1.
Complex plane
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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 part of a complex number represented by a displacement along the x-axis. The concept of the plane allows a geometric interpretation of complex numbers. Under addition, they add like vectors, in particular, multiplication by a complex number of modulus 1 acts as a rotation. The complex plane is known as the Argand plane. These are named after Jean-Robert Argand, although they were first described by Norwegian-Danish land surveyor, Argand diagrams are frequently used to plot the positions of the poles and zeroes of a function in the complex plane. In this customary notation the number z corresponds to the point in the Cartesian plane. In the Cartesian plane the point can also be represented in coordinates as = =. In the Cartesian plane it may be assumed that the arctangent takes values from −π/2 to π/2, and some care must be taken to define the real arctangent function for points when x ≤0. Here |z| is the value or modulus of the complex number z, θ, the argument of z, is usually taken on the interval 0 ≤ θ < 2π. Notice that without the constraint on the range of θ, the argument of z is multi-valued, because the exponential function is periodic. Thus, if θ is one value of arg, the values are given by arg = θ + 2nπ. The theory of contour integration comprises a part of complex analysis. In this context the direction of travel around a curve is important – reversing the direction in which the curve is traversed multiplies the value of the integral by −1. By convention the direction is counterclockwise. Almost all of complex analysis is concerned with complex functions – that is, here it is customary to speak of the domain of f as lying in the z-plane, while referring to the range or image of f as a set of points in the w-plane. In symbols we write z = x + i y, f = w = u + i v and it can be useful to think of the complex plane as if it occupied the surface of a sphere. We can establish a correspondence between the points on the surface of the sphere minus the north pole and the points in the complex plane as follows
2.
Cartesian plane
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Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair. The coordinates can also be defined as the positions of the projections of the point onto the two axis, expressed as signed distances from the origin. One can use the principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes. In general, n Cartesian coordinates specify the point in an n-dimensional Euclidean space for any dimension n and these coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes. The invention of Cartesian coordinates in the 17th century by René Descartes revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes can be described by Cartesian equations, algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2, centered at the origin of the plane, a familiar example is the concept of the graph of a function. Cartesian coordinates are also tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering. They are the most common system used in computer graphics, computer-aided geometric design. Nicole Oresme, a French cleric and friend of the Dauphin of the 14th Century, used similar to Cartesian coordinates well before the time of Descartes. The adjective Cartesian refers to the French mathematician and philosopher René Descartes who published this idea in 1637 and it was independently discovered by Pierre de Fermat, who also worked in three dimensions, although Fermat did not publish the discovery. Both authors used a single axis in their treatments and have a length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes La Géométrie was translated into Latin in 1649 by Frans van Schooten and these commentators introduced several concepts while trying to clarify the ideas contained in Descartes work. Many other coordinate systems have developed since Descartes, such as the polar coordinates for the plane. The development of the Cartesian coordinate system would play a role in the development of the Calculus by Isaac Newton. The two-coordinate description of the plane was later generalized into the concept of vector spaces. Choosing a Cartesian coordinate system for a one-dimensional space – that is, for a straight line—involves choosing a point O of the line, a unit of length, and an orientation for the line. An orientation chooses which of the two half-lines determined by O is the positive, and which is negative, we say that the line is oriented from the negative half towards the positive half
3.
Quadratic equation
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If a =0, then the equation is linear, not quadratic. The numbers a, b, and c are the coefficients of the equation, and may be distinguished by calling them, respectively, the coefficient, the linear coefficient. Because the quadratic equation involves only one unknown, it is called univariate, solutions to problems equivalent to the quadratic equation were known as early as 2000 BC. A quadratic equation with real or complex coefficients has two solutions, called roots and these two solutions may or may not be distinct, and they may or may not be real. It may be possible to express a quadratic equation ax2 + bx + c =0 as a product =0. In some cases, it is possible, by inspection, to determine values of p, q, r. If the quadratic equation is written in the form, then the Zero Factor Property states that the quadratic equation is satisfied if px + q =0 or rx + s =0. Solving these two linear equations provides the roots of the quadratic, for most students, factoring by inspection is the first method of solving quadratic equations to which they are exposed. As an example, x2 + 5x +6 factors as, the more general case where a does not equal 1 can require a considerable effort in trial and error guess-and-check, assuming that it can be factored at all by inspection. Except for special cases such as where b =0 or c =0 and this means that the great majority of quadratic equations that arise in practical applications cannot be solved by factoring by inspection. The process of completing the square makes use of the identity x 2 +2 h x + h 2 =2. Starting with an equation in standard form, ax2 + bx + c =0 Divide each side by a. Subtract the constant term c/a from both sides, add the square of one-half of b/a, the coefficient of x, to both sides. This completes the square, converting the left side into a perfect square, write the left side as a square and simplify the right side if necessary. Produce two linear equations by equating the square root of the side with the positive and negative square roots of the right side. Completing the square can be used to derive a formula for solving quadratic equations. The mathematical proof will now be briefly summarized and it can easily be seen, by polynomial expansion, that the following equation is equivalent to the quadratic equation,2 = b 2 −4 a c 4 a 2. Taking the square root of both sides, and isolating x, gives, x = − b ± b 2 −4 a c 2 a and these result in slightly different forms for the solution, but are otherwise equivalent
4.
Real number
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In mathematics, a real number is a value that represents a quantity along a line. The adjective real in this context was introduced in the 17th century by René Descartes, the real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2. Included within the irrationals are the numbers, such as π. Real numbers can be thought of as points on a long line called the number line or real line. Any real number can be determined by a possibly infinite decimal representation, such as that of 8.632, the real line can be thought of as a part of the complex plane, and complex numbers include real numbers. These descriptions of the numbers are not sufficiently rigorous by the modern standards of pure mathematics. All these definitions satisfy the definition and are thus equivalent. The statement that there is no subset of the reals with cardinality greater than ℵ0. Simple fractions were used by the Egyptians around 1000 BC, the Vedic Sulba Sutras in, c.600 BC, around 500 BC, the Greek mathematicians led by Pythagoras realized the need for irrational numbers, in particular the irrationality of the square root of 2. Arabic mathematicians merged the concepts of number and magnitude into a general idea of real numbers. In the 16th century, Simon Stevin created the basis for modern decimal notation, in the 17th century, Descartes introduced the term real to describe roots of a polynomial, distinguishing them from imaginary ones. In the 18th and 19th centuries, there was work on irrational and transcendental numbers. Johann Heinrich Lambert gave the first flawed proof that π cannot be rational, Adrien-Marie Legendre completed the proof, Évariste Galois developed techniques for determining whether a given equation could be solved by radicals, which gave rise to the field of Galois theory. Charles Hermite first proved that e is transcendental, and Ferdinand von Lindemann, lindemanns proof was much simplified by Weierstrass, still further by David Hilbert, and has finally been made elementary by Adolf Hurwitz and Paul Gordan. The development of calculus in the 18th century used the set of real numbers without having defined them cleanly. The first rigorous definition was given by Georg Cantor in 1871, in 1874, he showed that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite. Contrary to widely held beliefs, his first method was not his famous diagonal argument, the real number system can be defined axiomatically up to an isomorphism, which is described hereafter. Another possibility is to start from some rigorous axiomatization of Euclidean geometry, from the structuralist point of view all these constructions are on equal footing
5.
Complex number
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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, satisfying the equation i2 = −1. In this expression, a is the part and b is the imaginary part of the complex number. If z = a + b i, then ℜ z = a, ℑ z = b, Complex numbers extend the concept of the one-dimensional number line to the two-dimensional 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 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. In this way, the numbers are a field extension of the ordinary real numbers. As well as their use within mathematics, complex numbers have applications in many fields, including physics, chemistry, biology, economics, electrical engineering. The Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers and he called them fictitious during his attempts to find solutions to cubic equations in the 16th century. Complex numbers allow solutions to equations that have no solutions in real numbers. For example, the equation 2 = −9 has no real solution, Complex numbers provide a solution to this problem. The idea is to extend the real numbers with the unit i where i2 = −1. According to the theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. A complex number is a number of the form a + bi, for example, −3.5 + 2i is a complex number. The real number a is called the part of the complex number a + bi. By this convention the imaginary part does not include the unit, hence b. The real part of a number z is denoted by Re or ℜ. For example, Re = −3.5 Im =2, hence, in terms of its real and imaginary parts, a complex number z is equal to Re + Im ⋅ i. This expression is known as the Cartesian form of z. A real number a can be regarded as a number a + 0i whose imaginary part is 0
6.
