1.
Robert Recorde
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Robert Recorde was a Welsh physician and mathematician. He invented the sign and also introduced the pre-existing plus sign to English speakers in 1557. A member of a family of Tenby, Wales, born in 1512, Recorde entered the University of Oxford about 1525. Having adopted medicine as a profession, he went to the University of Cambridge to take the degree of M. D. in 1545 and he afterwards returned to Oxford, where he publicly taught mathematics, as he had done prior to going to Cambridge. It appears that he went to London, and acted as physician to King Edward VI and to Queen Mary. He was also controller of the Royal Mint and served as Comptroller of Mines and Monies in Ireland, after being sued for defamation by a political enemy, he was arrested for debt and died in the Kings Bench Prison, Southwark, by the middle of June 1558. The Pathway to Knowledge, containing the First Principles of Geometry, a book explaining Ptolemaic astronomy while mentioning the Copernican heliocentric model in passing. The Whetstone of Witte, whiche is the seconde parte of Arithmeteke, containing the extraction of rootes, the practise, with the rule of equation. This was the book in which the sign was introduced. With the publication of this book Recorde is credited with introducing algebra into England, a medical work, The Urinal of Physick, frequently reprinted. Sherburne states that Recorde also published Cosmographiae isagoge, and that he wrote books entitled De Arte faciendi Horologium, recordes chief contributions to the progress of algebra were in the way of systematising its notation. This article incorporates text from a now in the public domain, Chisholm, Hugh, ed. Recorde. The World of Mathematics Vol.1 Commentary on Robert Recorde Jourdain, the Nature of Mathematics Roberts, Gareth, and Fenny Smith, eds. Robert Recorde, The Life and Times of a Tudor Mathematician 232 pages Williams, Jack, Robert Recorde, Tudor Polymath, Expositor, the Mathematical Gazette Vol.60 No.411 Mar 1976 p 59-61 Roberts, Gordon, Robert Recorde, Tudor Scholar and Mathematician
2.
Grafiek (wiskunde)
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In mathematics, the graph of a function f is the collection of all ordered pairs. If the function x is a scalar, the graph is a two-dimensional graph. If the function x is an ordered pair of real numbers, the graph is the collection of all ordered triples. Graphing on a Cartesian plane is referred to as curve sketching. The graph of a function on real numbers may be mapped directly to the representation of the function. The concept of the graph of a function is generalized to the graph of a relation, note that although a function is always identified with its graph, they are not the same because it will happen that two functions with different codomain could have the same graph. For example, the cubic polynomial mentioned below is a surjection if its codomain is the real numbers, to test whether a graph of a curve is a function of x, one uses the vertical line test. To test whether a graph of a curve is a function of y, if the function has an inverse, the graph of the inverse can be found by reflecting the graph of the original function over the line y = x. In science, engineering, technology, finance, and other areas, in the simplest case one variable is plotted as a function of another, typically using rectangular axes, see Plot for details. In the modern foundation of mathematics known as set theory, a function, F = { a, if x =1, d, if x =2, c, if x =3, is. The graph of the polynomial on the real line f = x 3 −9 x is. If this set is plotted on a Cartesian plane, the result is a curve, the graph of the trigonometric function f = sin cos is. If this set is plotted on a three dimensional Cartesian coordinate system, the result is a surface, oftentimes it is helpful to show with the graph, the gradient of the function and several level curves. The level curves can be mapped on the surface or can be projected on the bottom plane. The second figure shows such a drawing of the graph of the function, f = −2 Given a function f of n variables, x 1, …, x n, the normal to the graph is. This is seen by considering the graph as a set of the function g = f − z. The graph of a function is contained in a Cartesian product of sets, fibre bundles arent cartesian products, but appear to be up close. There is a notion of a graph on a fibre bundle called a section
3.
Wiskunde
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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
4.
Letter
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A letter is a grapheme in an alphabetic system of writing, such as the Greek alphabet and its descendants. Letters also appear in abjads and abugidas, letters broadly denote phonemes in the spoken form of the language, although there is rarely a consistent exact correspondence between letters and phonemes. Written signs in writing systems are best called syllabograms or logograms. Letter, borrowed from Old French lettre, entered Middle English around AD1200, letter derives from Latin littera, which may have derived, via Etruscan, from the Greek διφθέρα. The Middle English plural lettres could refer to an epistle or written document, use of the singular letter to refer to a written document emerged in the 14th century. As symbols that denote segmental speech, letters are associated with phonetics, in a purely phonemic alphabet, a single phoneme is denoted by a single letter, but in history and practice letters often denote more than one phoneme. A pair of letters designating a single phoneme is called a digraph, examples of digraphs in English include ch, sh and th. A phoneme can also be represented by three letters, called a trigraph, an example is the combination sch in German. A letter may also be associated more than one phoneme. As an example of positional effects, the Spanish letter c is pronounced before a, o, or u, letters also have specific names associated with them. These names may differ with language, dialect and history, Z, for example, is usually called zed in all English-speaking countries except the U. S. where it is named zee. Letters, as elements of alphabets, have prescribed orders and this may generally be known as alphabetical order though collation is the science devoted to the complex task of ordering and sorting of letters and words in different languages. In Spanish, for instance, ñ is a letter being sorted after n. In English, n and ñ are sorted alike, letters may also have numerical value. This is true of Roman numerals and the letters of other writing systems, in English, Arabic numerals are typically used instead of letters. Letters may be used as words, the words a and I are the most common English letter-words. Sometimes O is used for Oh in poetic situations, in extremely informal cases of writing individual letters may replace words, e. g. u may be used instead of you in English, when the letter name is pronounced as a homophone of the word. Nearly all alphabets in the world today either descend directly from development or were inspired by its design
5.
