31 March 1596 – 11 February 1650) was a French philosopher, mathematician, and scientist.

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Representation theory

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures.

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A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element.

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Richard Dedekind

Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to abstract algebra (particularly ring theory), axiomatic foundation for the natural numbers, algebraic number theory and the definition of the real numbers.

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Riemann sphere

In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane, the complex plane plus a point at infinity.

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Riemann zeta function

The Riemann zeta function or Euler–Riemann zeta function,, is a function of a complex variable s that analytically continues the sum of the Dirichlet series which converges when the real part of is greater than 1.

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Right angle

In geometry and trigonometry, a right angle is an angle of exactly 90° (degrees), corresponding to a quarter turn.

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Ring (mathematics)

In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra.

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Roger Penrose

Sir Roger Penrose (born 8 August 1931) is an English mathematical physicist, mathematician and philosopher of science.

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Root locus

In control theory and stability theory, root locus analysis is a graphical method for examining how the roots of a system change with variation of a certain system parameter, commonly a gain within a feedback system.

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Root of unity

In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that gives 1 when raised to some positive integer power.

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Rotation matrix

In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space.

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Royal Danish Academy of Fine Arts

The Royal Danish Academy of Fine Arts (Det Kongelige Danske Kunstakademi) has provided education in the arts for more than 250 years, playing its part in the development of the art of Denmark.

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Schrödinger equation

In quantum mechanics, the Schrödinger equation is a mathematical equation that describes the changes over time of a physical system in which quantum effects, such as wave–particle duality, are significant.

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In abstract algebra, the sedenions form a 16-dimensional noncommutative and nonassociative algebra over the reals obtained by applying the Cayley–Dickson construction to the octonions.

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Series (mathematics)

In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity.

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Set (mathematics)

In mathematics, a set is a collection of distinct objects, considered as an object in its own right.

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Sign function

In mathematics, the sign function or signum function (from signum, Latin for "sign") is an odd mathematical function that extracts the sign of a real number.

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Signal processing

Signal processing concerns the analysis, synthesis, and modification of signals, which are broadly defined as functions conveying "information about the behavior or attributes of some phenomenon", such as sound, images, and biological measurements.

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In mathematics, the sine is a trigonometric function of an angle.

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Sine wave

A sine wave or sinusoid is a mathematical curve that describes a smooth periodic oscillation.

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In physics, sound is a vibration that typically propagates as an audible wave of pressure, through a transmission medium such as a gas, liquid or solid.

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In physics, spacetime is any mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum.

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Special relativity

In physics, special relativity (SR, also known as the special theory of relativity or STR) is the generally accepted and experimentally well-confirmed physical theory regarding the relationship between space and time.

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In geometry and physics, spinors are elements of a (complex) vector space that can be associated with Euclidean space.

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Split-complex number

In abstract algebra, a split complex number (or hyperbolic number, also perplex number, double number) has two real number components x and y, and is written z.

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Square (algebra)

In mathematics, a square is the result of multiplying a number by itself.

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Square matrix

In mathematics, a square matrix is a matrix with the same number of rows and columns.

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Square root

In mathematics, a square root of a number a is a number y such that; in other words, a number y whose square (the result of multiplying the number by itself, or) is a. For example, 4 and −4 are square roots of 16 because.

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Square root of 2

The square root of 2, or the (1/2)th power of 2, written in mathematics as or, is the positive algebraic number that, when multiplied by itself, gives the number 2.

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Steiner inellipse

In geometry, the Steiner inellipse,Weisstein, E. "Steiner Inellipse" — From MathWorld, A Wolfram Web Resource, http://mathworld.wolfram.com/SteinerInellipse.html.

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Stereographic projection

In geometry, the stereographic projection is a particular mapping (function) that projects a sphere onto a plane.

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Subtraction is an arithmetic operation that represents the operation of removing objects from a collection.

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Taylor series

In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.

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In mathematics, tensors are geometric objects that describe linear relations between geometric vectors, scalars, and other tensors.

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Time domain

Time domain is the analysis of mathematical functions, physical signals or time series of economic or environmental data, with respect to time.

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Tom M. Apostol

Tom Mike Apostol (August 20, 1923 – May 8, 2016) was an American analytic number theorist and professor at the California Institute of Technology, best known as the author of widely used mathematical textbooks.

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Topological ring

In mathematics, a topological ring is a ring R which is also a topological space such that both the addition and the multiplication are continuous as maps where R × R carries the product topology.

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Topological space

In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.

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In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.

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Total order

In mathematics, a linear order, total order, simple order, or (non-strict) ordering is a binary relation on some set X, which is antisymmetric, transitive, and a connex relation.

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Transcendence degree

In abstract algebra, the transcendence degree of a field extension L /K is a certain rather coarse measure of the "size" of the extension.

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In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal, that is it switches the row and column indices of the matrix by producing another matrix denoted as AT (also written A′, Atr, tA or At).

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A triangle is a polygon with three edges and three vertices.

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Triangle inequality

In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.

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Trigonometric functions

In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are functions of an angle.

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Tristan Needham

Tristan Needham is a mathematician and professor of mathematics at University of San Francisco.

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Turn (geometry)

A turn is a unit of plane angle measurement equal to 2pi radians, 360 degrees or 400 gradians.

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Two-dimensional space

Two-dimensional space or bi-dimensional space is a geometric setting in which two values (called parameters) are required to determine the position of an element (i.e., point).

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Vector space

A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.

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Video is an electronic medium for the recording, copying, playback, broadcasting, and display of moving visual media.

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Voltage, electric potential difference, electric pressure or electric tension (formally denoted or, but more often simply as V or U, for instance in the context of Ohm's or Kirchhoff's circuit laws) is the difference in electric potential between two points.

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A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases, and then decreases back to zero.

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Wick rotation

In physics, Wick rotation, named after Gian Carlo Wick, is a method of finding a solution to a mathematical problem in Minkowski space from a solution to a related problem in Euclidean space by means of a transformation that substitutes an imaginary-number variable for a real-number variable.

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William Rowan Hamilton

Sir William Rowan Hamilton MRIA (4 August 1805 – 2 September 1865) was an Irish mathematician who made important contributions to classical mechanics, optics, and algebra.

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Winding number

In mathematics, the winding number of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point.

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Zero of a function

In mathematics, a zero, also sometimes called a root, of a real-, complex- or generally vector-valued function f is a member x of the domain of f such that f(x) vanishes at x; that is, x is a solution of the equation f(x).

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Zeros and poles

In mathematics, a zero of a function is a value such that.

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(ε, δ)-definition of limit

In calculus, the (ε, δ)-definition of limit ("epsilon–delta definition of limit") is a formalization of the notion of limit.

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2 × 2 real matrices

In mathematics, the associative algebra of real matrices is denoted by M(2, R).

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[1] https://en.wikipedia.org/wiki/Complex_number

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