Addition
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Addition is one of the four basic operations of arithmetic, with the others being subtraction, multiplication and division. The addition of two numbers is the total amount of those quantities combined. For example, in the picture on the right, there is a combination of three apples and two together, making a total of five apples. This observation is equivalent to the mathematical expression 3 +2 =5 i. e.3 add 2 is equal to 5, besides counting fruits, addition can also represent combining other physical objects. In arithmetic, rules for addition involving fractions and negative numbers have been devised amongst others, in algebra, addition is studied more abstractly. It is commutative, meaning that order does not matter, and it is associative, repeated addition of 1 is the same as counting, addition of 0 does not change a number. Addition also obeys predictable rules concerning related operations such as subtraction and multiplication, performing addition is one of the simplest numerical tasks. Addition of very small numbers is accessible to toddlers, the most basic task,1 +1, can be performed by infants as young as five months and even some members of other animal species. In primary education, students are taught to add numbers in the system, starting with single digits. Mechanical aids range from the ancient abacus to the modern computer, Addition is written using the plus sign + between the terms, that is, in infix notation. The result is expressed with an equals sign, for example, 3½ =3 + ½ =3.5. This notation can cause confusion since in most other contexts juxtaposition denotes multiplication instead, the sum of a series of related numbers can be expressed through capital sigma notation, which compactly denotes iteration. For example, ∑ k =15 k 2 =12 +22 +32 +42 +52 =55. The numbers or the objects to be added in addition are collectively referred to as the terms, the addends or the summands. This is to be distinguished from factors, which are multiplied, some authors call the first addend the augend. In fact, during the Renaissance, many authors did not consider the first addend an addend at all, today, due to the commutative property of addition, augend is rarely used, and both terms are generally called addends. All of the above terminology derives from Latin, using the gerundive suffix -nd results in addend, thing to be added. Likewise from augere to increase, one gets augend, thing to be increased, sum and summand derive from the Latin noun summa the highest, the top and associated verb summare
7.
Multiplication
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Multiplication is one of the four elementary, mathematical operations of arithmetic, with the others being addition, subtraction and division. Multiplication can also be visualized as counting objects arranged in a rectangle or as finding the area of a rectangle whose sides have given lengths, the area of a rectangle does not depend on which side is measured first, which illustrates the commutative property. The product of two measurements is a new type of measurement, for multiplying the lengths of the two sides of a rectangle gives its area, this is the subject of dimensional analysis. The inverse operation of multiplication is division, for example, since 4 multiplied by 3 equals 12, then 12 divided by 3 equals 4. Multiplication by 3, followed by division by 3, yields the original number, Multiplication is also defined for other types of numbers, such as complex numbers, and more abstract constructs, like matrices. For these more abstract constructs, the order that the operands are multiplied sometimes does matter, a listing of the many different kinds of products that are used in mathematics is given in the product page. In arithmetic, multiplication is often written using the sign × between the terms, that is, in infix notation, there are other mathematical notations for multiplication, Multiplication is also denoted by dot signs, usually a middle-position dot,5 ⋅2 or 5. 2 The middle dot notation, encoded in Unicode as U+22C5 ⋅ dot operator, is standard in the United States, the United Kingdom, when the dot operator character is not accessible, the interpunct is used. In other countries use a comma as a decimal mark. In algebra, multiplication involving variables is often written as a juxtaposition, the notation can also be used for quantities that are surrounded by parentheses. In matrix multiplication, there is a distinction between the cross and the dot symbols. The cross symbol generally denotes the taking a product of two vectors, yielding a vector as the result, while the dot denotes taking the dot product of two vectors, resulting in a scalar. In computer programming, the asterisk is still the most common notation and this is due to the fact that most computers historically were limited to small character sets that lacked a multiplication sign, while the asterisk appeared on every keyboard. This usage originated in the FORTRAN programming language, the numbers to be multiplied are generally called the factors. The number to be multiplied is called the multiplicand, while the number of times the multiplicand is to be multiplied comes from the multiplier. Usually the multiplier is placed first and the multiplicand is placed second, however sometimes the first factor is the multiplicand, additionally, there are some sources in which the term multiplicand is regarded as a synonym for factor. In algebra, a number that is the multiplier of a variable or expression is called a coefficient, the result of a multiplication is called a product. A product of integers is a multiple of each factor, for example,15 is the product of 3 and 5, and is both a multiple of 3 and a multiple of 5
8.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
9.
Zero of a function
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In other words, a zero of a function is an input value that produces an output of zero. A root of a polynomial is a zero of the polynomial function. If the function maps real numbers to real numbers, its zeroes are the x-coordinates of the points where its graph meets the x-axis, an alternative name for such a point in this context is an x-intercept. Every equation in the unknown x may be rewritten as f =0 by regrouping all terms in the left-hand side and it follows that the solutions of such an equation are exactly the zeros of the function f. Every real polynomial of odd degree has an odd number of roots, likewise. Consequently, real odd polynomials must have at least one real root, the fundamental theorem of algebra states that every polynomial of degree n has n complex roots, counted with their multiplicities. The non-real roots of polynomials with real coefficients come in conjugate pairs, vietas formulas relate the coefficients of a polynomial to sums and products of its roots. Computing roots of functions, for polynomial functions, frequently requires the use of specialised or approximation techniques. However, some functions, including all those of degree no greater than 4. In topology and other areas of mathematics, the set of a real-valued function f, X → R is the subset f −1 of X. Zero sets are important in many areas of mathematics. One area of importance is algebraic geometry, where the first definition of an algebraic variety is through zero-sets. For instance, for each set S of polynomials in k, one defines the zero-locus Z to be the set of points in An on which the functions in S simultaneously vanish, that is to say Z =. Then a subset V of An is called an algebraic set if V = Z for some S. These affine algebraic sets are the building blocks of algebraic geometry. Zero Pole Fundamental theorem of algebra Newtons method Sendovs conjecture Mardens theorem Vanish at infinity Zero crossing Weisstein, Eric W. Root
10.
Polynomial
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In mathematics, a polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents. 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 variety of areas of mathematics and science. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, central concepts in algebra, the word polynomial joins two diverse roots, the Greek poly, meaning many, and the Latin nomen, or name. It was derived from the binomial by replacing the Latin root bi- with the Greek poly-. The word polynomial was first used in the 17th century, the x occurring in a polynomial is commonly called either a variable or an indeterminate. When the polynomial is considered as an expression, x is a symbol which does not have any value. It is thus correct to call it an indeterminate. However, when one considers the function defined by the polynomial, then x represents the argument of the function, many authors use these two words interchangeably. It is a convention to use uppercase letters for the indeterminates. However one may use it over any domain where addition and multiplication are defined, in particular, when a is the indeterminate x, then the image of x by this function is the polynomial P itself. This equality allows writing let P be a polynomial as a shorthand for let P be a polynomial in the indeterminate x. A polynomial is an expression that can be built from constants, the word indeterminate means that x represents no particular value, although any value may be substituted for it. The mapping that associates the result of substitution to the substituted value is a function. This can be expressed concisely by using summation notation, ∑ k =0 n a k x k That is. Each term consists of the product of a number—called the coefficient of the term—and a finite number of indeterminates, because x = x1, the degree of an indeterminate without a written exponent is one. A term and a polynomial with no indeterminates are called, respectively, a constant term, the degree of a constant term and of a nonzero constant polynomial is 0. The degree of the polynomial,0, is generally treated as not defined
11.
