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Complex Numbers
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is a solution of the equation x2 = −1, which is called an imaginary number because there is no real number that satisfies this equation. For the complex number a + bi, a is called the real part, and b is called the imaginary part. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, and are fundamental in many aspects of the scientific description of the natural world.[1][2] The complex number system can be defined as the algebraic extension of the ordinary real numbers by an imaginary number i.[3] This means that complex numbers can be added, subtracted, and multiplied, as polynomials in the variable i, with the rule i2 = −1 imposed. Furthermore, complex numbers can also be divided by nonzero complex numbers
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Argand Diagram
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. It can be thought of as a modified Cartesian plane, with the real part of a complex number represented by a displacement along the x-axis, and the imaginary part by a displacement along the y-axis.[1] The concept of the complex plane allows a geometric interpretation of complex numbers. Under addition, they add like vectors. The multiplication of two complex numbers can be expressed most easily in polar coordinates—the magnitude or modulus of the product is the product of the two absolute values, or moduli, and the angle or argument of the product is the sum of the two angles, or arguments
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Casus Irreducibilis
In algebra, casus irreducibilis ( Latin
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. Specifically, if a cubic polynomial is irreducible over the rational numbers and has three real roots, then in order to express the roots with radicals, one must introduce complex-valued expressions, even though the resulting expressions are ultimately real-valued
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Vector (geometric)
In mathematics, physics, and engineering, a Euclidean vector (sometimes called a geometric[1] or spatial vector,[2] or—as here—simply a vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors according to vector algebra. A Euclidean vector
Euclidean vector
is frequently represented by a line segment with a definite direction, or graphically as an arrow, connecting an initial point A with a terminal point B,[3] and denoted by A B → . displaystyle overrightarrow AB
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Cartesian Coordinate System
A Cartesian coordinate system
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. Each reference line is called a coordinate axis or just axis (plural axes) of the system, and the point where they meet is its origin, at ordered pair (0, 0)
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Jean-Robert Argand
Jean-Robert Argand (July 18, 1768 – August 13, 1822) was an amateur mathematician. In 1806, while managing a bookstore in Paris, he published the idea of geometrical interpretation of complex numbers known as the Argand diagram
Argand diagram
and is known for the first rigorous proof of the Fundamental Theorem of Algebra. Contents1 Life 2 See also 3 References 4 External links 5 Further readingLife[edit] Jean-Robert Argand was born in Geneva, Switzerland
Switzerland
to Jacques Argand and Eve Carnac. His background and education are mostly unknown
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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.[1] Namely, it is the imaginary rotation that is needed to move the object from a reference placement to its current placement. A rotation may not be enough to reach the current placement. It may be necessary to add an imaginary translation, called the object's location (or position, or linear position). The location and orientation together fully describe how the object is placed in space
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Turn (geometry)
A turn is a unit of plane angle measurement equal to 2π radians, 360 degrees or 400 gradians. A turn is also referred to as a cycle (abbreviated cyc), revolution (abbreviated rev), complete rotation (abbreviated rot) or full circle. Subdivisions of a turn include half turns, quarter turns, centiturns, milliturns, points, etc.Contents1 Subdivision of turns 2 History 3 Unit conversion 4 Tau
Tau
proposals 5 Examples of use 6 Kinematics
Kinematics
of turns 7 See also 8 Notes and references 9 External linksSubdivision of turns[edit] A turn can be divided in 100 centiturns or 1000 milliturns, with each milliturn corresponding to an angle of 0.36°, which can also be written as 21′ 36″. A protractor divided in centiturns is normally called a percentage protractor. Binary fractions of a turn are also used. Sailors have traditionally divided a turn into 32 compass points
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Right Angle
In geometry and trigonometry, a right angle is an angle of exactly 90° (degrees),[1] corresponding to a quarter turn.[2] If a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles.[3] The term is a calque of Latin angulus rectus; here rectus means "upright", referring to the vertical perpendicular to a horizontal base line. Closely related and important geometrical concepts are perpendicular lines, meaning lines that form right angles at their point of intersection, and orthogonality, which is the property of forming right angles, usually applied to vectors. The presence of a right angle in a triangle is the defining factor for right triangles,[4] making the right angle basic to trigonometry.Contents1 In elementary geometry 2 Symbols 3 Euclid 4 Conversion to other units 5 Rule of 3-4-5 6 Thales' theorem 7 See also 8 ReferencesIn elementary geometry[edit] A rectangle is a quadrilateral with four right angles
