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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 the imaginary unit (which satisfies the equation i2 = −1). In this expression, a is called the real part of the complex number, and b is called the imaginary part. If z = a + b i {displaystyle z=a+bi} , then we write Re ( z ) = a , {displaystyle operatorname {Re} (z)=a,} and Im ( z ) = b . {displaystyle operatorname {Im} (z)=b.} Complex numbers extend the concept of the onedimensional number line to the twodimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point (a, b) in the complex plane. A complex number whose real part is zero is said to be purely imaginary , whereas a complex number whose imaginary part is zero is a real number [...More...]  "Complex Numbers" on: Wikipedia Yahoo 

Irreducible Polynomial In mathematics , an IRREDUCIBLE POLYNOMIAL is, roughly speaking, a nonconstant polynomial that cannot be factored into the product of two nonconstant 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. One says that the polynomial x2 − 2 is irreducible over the integers but not over the reals [...More...]  "Irreducible Polynomial" on: Wikipedia Yahoo 

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_{n1}x^{n1}+cdots +a_{0}=0} with integer coefficients. Solutions of the equation are roots (equivalently, zeroes) of the polynomial on the left side of the equation. If a0 and an are nonzero, then each rational solution x, when written as a fraction x = p/q in lowest terms (i.e., the greatest common divisor of p and q is 1), satisfies * p is an integer factor of the constant term a0, and * q is an integer factor of the leading coefficient an.The rational root theorem is a special case (for a single linear factor) of Gauss\'s lemma on the factorization of polynomials. The INTEGRAL ROOT THEOREM is a special case of the rational root theorem if the leading coefficient an = 1 [...More...]  "Rational Root Test" on: Wikipedia Yahoo 

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 realvalued. This was proven by Pierre Wantzel in 1843. One can decide whether a given irreducible cubic polynomial is in casus irreducibilis using the discriminant D, via Cardano\'s formula . Let the cubic equation be given by a x 3 + b x 2 + c x + d = 0. {displaystyle ax^{3}+bx^{2}+cx+d=0.,} Then the discriminant D appearing in the algebraic solution is given by D = 18 a b c d 4 b 3 d + b 2 c 2 4 a c 3 27 a 2 d 2 [...More...]  "Casus Irreducibilis" on: Wikipedia Yahoo 

Fundamental Theorem Of Algebra The FUNDAMENTAL THEOREM OF ALGEBRA states that every nonconstant singlevariable polynomial with complex coefficients has at least one complex root . This includes polynomials with real coefficients, since every real number is a complex number with an imaginary part equal to zero. Equivalently (by definition), the theorem states that the field of complex numbers is algebraically closed . The theorem is also stated as follows: every nonzero, singlevariable, degree n polynomial with complex coefficients has, counted with multiplicity , exactly n complex roots. The equivalence of the two statements can be proven through the use of successive polynomial division . In spite of its name, there is no purely algebraic proof of the theorem, since any proof must use the completeness of the reals (or some other equivalent formulation of completeness), which is not an algebraic concept [...More...]  "Fundamental Theorem Of Algebra" on: Wikipedia Yahoo 

Polynomial In mathematics , a POLYNOMIAL is an expression consisting of variables (or indeterminates ) and coefficients , that involves only the operations of addition , subtraction , multiplication , and nonnegative integer exponents of variables. An example of a polynomial of a single indeterminate x is x2 − 4x + 7. An example in three variables is x3 + 2xyz2 − yz + 1. Polynomials appear in a wide variety of areas of mathematics and science. For example, they are used to form polynomial equations , which encode a wide range of problems, from elementary word problems to complicated problems in the sciences; they are used to define POLYNOMIAL FUNCTIONS, which appear in settings ranging from basic chemistry and physics to economics and social science ; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties , central concepts in algebra and algebraic geometry [...More...]  "Polynomial" on: Wikipedia Yahoo 

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 [...More...]  "Negative Numbers" on: Wikipedia Yahoo 

Trigonometric Functions In mathematics , the TRIGONOMETRIC FUNCTIONS (also called CIRCULAR FUNCTIONS, ANGLE FUNCTIONS or GONIOMETRIC FUNCTIONS ) 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 xaxis, the sine of the angle gives the length of the ycomponent (the opposite to the angle or the rise) of the triangle, the cosine gives the length of the xcomponent (the adjacent of the angle or the run), and the tangent function gives the slope (ycomponent divided by the xcomponent). More precise definitions are detailed below [...More...]  "Trigonometric Functions" on: Wikipedia Yahoo 