Imaginary number
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An imaginary number is a complex number that can be written as a real number multiplied by the imaginary unit i, which is defined by its property i2 = −1. The square of an imaginary number bi is −b2, for example, 5i is an imaginary number, and its square is −25. Zero is considered to be real and imaginary. Originally coined in the 17th century as a term and regarded as fictitious or useless. Some authors use the term pure imaginary number to denote what is called here an imaginary number, the concept had appeared in print earlier, for instance in work by Gerolamo Cardano. At the time imaginary numbers, as well as numbers, were poorly understood and regarded by some as fictitious or useless. The use of numbers was not widely accepted until the work of Leonhard Euler. The geometric significance of numbers as points in a plane was first described by Caspar Wessel. This idea first surfaced with the articles by James Cockle beginning in 1848, geometrically, imaginary numbers are found on the vertical axis of the complex number plane, allowing them to be presented perpendicular to the real axis. One way of viewing imaginary numbers is to consider a number line, positively increasing in magnitude to the right. This vertical axis is called the imaginary axis and is denoted iℝ, I. In this representation, multiplication by –1 corresponds to a rotation of 180 degrees about the origin, note that a 90-degree rotation in the negative direction also satisfies this interpretation. This reflects the fact that −i also solves the equation x2 = −1, in general, multiplying by a complex number is the same as rotating around the origin by the complex numbers argument, followed by a scaling by its magnitude. Care must be used when working with numbers expressed as the principal values of the square roots of negative numbers. For example,6 =36 = ≠ −4 −9 = =6 i 2 = −6, Imaginary unit de Moivres formula NaN Octonion Quaternion Nahin, Paul. An Imaginary Tale, the Story of the Square Root of −1, explains many applications of imaginary expressions. How can one show that imaginary numbers really do exist, – an article that discusses the existence of imaginary numbers. In our time, Imaginary numbers Discussion of imaginary numbers on BBC Radio 4, 5Numbers programme 4 BBC Radio 4 programme Why Use Imaginary Numbers
12.
Square (algebra)
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In mathematics, a square is the result of multiplying a number by itself. The verb to square is used to denote this operation, squaring is the same as raising to the power 2, and is denoted by a superscript 2, for instance, the square of 3 may be written as 32, which is the number 9. In some cases when superscripts are not available, as for instance in programming languages or plain text files, the adjective which corresponds to squaring is quadratic. The square of an integer may also be called a number or a perfect square. In algebra, the operation of squaring is often generalized to polynomials, other expressions, for instance, the square of the linear polynomial x +1 is the quadratic polynomial x2 + 2x +1. One of the important properties of squaring, for numbers as well as in other mathematical systems, is that. That is, the function satisfies the identity x2 =2. This can also be expressed by saying that the function is an even function. The squaring function preserves the order of numbers, larger numbers have larger squares. In other words, squaring is a function on the interval. Hence, zero is its global minimum, the only cases where the square x2 of a number is less than x occur when 0 < x <1, that is, when x belongs to an open interval. This implies that the square of an integer is never less than the original number, every positive real number is the square of exactly two numbers, one of which is strictly positive and the other of which is strictly negative. Zero is the square of one number, itself. For this reason, it is possible to define the square root function, no square root can be taken of a negative number within the system of real numbers, because squares of all real numbers are non-negative. There are several uses of the squaring function in geometry. The name of the squaring function shows its importance in the definition of the area, the area depends quadratically on the size, the area of a shape n times larger is n2 times greater. The squaring function is related to distance through the Pythagorean theorem and its generalization, Euclidean distance is not a smooth function, the three-dimensional graph of distance from a fixed point forms a cone, with a non-smooth point at the tip of the cone. However, the square of the distance, which has a paraboloid as its graph, is a smooth, the dot product of a Euclidean vector with itself is equal to the square of its length, v⋅v = v2
13.
Square root
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In mathematics, a square root of a number a is a number y such that y2 = a, in other words, a number y whose square is a. For example,4 and −4 are square roots of 16 because 42 =2 =16, every nonnegative real number a has a unique nonnegative square root, called the principal square root, which is denoted by √a, where √ is called the radical sign or radix. For example, the square root of 9 is 3, denoted √9 =3. The term whose root is being considered is known as the radicand, the radicand is the number or expression underneath the radical sign, in this example 9. Every positive number a has two roots, √a, which is positive, and −√a, which is negative. Together, these two roots are denoted ± √a, although the principal square root of a positive number is only one of its two square roots, the designation the square root is often used to refer to the principal square root. For positive a, the square root can also be written in exponent notation. Square roots of numbers can be discussed within the framework of complex numbers. In Ancient India, the knowledge of theoretical and applied aspects of square and square root was at least as old as the Sulba Sutras, a method for finding very good approximations to the square roots of 2 and 3 are given in the Baudhayana Sulba Sutra. Aryabhata in the Aryabhatiya, has given a method for finding the root of numbers having many digits. It was known to the ancient Greeks that square roots of positive numbers that are not perfect squares are always irrational numbers, numbers not expressible as a ratio of two integers. This is the theorem Euclid X,9 almost certainly due to Theaetetus dating back to circa 380 BC, the particular case √2 is assumed to date back earlier to the Pythagoreans and is traditionally attributed to Hippasus. Mahāvīra, a 9th-century Indian mathematician, was the first to state that square roots of negative numbers do not exist, a symbol for square roots, written as an elaborate R, was invented by Regiomontanus. An R was also used for Radix to indicate square roots in Gerolamo Cardanos Ars Magna, according to historian of mathematics D. E. Smith, Aryabhatas method for finding the root was first introduced in Europe by Cataneo in 1546. According to Jeffrey A. Oaks, Arabs used the letter jīm/ĝīm, the letter jīm resembles the present square root shape. Its usage goes as far as the end of the century in the works of the Moroccan mathematician Ibn al-Yasamin. The symbol √ for the root was first used in print in 1525 in Christoph Rudolffs Coss
14.
Multiplicity (mathematics)
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In mathematics, the multiplicity of a member of a multiset is the number of times it appears in the multiset. For example, the number of times a given polynomial equation has a root at a given point, the notion of multiplicity is important to be able to count correctly without specifying exceptions. Hence the expression, counted with multiplicity, if multiplicity is ignored, this may be emphasized by counting the number of distinct elements, as in the number of distinct roots. However, whenever a set is formed, multiplicity is automatically ignored, without requiring use of the term distinct. In the prime factorization, for example,60 =2 ×2 ×3 ×5 the multiplicity of the prime factor 2 is 2, thus,60 has 4 prime factors, but only 3 distinct prime factors. Let F be a field and p be a polynomial in one variable, an element a ∈ F is called a root of multiplicity k of p if there is a polynomial s such that s ≠0 and p = ks. If k =1, then a is called a simple root, if k ≥2, then a is called a multiple root. For instance, the polynomial p = x3 + 2x2 − 7x +4 has 1 and −4 as roots and this means that 1 is a root of multiplicity 2, and −4 is a simple root. Multiplicity can be thought of as How many times does the solution appear in the original equation, the derivative of a polynomial has a multiplicity n -1 at a root of multiplicity n of the polynomial. The discriminant of a polynomial is zero if and only if the polynomial has a multiple root, the graph of a polynomial function y = f intersects the x-axis at the real roots of the polynomial. The graph is tangent to this axis at the roots of f. The graph crosses the x-axis at roots of odd multiplicity and bounces off the x-axis at roots of even multiplicity, a non-zero polynomial function is always non-negative if and only if all its roots have an even multiplicity and there exists x0 such that f >0. In algebraic geometry, the intersection of two sub-varieties of a variety is a finite union of irreducible varieties. To each component of such an intersection is attached an intersection multiplicity and this notion is local in the sense that it may be defined by looking what occurs in a neighborhood of any generic point of this component. It follows that without loss of generality, we may consider, for defining the intersection multiplicity, thus, given two affine varieties V1 and V2, let us consider an irreducible component W of the intersection of V1 and V2. Let d be the dimension of W, and P be any point of W. This ring is thus a finite dimensional vector space over the ground field and its dimension is the intersection multiplicity of V1 and V2 at W. This definition allows to state precisely Bézouts theorem and its generalizations and this definition generalizes the multiplicity of a root of a polynomial in the following way
15.
Iota
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Iota /aɪˈoʊtə/ is the ninth letter of the Greek alphabet. It was derived from the Phoenician letter Yodh, letters that arose from this letter include the Latin I and J, the Cyrillic І, Yi, and Je, and iotated letters. In the system of Greek numerals, iota has a value of 10, in ancient Greek it occurred in both long and short versions, but this distinction was lost in Koine Greek. Iota participated as the element in falling diphthongs, with both long and short vowels as the first element. Examples include ᾼ ᾳ ῌ ῃ ῼ ῳ, the former diphthongs became digraphs for simple vowels in Koine Greek. This refers to iota, the smallest letter, or possibly Yodh, י, the word jot derives from iota. The German, Portuguese and Spanish name for the letter J is derived from iota, in some programming languages iota is used to represent and generate an array of consecutive integers. For example, in APL ι4 gives 1234, the lowercase iota symbol is sometimes used to write the imaginary unit, but more often Roman i or j is used. In mathematics, the map of one space into another is sometimes denoted by the lowercase iota. In logic, the lowercase iota denotes the definite descriptor, the lowercase iota symbol has Unicode code point U+03B9 and the uppercase U+0399. Greek Iota / Ypogegrammeni Coptic Iaude Latin Iota Cyrillic Iota Technical Iota Mathematical Iota These characters are used only as mathematical symbols, stylized Greek text should be encoded using the normal Greek letters, with markup and formatting to indicate text style
16.