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Trigonometric Functions
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions[1][2]) are functions of an angle. They relate the angles of a triangle to the lengths of its sides. Trigonometric functions
Trigonometric functions
are important in the study of triangles and modeling periodic phenomena, among many other applications. The most familiar trigonometric functions are the sine, cosine, and tangent. In the context of the standard unit circle (a circle with radius 1 unit), where a triangle is formed by a ray starting at the origin and making some angle with the x-axis, the sine of the angle gives the length of the y-component (the opposite to the angle or the rise) of the triangle, the cosine gives the length of the x-component (the adjacent of the angle or the run), and the tangent function gives the slope (y-component divided by the x-component). More precise definitions are detailed below
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Negative Numbers
In mathematics, a negative number is a real number that is less than zero. Negative numbers represent opposites. If positive represents a movement to the right, negative represents a movement to the left. If positive represents above sea level, then negative represents below sea level. If positive represents a deposit, negative represents a withdrawal. They are often used to represent the magnitude of a loss or deficiency. A debt that is owed may be thought of as a negative asset, a decrease in some quantity may be thought of as a negative increase. If a quantity may have either of two opposite senses, then one may choose to distinguish between those senses—perhaps arbitrarily—as positive and negative. In the medical context of fighting a tumor, an expansion could be thought of as a negative shrinkage. Negative numbers are used to describe values on a scale that goes below zero, such as the Celsius and Fahrenheit
Fahrenheit
scales for temperature
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Rational Root Test
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 a n x n + a n − 1 x n − 1 + ⋯ + a 0 = 0 displaystyle a_ n x^ n +a_ n-1 x^ n-1 +cdots +a_ 0 =0 with integer coefficients
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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. The property of irreducibility depends on the nature of the coefficients that are accepted for the possible factors, that is, the field or ring to which the coefficients of the polynomial and its possible factors are supposed to belong. For example, the polynomial x2 − 2 is a polynomial with integer coefficients, but, as every integer is also a real number, it is also a polynomial with real coefficients. It is irreducible if it is considered as a polynomial with integer coefficients, but it factors as ( x − 2 ) ( x + 2 ) displaystyle (x- sqrt 2 )(x+ sqrt 2 ) if it is considered as a polynomial with real coefficients
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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[x], the ring of polynomials in the variable x with coefficients in F.Contents1 Examples 2 Equivalent properties2.1 The only irreducible polynomials are those of degree one 2.2 Every polynomial is a product of first degree polynomials 2.3 Polynomials of prime degree have roots 2.4 The field has no proper algebraic extension 2.5 The field has no proper finite extension 2.6 Every endomorphism of Fn has some eigenvector 2.7 Decomposition of rational expressions 2.8 Relatively prime polynomials and roots3 Other properties 4 Notes 5 ReferencesExamples[edit] As an example, the field of real numbers is not algebraically closed, because the polynomial equation x2 + 1 = 0  has no solution in real numbers, even though all its coefficients (1 and 0) are real
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Ordered Pair
In mathematics, an ordered pair (a, b) is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (a, b) is different from the ordered pair (b, a) unless a = b. (In contrast, the unordered pair a, b equals the unordered pair b, a .) Ordered pairs are also called 2-tuples, or sequences (sometimes, lists in a computer science context) of length 2; ordered pairs of scalars are also called 2-dimensional vectors. The entries of an ordered pair can be other ordered pairs, enabling the recursive definition of ordered n-tuples (ordered lists of n objects). For example, the ordered triple (a,b,c) can be defined as (a, (b,c)), i.e., as one pair nested in another. In the ordered pair (a, b), the object a is called the first entry, and the object b the second entry of the pair
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Root Of A Function
In mathematics, a zero, also sometimes called a root, of a real-, complex- or generally vector-valued function f displaystyle f is a member x displaystyle x of the domain of f displaystyle f such that f ( x ) displaystyle f(x) vanishes at x displaystyle x ; that is, x displaystyle x is a solution of the equation f ( x ) = 0 displaystyle f(x)=0 . In other words, a "zero" of a function is an input value that produces an output of 0 displaystyle 0 .[1] A root of a polynomial is a zero of the corresponding polynomial function
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