JeanRobert Argand JEANROBERT ARGAND (July 18, 1768 – August 13, 1822) was an amateur mathematician . In 1806, while managing a bookstore in Paris 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 . CONTENTS * 1 Life * 2 See also * 3 References * 4 External links * 5 Further reading LIFE JeanRobert Argand was born in Geneva Geneva , Switzerland Switzerland to Jacques Argand and Eve Carnac. His background and education are mostly unknown. Since his knowledge of mathematics was selftaught and he did not belong to any mathematical organizations, he likely pursued mathematics as a hobby rather than a profession [...More...]  "JeanRobert Argand" on: Wikipedia Yahoo 

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 is in. 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. The abovementioned imaginary rotation and translation may be thought to occur in any order, as the orientation of an object does not change when it translates, and its location does not change when it rotates. Euler\'s rotation theorem shows that in three dimensions any orientation can be reached with a single rotation around a fixed axis [...More...]  "Orientation (geometry)" on: Wikipedia Yahoo 

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 REVOLUTION or COMPLETE ROTATION or FULL CIRCLE or CYCLE or REV or ROT. A turn can be subdivided in many different ways: into half turns, quarter turns, centiturns, milliturns, binary angles, points etc. CONTENTS * 1 Subdivision of turns * 2 History * 3 Unit conversion * 4 Tau proposal * 5 Examples of use * 6 Kinematics Kinematics of turns * 7 See also * 8 Notes and references * 9 External links SUBDIVISION OF TURNSA 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 [...More...]  "Turn (geometry)" on: Wikipedia Yahoo 

Right Angle In geometry and trigonometry , a RIGHT ANGLE is an angle of exactly 90° (degrees) , corresponding to a quarter turn . If a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles. 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 , making the right angle basic to trigonometry [...More...]  "Right Angle" on: Wikipedia Yahoo 

Algebraically Closed Field In abstract algebra , an ALGEBRAICALLY CLOSED FIELD F contains a root for every nonconstant polynomial in F, the ring of polynomials in the variable x with coefficients in F. CONTENTS * 1 Examples * 2 Equivalent properties * 2.1 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 roots * 3 Other properties * 4 Notes * 5 References EXAMPLESAs 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 [...More...]  "Algebraically Closed Field" on: Wikipedia Yahoo 

Root Of A Function In mathematics , a ZERO, also sometimes called a ROOT, of a real, complex or generally vectorvalued 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) = 0. In other words, a "zero" of a function is an input value that produces an output of zero (0). A ROOT of a polynomial is a zero of the corresponding polynomial function . The fundamental theorem of algebra shows that any nonzero polynomial has a number of roots at most equal to its degree and that the number of roots and the degree are equal when one considers the complex roots (or more generally the roots in an algebraically closed extension ) counted with their multiplicities . For example, the polynomial f of degree two, defined by f ( x ) = x 2 5 x + 6 {displaystyle f(x)=x^{2}5x+6} has the two roots 2 and 3, since f ( 2 ) = 2 2 5 2 + 6 = 0 a n d f ( 3 ) = 3 2 5 3 + 6 = 0 [...More...]  "Root Of A Function" on: Wikipedia Yahoo 

Reflection Symmetry REFLECTION SYMMETRY, LINE SYMMETRY, MIRROR SYMMETRY, MIRRORIMAGE SYMMETRY, is symmetry with respect to reflection . That is, a figure which does not change upon undergoing a reflection has reflectional symmetry. In 2D there is a line/axis of symmetry, in 3D a plane of symmetry. An object or figure which is indistinguishable from its transformed image is called mirror symmetric . CONTENTS * 1 Symmetric function * 2 Symmetric geometrical shapes * 3 Mathematical equivalents * 4 Advanced types of reflection symmetry * 5 In nature * 6 In architecture * 7 See also * 8 References * 9 Bibliography * 9.1 General * 9.2 Advanced * 10 External links SYMMETRIC FUNCTION A normal distribution bell curve is an example symmetric function In formal terms, a mathematical object is symmetric with respect to a given operation such as reflection, rotation or translation , if, when applied to the object, this operation preserves some property of the object [...More...]  "Reflection Symmetry" on: Wikipedia Yahoo 

Complex Conjugate In mathematics , the COMPLEX CONJUGATE of a complex number is the number with equal real part and imaginary part equal in magnitude but opposite in sign . For example, the complex conjugate of 3 + 4i is 3 − 4i. In polar form , the conjugate of e i {displaystyle rho e^{iphi }} is e i {displaystyle rho e^{iphi }} . This can be shown using Euler\'s formula . Complex conjugates are important for finding roots of polynomials . According to the complex conjugate root theorem , if a complex number is a root to a polynomial in one variable with real coefficients (such as the quadratic equation or the cubic equation ), so is its conjugate. CONTENTS * 1 Notation * 2 Properties * 3 Use as a variable * 4 Generalizations * 5 See also * 6 Notes * 7 References NOTATIONThe complex conjugate of a complex number z {displaystyle z} is written as z {displaystyle {overline {z}}} or z {displaystyle z^{*}!} [...More...]  "Complex Conjugate" on: Wikipedia Yahoo 