Electrical engineering
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Electrical engineering is a field of engineering that generally deals with the study and application of electricity, electronics, and electromagnetism. This field first became an occupation in the later half of the 19th century after commercialization of the electric telegraph, the telephone. Subsequently, broadcasting and recording media made electronics part of daily life, the invention of the transistor, and later the integrated circuit, brought down the cost of electronics to the point they can be used in almost any household object. Electrical engineers typically hold a degree in engineering or electronic engineering. Practicing engineers may have professional certification and be members of a professional body, such bodies include the Institute of Electrical and Electronics Engineers and the Institution of Engineering and Technology. Electrical engineers work in a wide range of industries and the skills required are likewise variable. These range from basic circuit theory to the management skills required of a project manager, the tools and equipment that an individual engineer may need are similarly variable, ranging from a simple voltmeter to a top end analyzer to sophisticated design and manufacturing software. Electricity has been a subject of scientific interest since at least the early 17th century and he also designed the versorium, a device that detected the presence of statically charged objects. In the 19th century, research into the subject started to intensify, Electrical engineering became a profession in the later 19th century. Practitioners had created an electric telegraph network and the first professional electrical engineering institutions were founded in the UK. Over 50 years later, he joined the new Society of Telegraph Engineers where he was regarded by other members as the first of their cohort, Practical applications and advances in such fields created an increasing need for standardised units of measure. They led to the standardization of the units volt, ampere, coulomb, ohm, farad. This was achieved at a conference in Chicago in 1893. During these years, the study of electricity was considered to be a subfield of physics. Thats because early electrical technology was electromechanical in nature, the Technische Universität Darmstadt founded the worlds first department of electrical engineering in 1882. The first course in engineering was taught in 1883 in Cornell’s Sibley College of Mechanical Engineering. It was not until about 1885 that Cornell President Andrew Dickson White established the first Department of Electrical Engineering in the United States, in the same year, University College London founded the first chair of electrical engineering in Great Britain. Professor Mendell P. Weinbach at University of Missouri soon followed suit by establishing the engineering department in 1886
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Control engineering
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Control engineering or control systems engineering is the engineering discipline that applies control theory to design systems with desired behaviors. When a device is designed to perform without the need of human inputs for correction it is called automatic control, multi-disciplinary in nature, control systems engineering activities focus on implementation of control systems mainly derived by mathematical modeling of systems of a diverse range. Modern day control engineering is a new field of study that gained significant attention during the 20th century with the advancement of technology. It can be defined or classified as practical application of control theory. Control engineering has a role in a wide range of control systems. Automatic control systems were first developed over two years ago. The first feedback control device on record is thought to be the ancient Ktesibioss water clock in Alexandria and it kept time by regulating the water level in a vessel and, therefore, the water flow from that vessel. This certainly was a device as water clocks of similar design were still being made in Baghdad when the Mongols captured the city in 1258 A. D. A variety of devices have been used over the centuries to accomplish useful tasks or simply to just entertain. In his 1868 paper On Governors, James Clerk Maxwell was able to explain instabilities exhibited by the governor using differential equations to describe the control system. This demonstrated the importance and usefulness of models and methods in understanding complex phenomena. Elements of control theory had appeared earlier but not as dramatically and convincingly as in Maxwells analysis, Control theory made significant strides over the next century. In the very first control relationships, a current output was represented by a control input. However, not having adequate technology to implement electrical control systems, designers were left with the option of less efficient, a very effective mechanical controller that is still widely used in some hydro plants is the governor. There are two divisions in control theory, namely, classical and modern, which have direct implications for the control engineering applications. The scope of control theory is limited to single-input and single-output system design. Many systems may be assumed to have an order and single variable system response in the time domain. A controller designed using classical theory often requires on-site tuning due to incorrect design approximations, the most common controllers designed using classical control theory are PID controllers
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Electric current
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An electric current is a flow of electric charge. In electric circuits this charge is carried by moving electrons in a wire. It can also be carried by ions in an electrolyte, or by both ions and electrons such as in an ionised gas. The SI unit for measuring a current is the ampere. Electric current is measured using a device called an ammeter, electric currents cause Joule heating, which creates light in incandescent light bulbs. They also create magnetic fields, which are used in motors, inductors and generators, the particles that carry the charge in an electric current are called charge carriers. In metals, one or more electrons from each atom are loosely bound to the atom and these conduction electrons are the charge carriers in metal conductors. The conventional symbol for current is I, which originates from the French phrase intensité de courant, current intensity is often referred to simply as current. The I symbol was used by André-Marie Ampère, after whom the unit of current is named, in formulating the eponymous Ampères force law. The notation travelled from France to Great Britain, where it became standard, in a conductive material, the moving charged particles which constitute the electric current are called charge carriers. In other materials, notably the semiconductors, the carriers can be positive or negative. Positive and negative charge carriers may even be present at the same time, a flow of positive charges gives the same electric current, and has the same effect in a circuit, as an equal flow of negative charges in the opposite direction. Since current can be the flow of positive or negative charges. The direction of current is arbitrarily defined as the same direction as positive charges flow. This is called the direction of current I. If the current flows in the direction, the variable I has a negative value. When analyzing electrical circuits, the direction of current through a specific circuit element is usually unknown. Consequently, the directions of currents are often assigned arbitrarily
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Rectangular coordinate system
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Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair. The coordinates can also be defined as the positions of the projections of the point onto the two axis, expressed as signed distances from the origin. One can use the principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes. In general, n Cartesian coordinates specify the point in an n-dimensional Euclidean space for any dimension n and these coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes. The invention of Cartesian coordinates in the 17th century by René Descartes revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes can be described by Cartesian equations, algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2, centered at the origin of the plane, a familiar example is the concept of the graph of a function. Cartesian coordinates are also tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering. They are the most common system used in computer graphics, computer-aided geometric design. Nicole Oresme, a French cleric and friend of the Dauphin of the 14th Century, used similar to Cartesian coordinates well before the time of Descartes. The adjective Cartesian refers to the French mathematician and philosopher René Descartes who published this idea in 1637 and it was independently discovered by Pierre de Fermat, who also worked in three dimensions, although Fermat did not publish the discovery. Both authors used a single axis in their treatments and have a length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes La Géométrie was translated into Latin in 1649 by Frans van Schooten and these commentators introduced several concepts while trying to clarify the ideas contained in Descartes work. Many other coordinate systems have developed since Descartes, such as the polar coordinates for the plane. The development of the Cartesian coordinate system would play a role in the development of the Calculus by Isaac Newton. The two-coordinate description of the plane was later generalized into the concept of vector spaces. Choosing a Cartesian coordinate system for a one-dimensional space – that is, for a straight line—involves choosing a point O of the line, a unit of length, and an orientation for the line. An orientation chooses which of the two half-lines determined by O is the positive, and which is negative, we say that the line is oriented from the negative half towards the positive half
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Polar form
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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, satisfying the equation i2 = −1. In this expression, a is the part and b is the imaginary part of the complex number. If z = a + b i, then ℜ z = a, ℑ z = b, Complex numbers extend the concept of the one-dimensional number line to the two-dimensional 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 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. In this way, the numbers are a field extension of the ordinary real numbers. As well as their use within mathematics, complex numbers have applications in many fields, including physics, chemistry, biology, economics, electrical engineering. The Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers and he called them fictitious during his attempts to find solutions to cubic equations in the 16th century. Complex numbers allow solutions to equations that have no solutions in real numbers. For example, the equation 2 = −9 has no real solution, Complex numbers provide a solution to this problem. The idea is to extend the real numbers with the unit i where i2 = −1. According to the theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. A complex number is a number of the form a + bi, for example, −3.5 + 2i is a complex number. The real number a is called the part of the complex number a + bi. By this convention the imaginary part does not include the unit, hence b. The real part of a number z is denoted by Re or ℜ. For example, Re = −3.5 Im =2, hence, in terms of its real and imaginary parts, a complex number z is equal to Re + Im ⋅ i. This expression is known as the Cartesian form of z. A real number a can be regarded as a number a + 0i whose imaginary part is 0
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Absolute value
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In mathematics, the absolute value or modulus |x| of a real number x is the non-negative value of x without regard to its sign. Namely, |x| = x for a x, |x| = −x for a negative x. For example, the value of 3 is 3. The absolute value of a number may be thought of as its distance from zero, generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, a value is also defined for the complex numbers. The absolute value is related to the notions of magnitude, distance. The term absolute value has been used in this sense from at least 1806 in French and 1857 in English, the notation |x|, with a vertical bar on each side, was introduced by Karl Weierstrass in 1841. Other names for absolute value include numerical value and magnitude, in programming languages and computational software packages, the absolute value of x is generally represented by abs, or a similar expression. Thus, care must be taken to interpret vertical bars as an absolute value sign only when the argument is an object for which the notion of an absolute value is defined. For any real number x the value or modulus of x is denoted by |x| and is defined as | x | = { x, if x ≥0 − x. As can be seen from the definition, the absolute value of x is always either positive or zero. Indeed, the notion of a distance function in mathematics can be seen to be a generalisation of the absolute value of the difference. Since the square root notation without sign represents the square root. This identity is used as a definition of absolute value of real numbers. The absolute value has the four fundamental properties, The properties given by equations - are readily apparent from the definition. To see that equation holds, choose ε from so that ε ≥0, some additional useful properties are given below. These properties are either implied by or equivalent to the properties given by equations -, for example, Absolute value is used to define the absolute difference, the standard metric on the real numbers. Since the complex numbers are not ordered, the definition given above for the absolute value cannot be directly generalised for a complex number
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Argument (complex analysis)
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In mathematics, arg is a function operating on complex numbers. It gives the angle between the real axis to the line joining the point to the origin, shown as φ in figure 1. The numeric value is given by the angle in radians and is positive if measured counterclockwise, algebraically, as any real quantity φ such that z = r = r e i φ for some positive real r. The quantity r is the modulus of z, denoted |z|, the names magnitude for the modulus and phase for the argument are sometimes used equivalently. Similarly, from the periodicity of sin and cos, the definition also has this property. Because a complete rotation around the leaves a complex number unchanged. This is shown in figure 3, a representation of the multi-valued function and this represents an angle of up to half a complete circle from the positive real axis in either direction. Some authors define the range of the value as being in the closed-open interval, Arg. The principal value Arg of a number given as x + iy is normally available in math libraries of many programming languages using the function atan2 or some language-specific variant. The value of atan2 is the value in the range. If z ≠0 and n is any integer, then Arg ≡ n Arg ( mod, Arg = Arg − Arg = −3 π4 − π2 = −5 π4 =3 π4 ( mod
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Quadratic function
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A univariate quadratic function has the form f = a x 2 + b x + c, a ≠0 in the single variable x. The graph of a quadratic function is a parabola whose axis of symmetry is parallel to the y-axis. If the quadratic function is set equal to zero, then the result is a quadratic equation, the solutions to the univariate equation are called the roots of the univariate function. In general there can be a large number of variables, in which case the resulting surface is called a quadric. The adjective quadratic comes from the Latin word quadrātum, a term like x2 is called a square in algebra because it is the area of a square with side x. In general, a prefix indicates the number 4. Quadratum is the Latin word for square, the coefficients of a polynomial are often taken to be real or complex numbers, but in fact, a polynomial may be defined over any ring. When using the quadratic polynomial, authors sometimes mean having degree exactly 2. If the degree is less than 2, this may be called a degenerate case, usually the context will establish which of the two is meant. Sometimes the word order is used with the meaning of degree, a quadratic polynomial may involve a single variable x, or multiple variables such as x, y, and z. Any single-variable quadratic polynomial may be written as a x 2 + b x + c, where x is the variable, and a, b, and c represent the coefficients. In elementary algebra, such polynomials often arise in the form of a quadratic equation a x 2 + b x + c =0, each quadratic polynomial has an associated quadratic function, whose graph is a parabola. Such polynomials are fundamental to the study of sections, which are characterized by equating the expression for f to zero. Similarly, quadratic polynomials with three or more variables correspond to quadric surfaces and hypersurfaces, in linear algebra, quadratic polynomials can be generalized to the notion of a quadratic form on a vector space. F = a 2 + k is called the vertex form, the coefficient a is the same value in all three forms. To convert the standard form to factored form, one only the quadratic formula to determine the two roots r1 and r2. To convert the standard form to form, one needs a process called completing the square. To convert the factored form to form, one needs to multiply
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Additive inverse
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In mathematics, the additive inverse of a number a is the number that, when added to a, yields zero. This number is known as the opposite, sign change. For a real number, it reverses its sign, the opposite to a number is negative. Zero is the inverse of itself. The additive inverse of a is denoted by unary minus, −a. For example, the inverse of 7 is −7, because 7 + =0. The additive inverse is defined as its inverse element under the operation of addition. As for any operation, double additive inverse has no net effect. For a number and, generally, in any ring, the inverse can be calculated using multiplication by −1. Examples of rings of numbers are integers, rational numbers, real numbers, Additive inverse is closely related to subtraction, which can be viewed as an addition of the opposite, a − b = a +. Conversely, additive inverse can be thought of as subtraction from zero, if such an operation admits an identity element o, then this element is unique. For a given x , if there exists x′ such that x + x′ = o , if + is associative, then an additive inverse is unique. To see this, let x′ and x″ each be additive inverses of x, for example, since addition of real numbers is associative, each real number has a unique additive inverse. All the following examples are in fact abelian groups, complex numbers, on the complex plane, this operation rotates a complex number 180 degrees around the origin. Addition of real- and complex-valued functions, here, the inverse of a function f is the function −f defined by = − f , for all x, such that f + = o . More generally, what precedes applies to all functions with values in a group, sequences, matrices. In a vector space the additive inverse −v is often called the vector of v, it has the same magnitude as the original. Additive inversion corresponds to multiplication by −1
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Multiplicative inverse
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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. The multiplicative inverse of a fraction a/b is b/a, for the multiplicative inverse of a real number, divide 1 by the number. For example, the reciprocal of 5 is one fifth, the reciprocal function, the function f that maps x to 1/x, is one of the simplest examples of a function which is its own inverse. In the phrase multiplicative inverse, the qualifier multiplicative is often omitted, multiplicative inverses can be defined over many mathematical domains as well as numbers. In these cases it can happen that ab ≠ ba, then inverse typically implies that an element is both a left and right inverse. The notation f −1 is sometimes used for the inverse function of the function f. For example, the multiplicative inverse 1/ = −1 is the cosecant of x, only for linear maps are they strongly related. The terminology difference reciprocal versus inverse is not sufficient to make this distinction, since many authors prefer the opposite naming convention, in the real numbers, zero does not have a reciprocal because no real number multiplied by 0 produces 1. With the exception of zero, reciprocals of every real number are real, reciprocals of every rational number are rational, the property that every element other than zero has a multiplicative inverse is part of the definition of a field, of which these are all examples. On the other hand, no other than 1 and −1 has an integer reciprocal. In modular arithmetic, the multiplicative inverse of a is also defined. This multiplicative inverse exists if and only if a and n are coprime, for example, the inverse of 3 modulo 11 is 4 because 4 ·3 ≡1. The extended Euclidean algorithm may be used to compute it, the sedenions are an algebra in which every nonzero element has a multiplicative inverse, but which nonetheless has divisors of zero, i. e. nonzero elements x, y such that xy =0. A square matrix has an inverse if and only if its determinant has an inverse in the coefficient ring, the linear map that has the matrix A−1 with respect to some base is then the reciprocal function of the map having A as matrix in the same base. Thus, the two notions of the inverse of a function are strongly related in this case, while they must be carefully distinguished in the general case. A ring in which every element has a multiplicative inverse is a division ring. As mentioned above, the reciprocal of every complex number z = a + bi is complex. In particular, if ||z||=1, then 1 / z = z ¯, consequently, the imaginary units, ±i, have additive inverse equal to multiplicative inverse, and are the only complex numbers with this property
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Field (mathematics)
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In mathematics, a field is a set on which are defined addition, subtraction, multiplication, and division, which behave as they do when applied to rational and real numbers. A field is thus an algebraic structure, which is widely used in algebra, number theory. The best known fields are the field of numbers. In addition, the field of numbers is widely used, not only in mathematics. Finite fields are used in most cryptographic protocols used for computer security, any field may be used as the scalars for a vector space, which is the standard general context for linear algebra. Formally, a field is a set together with two operations the addition and the multiplication, which have the properties, called axioms of fields. An operation is a mapping that associates an element of the set to every pair of its elements, the result of the addition of a and b is called the sum of a and b and denoted a + b. Similarly, the result of the multiplication of a and b is called the product of a and b, associativity of addition and multiplication For all a, b and c in F, one has a + = + c and a · = · c. Commutativity of addition and multiplication For all a and b in F one has a + b = b + a and a · b = b · a. Existence of additive and multiplicative identity elements There exists an element 0 in F, called the identity, such that for all a in F. There is an element 1, different from 0 and called the identity, such that for all a in F. Existence of additive inverses and multiplicative inverses For every a in F, there exists an element in F, denoted −a, such that a + =0. For every a ≠0 in F, there exists an element in F, denoted a−1, 1/a, or 1/a, distributivity of multiplication over addition For all a, b and c in F, one has a · = +. The elements 0 and 1 being required to be distinct, a field has, at least, for every a in F, one has − a = ⋅ a. Thus, the inverse of every element is known as soon as one knows the additive inverse of 1. A subtraction and a division are defined in every field by a − b = a +, a subfield E of a field F is a subset of F that contains 1, and is closed under addition, multiplication, additive inverse and multiplicative inverse of a nonzero element. It is straightforward to verify that a subfield is indeed a field, two groups are associated to every field. The field itself is a group under addition, when considering this group structure rather the field structure, one talks of the additive group of the field
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Up to
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In mathematics, the phrase up to appears in discussions about the elements of a set, and the conditions under which subsets of those elements may be considered equivalent. The statement elements a and b of set S are equivalent up to X means that a and b are equivalent if criterion X is ignored and that is, a and b can be transformed into one another if a transform corresponding to X is applied. Looking at the entire set S, when X is ignored the elements can be arranged in subsets whose elements are equivalent, such subsets are called equivalence classes. If X is some property or process, the phrase up to X means disregarding a possible difference in X, for instance the statement an integers prime factorization is unique up to ordering, means that the prime factorization is unique if we disregard the order of the factors. Further examples concerning up to isomorphism, up to permutations and up to rotations are described below, in informal contexts, mathematicians often use the word modulo for similar purposes, as in modulo isomorphism. The Tetris game does not allow reflections, so the former notation is likely to more natural. To add in the count, there is no formal notation. However, it is common to write there are seven reflecting tetrominos up to rotations, in this, Tetris provides an excellent example, as a reader might simply count 7 pieces ×4 rotations as 28, where some pieces have fewer than four rotation states. In the eight queens puzzle, if the eight queens are considered to be distinct, the regular n-gon, for given n, is unique up to similarity. In other words, if all similar n-gons are considered instances of the same n-gon, then there is only one regular n-gon. In group theory, for example, we may have a group G acting on a set X, another typical example is the statement that there are two different groups of order 4 up to isomorphism, or modulo isomorphism, there are two groups of order 4. This means that there are two classes of groups of order 4, assuming we consider groups to be equivalent if they are isomorphic. A hyperreal x and its standard part st are equal up to an infinitesimal difference, adequality All other things being equal Modulo Quotient set Quotient group Synecdoche Abuse of notation Up-to Techniques for Weak Bisimulation
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Isomorphism
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In mathematics, an isomorphism is a homomorphism or morphism that admits an inverse. Two mathematical objects are isomorphic if an isomorphism exists between them, an automorphism is an isomorphism whose source and target coincide. For most algebraic structures, including groups and rings, a homomorphism is an isomorphism if, in topology, where the morphisms are continuous functions, isomorphisms are also called homeomorphisms or bicontinuous functions. In mathematical analysis, where the morphisms are functions, isomorphisms are also called diffeomorphisms. A canonical isomorphism is a map that is an isomorphism. Two objects are said to be isomorphic if there is a canonical isomorphism between them. Isomorphisms are formalized using category theory, let R + be the multiplicative group of positive real numbers, and let R be the additive group of real numbers. The logarithm function log, R + → R satisfies log = log x + log y for all x, y ∈ R +, so it is a group homomorphism. The exponential function exp, R → R + satisfies exp = for all x, y ∈ R, the identities log exp x = x and exp log y = y show that log and exp are inverses of each other. Since log is a homomorphism that has an inverse that is also a homomorphism, because log is an isomorphism, it translates multiplication of positive real numbers into addition of real numbers. This facility makes it possible to real numbers using a ruler. Consider the group, the integers from 0 to 5 with addition modulo 6 and these structures are isomorphic under addition, if you identify them using the following scheme, ↦0 ↦1 ↦2 ↦3 ↦4 ↦5 or in general ↦ mod 6. For example, + =, which translates in the system as 1 +3 =4. Even though these two groups look different in that the sets contain different elements, they are indeed isomorphic, more generally, the direct product of two cyclic groups Z m and Z n is isomorphic to if and only if m and n are coprime. For example, R is an ordering ≤ and S an ordering ⊑, such an isomorphism is called an order isomorphism or an isotone isomorphism. If X = Y, then this is a relation-preserving automorphism, in a concrete category, such as the category of topological spaces or categories of algebraic objects like groups, rings, and modules, an isomorphism must be bijective on the underlying sets. In algebraic categories, an isomorphism is the same as a homomorphism which is bijective on underlying sets, in abstract algebra, two basic isomorphisms are defined, Group isomorphism, an isomorphism between groups Ring isomorphism, an isomorphism between rings. Just as the automorphisms of an algebraic structure form a group, letting a particular isomorphism identify the two structures turns this heap into a group
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Complex conjugate
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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 conjugate of 3 + 4i is 3 − 4i. In polar form, the conjugate of ρ e i ϕ is ρ e − i ϕ and this can be shown using Eulers 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, so is its conjugate. The complex conjugate of a number z is written as z ¯ or z ∗. The first notation avoids confusion with the notation for the transpose of a matrix. The second is preferred in physics, where dagger is used for the conjugate transpose, If a complex number is represented as a 2×2 matrix, the notations are identical. In some texts, the conjugate of a previous known number is abbreviated as c. c. A significant property of the conjugate is that a complex number is equal to its complex conjugate if its imaginary part is zero. The conjugate of the conjugate of a number z is z. The ultimate relation is the method of choice to compute the inverse of a number if it is given in rectangular coordinates. Exp = exp ¯ log = log ¯ if z is non-zero If p is a polynomial with real coefficients, thus, non-real roots of real polynomials occur in complex conjugate pairs. In general, if ϕ is a function whose restriction to the real numbers is real-valued. The map σ = z ¯ from C to C is a homeomorphism and antilinear, even though it appears to be a well-behaved function, it is not holomorphic, it reverses orientation whereas holomorphic functions locally preserve orientation. It is bijective and compatible with the operations, and hence is a field automorphism. As it keeps the real numbers fixed, it is an element of the Galois group of the field extension C / R and this Galois group has only two elements, σ and the identity on C. Thus the only two field automorphisms of C that leave the real numbers fixed are the identity map and complex conjugation. Similarly, for a fixed complex unit u = exp, the equation z − z 0 z ¯ − z 0 ¯ = u determines the line through z 0 in the direction of u
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Unit circle
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In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, the circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane. The unit circle is often denoted S1, the generalization to higher dimensions is the unit sphere, if is a point on the unit circles circumference, then | x | and | y | are the lengths of the legs of a right triangle whose hypotenuse has length 1. Thus, by the Pythagorean theorem, x and y satisfy the equation x 2 + y 2 =1. The interior of the circle is called the open unit disk. One may also use other notions of distance to define other unit circles, such as the Riemannian circle, see the article on mathematical norms for additional examples. The unit circle can be considered as the complex numbers. In quantum mechanics, this is referred to as phase factor, the equation x2 + y2 =1 gives the relation cos 2 + sin 2 =1. The unit circle also demonstrates that sine and cosine are periodic functions, triangles constructed on the unit circle can also be used to illustrate the periodicity of the trigonometric functions. First, construct a radius OA from the origin to a point P on the circle such that an angle t with 0 < t < π/2 is formed with the positive arm of the x-axis. Now consider a point Q and line segments PQ ⊥ OQ, the result is a right triangle △OPQ with ∠QOP = t. Because PQ has length y1, OQ length x1, and OA length 1, sin = y1 and cos = x1. Having established these equivalences, take another radius OR from the origin to a point R on the circle such that the same angle t is formed with the arm of the x-axis. Now consider a point S and line segments RS ⊥ OS, the result is a right triangle △ORS with ∠SOR = t. It can hence be seen that, because ∠ROQ = π − t, R is at in the way that P is at. The conclusion is that, since is the same as and is the same as, it is true that sin = sin and it may be inferred in a similar manner that tan = −tan, since tan = y1/x1 and tan = y1/−x1. A simple demonstration of the above can be seen in the equality sin = sin = 1/√2, when working with right triangles, sine, cosine, and other trigonometric functions only make sense for angle measures more than zero and less than π/2. However, when defined with the circle, these functions produce meaningful values for any real-valued angle measure – even those greater than 2π
31.
Orthogonal group
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Equivalently, it is the group of n×n orthogonal matrices, where the group operation is given by matrix multiplication, an orthogonal matrix is a real matrix whose inverse equals its transpose. An important subgroup of O is the orthogonal group, denoted SO. This group is called the rotation group, because, in dimensions 2 and 3. In low dimension, these groups have been studied, see SO, SO and SO. This is a subgroup of the linear group GL given by O = where QT is the transpose of Q and I is the identity matrix. This article mainly discusses the groups of quadratic forms that may be expressed over some bases as the dot product, over the reals. Over the reals, for any quadratic form, there is a basis. Thus the orthogonal group depends only on the numbers of 1 and of −1, and is denoted O, for details, see indefinite orthogonal group. The derived subgroup Ω of O is an often studied object because, the Cartan–Dieudonné theorem describes the structure of the orthogonal group for a non-singular form. The determinant of any orthogonal matrix is either 1 or −1, the orthogonal n-by-n matrices with determinant 1 form a normal subgroup of O known as the special orthogonal group SO, consisting of all proper rotations. By analogy with GL–SL, the group is sometimes called the general orthogonal group and denoted GO. The term rotation group can be used to either the special or general orthogonal group. When this distinction is to be emphasized, the groups may be denoted O and O, reserving n for the dimension of the space. The letters p or r are also used, indicating the rank of the corresponding Lie algebra, in odd dimension the corresponding Lie algebra is s o, while in even dimension the Lie algebra is s o. In two dimensions, O is the group of all rotations about the origin and all reflections along a line through the origin, SO is the group of all rotations about the origin. These groups are related, SO is a subgroup of O of index 2. More generally, in any number of dimensions an even number of reflections gives a rotation, therefore, the rotations define a subgroup of O, but the reflections do not define a subgroup. A reflection through the origin may be generated as a combination of one reflection along each of the axes, the reflection through the origin is not a reflection in the usual sense in even dimensions, but rather a rotation
32.
Nth root
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A root of degree 2 is called a square root and a root of degree 3, a cube root. Roots of higher degree are referred by using numbers, as in fourth root, twentieth root. For example,2 is a root of 4, since 22 =4. −2 is also a root of 4, since 2 =4. A real number or complex number has n roots of degree n. While the roots of 0 are not distinct, the n nth roots of any real or complex number are all distinct. If n is odd and x is real, one nth root is real and has the sign as x. Finally, if x is not real, then none of its nth roots is real. Roots are usually using the radical symbol or radix or √, with x or √ x denoting the square root, x 3 denoting the cube root, x 4 denoting the fourth root. In the expression x n, n is called the index, is the sign or radix. For example, −8 has three roots, −2,1 + i √3 and 1 − i √3. Out of these,1 + i √3 has the least argument,4 has two square roots,2 and −2, having arguments 0 and π respectively. So 2 is considered the root on account of having the lesser argument. An unresolved root, especially one using the symbol, is often referred to as a surd or a radical. Nth roots can also be defined for complex numbers, and the roots of 1 play an important role in higher mathematics. The origin of the root symbol √ is largely speculative, some sources imply that the symbol was first used by Arab mathematicians. One of those mathematicians was Abū al-Hasan ibn Alī al-Qalasādī, legend has it that it was taken from the Arabic letter ج, which is the first letter in the Arabic word جذر. However, many scholars, including Leonhard Euler, believe it originates from the letter r, the symbol was first seen in print without the vinculum in the year 1525 in Die Coss by Christoff Rudolff, a German mathematician
33.
Branch point
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Multi-valued functions are rigorously studied using Riemann surfaces, and the formal definition of branch points employs this concept. Branch points fall into three categories, algebraic branch points, transcendental branch points, and logarithmic branch points. Algebraic branch points most commonly arise from functions in which there is an ambiguity in the extraction of a root, such as solving the equation z = w2 for w as a function of z. Here the branch point is the origin, because the continuation of any solution around a closed loop containing the origin will result in a different function. Despite the algebraic branch point, the w is well-defined as a multiple-valued function and. This is in contrast to transcendental and logarithmic branch points, that is, points at which a function has nontrivial monodromy. In geometric function theory, unqualified use of the branch point typically means the former more restrictive kind. In other areas of analysis, the unqualified term may also refer to the more general branch points of transcendental type. Let Ω be an open set in the complex plane C and ƒ, Ω → C a holomorphic function. If ƒ is not constant, then the set of the points of ƒ, that is. So each critical point z0 of ƒ lies at the center of a disc B containing no other point of ƒ in its closure. Let γ be the boundary of B, taken with its positive orientation, the winding number of ƒ with respect to the point ƒ is a positive integer called the ramification index of z0. If the ramification index is greater than 1, then z0 is called a point of ƒ. Equivalently, z0 is a point if there exists a holomorphic function φ defined in a neighborhood of z0 such that ƒ = φk for some positive integer k >1. Typically, one is not interested in ƒ itself, but in its inverse function and it is common to abuse language and refers to a branch point w0 = ƒ of ƒ as a branch point of the global analytic function ƒ−1. More general definitions of branch points are possible for other kinds of multiple-valued global analytic functions, a unifying framework for dealing with such examples is supplied in the language of Riemann surfaces below. In particular, in more general picture, poles of order greater than 1 can also be considered ramification points. In terms of the global analytic function ƒ−1, branch points are those points around which there is nontrivial monodromy
34.
Root of unity
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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 n. Roots of unity are used in branches of mathematics, and are especially important in number theory, the theory of group characters. In field theory and ring theory the notion of root of unity also applies to any ring with an identity element. Any algebraically closed field has exactly n nth roots of unity if n is not divisible by the characteristic of the field, an nth root of unity, where n is a positive integer, is a number z satisfying the equation z n =1. Without further specification, the roots of unity are complex numbers, however the defining equation of roots of unity is meaningful over any field F, and this allows considering roots of unity in F. Whichever is the field F, the roots of unity in F are either numbers, if the characteristic of F is 0, or, otherwise. Conversely, every element in a finite field is a root of unity in that field. See Root of unity modulo n and Finite field for further details, an nth root of unity is primitive if it is not a kth root of unity for some smaller k, z k ≠1. Every nth root of unity z is a primitive ath root of unity for some a where 1 ≤ a ≤ n. In fact, if z1 =1 then z is a primitive first root of unity, otherwise if z2 =1 then z is a second root of unity. And, as z is a root of unity, one finds a first a such that za =1. If z is an nth root of unity and a ≡ b then za = zb, Therefore, given a power za of z, it can be assumed that 1 ≤ a ≤ n. Any integer power of an nth root of unity is also an nth root of unity, n = z k n = k =1 k =1. In particular, the reciprocal of an nth root of unity is its complex conjugate, let z be a primitive nth root of unity. Zn−1, zn = z0 =1 are all distinct, assume the contrary, that za = zb where 1 ≤ a < b ≤ n. But 0 < b − a < n, which contradicts z being primitive. Since an nth-degree polynomial equation can only have n distinct roots, from the preceding, it follows that if z is a primitive nth root of unity, z a = z b ⟺ a ≡ b. If z is not primitive there is only one implication, a ≡ b ⟹ z a = z b
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Regular polygon
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In Euclidean geometry, a regular polygon is a polygon that is equiangular and equilateral. Regular polygons may be convex or star, in the limit, a sequence of regular polygons with an increasing number of sides becomes a circle, if the perimeter is fixed, or a regular apeirogon, if the edge length is fixed. These properties apply to all regular polygons, whether convex or star, a regular n-sided polygon has rotational symmetry of order n. All vertices of a regular polygon lie on a common circle and that is, a regular polygon is a cyclic polygon. Together with the property of equal-length sides, this implies that every regular polygon also has a circle or incircle that is tangent to every side at the midpoint. Thus a regular polygon is a tangential polygon, a regular n-sided polygon can be constructed with compass and straightedge if and only if the odd prime factors of n are distinct Fermat primes. The symmetry group of a regular polygon is dihedral group Dn, D2, D3. It consists of the rotations in Cn, together with reflection symmetry in n axes that pass through the center, if n is even then half of these axes pass through two opposite vertices, and the other half through the midpoint of opposite sides. If n is odd then all pass through a vertex. All regular simple polygons are convex and those having the same number of sides are also similar. An n-sided convex regular polygon is denoted by its Schläfli symbol, for n <3 we have two degenerate cases, Monogon, degenerate in ordinary space. Digon, a line segment, degenerate in ordinary space. In certain contexts all the polygons considered will be regular, in such circumstances it is customary to drop the prefix regular. For instance, all the faces of uniform polyhedra must be regular, for n >2 the number of diagonals is n 2, i. e.0,2,5,9. for a triangle, square, pentagon, hexagon. The diagonals divide the polygon into 1,4,11,24, for a regular n-gon inscribed in a unit-radius circle, the product of the distances from a given vertex to all other vertices equals n. For a regular simple n-gon with circumradius R and distances di from a point in the plane to the vertices. For a regular n-gon, the sum of the distances from any interior point to the n sides is n times the apothem. This is a generalization of Vivianis theorem for the n=3 case, the sum of the perpendiculars from a regular n-gons vertices to any line tangent to the circumcircle equals n times the circumradius
36.
Modulo operation
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In computing, the modulo operation finds the remainder after division of one number by another. Given two positive numbers, a and n, a n is the remainder of the Euclidean division of a by n. Although typically performed with a and n both being integers, many computing systems allow other types of numeric operands, the range of numbers for an integer modulo of n is 0 to n −1. See modular arithmetic for an older and related convention applied in number theory, when either a or n is negative, the naive definition breaks down and programming languages differ in how these values are defined. In mathematics, the result of the operation is the remainder of the Euclidean division. Computers and calculators have various ways of storing and representing numbers, usually, in number theory, the positive remainder is always chosen, but programming languages choose depending on the language and the signs of a or n. Standard Pascal and ALGOL68 give a positive remainder even for negative divisors, a modulo 0 is undefined in most systems, although some do define it as a. Despite its widespread use, truncated division is shown to be inferior to the other definitions, when the result of a modulo operation has the sign of the dividend, it can lead to surprising mistakes. For special cases, on some hardware, faster alternatives exist, optimizing compilers may recognize expressions of the form expression % constant where constant is a power of two and automatically implement them as expression &. This can allow writing clearer code without compromising performance and this optimization is not possible for languages in which the result of the modulo operation has the sign of the dividend, unless the dividend is of an unsigned integer type. This is because, if the dividend is negative, the modulo will be negative, some modulo operations can be factored or expanded similar to other mathematical operations. This may be useful in cryptography proofs, such as the Diffie–Hellman key exchange, identity, mod n = a mod n. nx mod n =0 for all positive integer values of x. If p is a number which is not a divisor of b, then abp−1 mod p = a mod p. B−1 mod n denotes the multiplicative inverse, which is defined if and only if b and n are relatively prime. Distributive, mod n = mod n. ab mod n = mod n, division, a/b mod n = mod n, when the right hand side is defined. Inverse multiplication, mod n = a mod n, modulo and modulo – many uses of the word modulo, all of which grew out of Carl F. Gausss introduction of modular arithmetic in 1801. Modular exponentiation ^ Perl usually uses arithmetic modulo operator that is machine-independent, for examples and exceptions, see the Perl documentation on multiplicative operators. ^ Mathematically, these two choices are but two of the number of choices available for the inequality satisfied by a remainder
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Integer
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An integer is a number that can be written without a fractional component. For example,21,4,0, and −2048 are integers, while 9.75, 5 1⁄2, the set of integers consists of zero, the positive natural numbers, also called whole numbers or counting numbers, and their additive inverses. This is often denoted by a boldface Z or blackboard bold Z standing for the German word Zahlen, ℤ is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite. The integers form the smallest group and the smallest ring containing the natural numbers, in algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, the integers are the integers that are also rational numbers. Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, however, with the inclusion of the negative natural numbers, and, importantly,0, Z is also closed under subtraction. The integers form a ring which is the most basic one, in the following sense, for any unital ring. This universal property, namely to be an object in the category of rings. Z is not closed under division, since the quotient of two integers, need not be an integer, although the natural numbers are closed under exponentiation, the integers are not. The following lists some of the properties of addition and multiplication for any integers a, b and c. In the language of algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, in fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, not every integer has an inverse, e. g. there is no integer x such that 2x =1, because the left hand side is even. This means that Z under multiplication is not a group, all the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of algebraic structure. Only those equalities of expressions are true in Z for all values of variables, note that certain non-zero integers map to zero in certain rings. The lack of zero-divisors in the means that the commutative ring Z is an integral domain
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Principal value
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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. The simplest case arises in taking the root of a positive real number. For example,4 has two roots,2 and –2, of these the positive root,2, is considered the principal root and is denoted as 4. Consider the complex logarithm log z. It is defined as the complex number w such that e w = z Now, for example and this means we want to solve e w = i for w. But is it the only solution, of course, there are other solutions, which is evidenced by considering the position of i in the complex plane and in particular its argument arg i. We can rotate counterclockwise π/2 radians from 1 to reach i initially, so, we can conclude that i is also a solution for log i. It becomes clear that we can add any multiple of 2πi to our initial solution to all values for log i. But this has a consequence that may be surprising in comparison of real valued functions, each value of k determines what is known as a branch, a single-valued component of the multiple-valued log function. The branch corresponding to k=0 is known as the branch, and along this branch. In general, if f is multiple-valued, the branch of f is denoted p v f such that for z in the domain of f. Complex valued elementary functions can be multiple valued over some domains, the principal value of some of these functions can be obtained by decomposing the function into simpler ones whereby the principal value of the simple functions are straightforward to obtain. We have examined the function above, i. e. log z = ln | z | + i. Now, arg z is intrinsically multivalued, one often defines the argument of some complex number to be between -π and π, so we take this to be the principal value of the argument, and we write the argument function on this branch Arg z. Using Arg z instead of arg z, we obtain the value of the logarithm. To compute these values one can use functions, atan2 with principal value in the range (-π, π] atan with principal value in the range (-π/2, π/2] Principal branch Branch point
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Factorial
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In mathematics, the factorial of a non-negative integer n, denoted by n. is the product of all positive integers less than or equal to n. =5 ×4 ×3 ×2 ×1 =120, the value of 0. is 1, according to the convention for an empty product. The factorial operation is encountered in areas of mathematics, notably in combinatorics, algebra. Its most basic occurrence is the fact there are n. ways to arrange n distinct objects into a sequence. This fact was known at least as early as the 12th century, fabian Stedman, in 1677, described factorials as applied to change ringing. After describing a recursive approach, Stedman gives a statement of a factorial, Now the nature of these methods is such, the factorial function is formally defined by the product n. = ∏ k =1 n k, or by the relation n. = {1 if n =0. The factorial function can also be defined by using the rule as n. All of the above definitions incorporate the instance 0, =1, in the first case by the convention that the product of no numbers at all is 1. This is convenient because, There is exactly one permutation of zero objects, = n. ×, valid for n >0, extends to n =0. It allows for the expression of many formulae, such as the function, as a power series. It makes many identities in combinatorics valid for all applicable sizes, the number of ways to choose 0 elements from the empty set is =0. More generally, the number of ways to choose n elements among a set of n is = n. n, the factorial function can also be defined for non-integer values using more advanced mathematics, detailed in the section below. This more generalized definition is used by advanced calculators and mathematical software such as Maple or Mathematica, although the factorial function has its roots in combinatorics, formulas involving factorials occur in many areas of mathematics. There are n. different ways of arranging n distinct objects into a sequence, often factorials appear in the denominator of a formula to account for the fact that ordering is to be ignored. A classical example is counting k-combinations from a set with n elements, one can obtain such a combination by choosing a k-permutation, successively selecting and removing an element of the set, k times, for a total of n k _ = n ⋯ possibilities. This however produces the k-combinations in an order that one wishes to ignore, since each k-combination is obtained in k. different ways. This number is known as the coefficient, because it is also the coefficient of Xk in n