Alfabet
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An alphabet is a standard set of letters that is used to write one or more languages based upon the general principle that the letters represent phonemes of the spoken language. This is in contrast to other types of writing systems, such as syllabaries and logographies, the Proto-Canaanite script, later known as the Phoenician alphabet, is the first fully phonemic script. Thus the Phoenician alphabet is considered to be the first alphabet, the Phoenician alphabet is the ancestor of most modern alphabets, including Arabic, Greek, Latin, Cyrillic, Hebrew, and possibly Brahmic. Under a terminological distinction promoted by Peter T. Daniels, an alphabet is a script that represents both vowels and consonants as letters equally. In this narrow sense of the word the first true alphabet was the Greek alphabet, in other alphabetic scripts such as the original Phoenician, Hebrew or Arabic, letters predominantly or exclusively represent consonants, such a script is also called an abjad. A third type, called abugida or alphasyllabary, is one where vowels are shown by diacritics or modifications of consonantal letters, as in Devanagari. The Khmer alphabet is the longest, with 74 letters, there are dozens of alphabets in use today, the most popular being the Latin alphabet. Many languages use modified forms of the Latin alphabet, with additional letters formed using diacritical marks, while most alphabets have letters composed of lines, there are also exceptions such as the alphabets used in Braille. Alphabets are usually associated with an ordering of letters. This makes them useful for purposes of collation, specifically by allowing words to be sorted in alphabetical order and it also means that their letters can be used as an alternative method of numbering ordered items, in such contexts as numbered lists and number placements. The English word alphabet came into Middle English from the Late Latin word alphabetum, the Greek word was made from the first two letters, alpha and beta. The names for the Greek letters came from the first two letters of the Phoenician alphabet, aleph, which also meant ox, and bet, in the alphabet song in English, the term ABCs is used instead of the word alphabet. Knowing ones ABCs, in general, can be used as a metaphor for knowing the basics about anything, the history of the alphabet started in ancient Egypt. These glyphs were used as guides for logograms, to write grammatical inflections. Based on letter appearances and names, it is believed to be based on Egyptian hieroglyphs and this script had no characters representing vowels, although originally it probably was a syllabary, but unneeded symbols were discarded. An alphabetic cuneiform script with 30 signs including three that indicate the vowel was invented in Ugarit before the 15th century BC. This script was not used after the destruction of Ugarit, the Proto-Sinaitic script eventually developed into the Phoenician alphabet, which is conventionally called Proto-Canaanite before ca.1050 BC. The oldest text in Phoenician script is an inscription on the sarcophagus of King Ahiram and this script is the parent script of all western alphabets
6.
Oplossen van vergelijkingen
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When seeking a solution, one or more free variables are designated as unknowns. A solution is an assignment of expressions to the variables that makes the equality in the equation true. In other words, a solution is an expression or a collection of such that, when substituted for the unknowns. A problem of solving an equation may be numeric or symbolic, Solving an equation numerically means that only numbers represented explicitly as numerals, are admitted as solutions. Solving an equation means that expressions that may contain known variables or possibly also variables not in the original equation are admitted as solutions. It is also possible to take the variable y to be the unknown, or x and y can both be treated as unknowns, and then there are many solutions to the equation. Instantiating a symbolic solution with specific numbers always gives a solution, for example, a =0 gives =. Note that the distinction between variables and unknown variables is made in the statement of the problem, rather than the equation. However, in areas of mathematics the convention is to reserve some variables as known. When writing polynomials, the coefficients are taken to be known and the indeterminates to be unknown. Depending on the problem, the task may be to any solution or all solutions. The set of all solutions is called the solution set, in the example above, the solution = is also a parametrization of the solution set with the parameter being a. A wording such as an equation in x and y, or solve for x and y, implies that the unknowns are as indicated, in these cases x and y. In one general case, we have a such as ƒ = c, where x1. xn are the unknowns. Its solutions are the members of the inverse image ƒ −1 =, note that the set of solutions can be the empty set, a singleton, finite, or infinite. One particular solution is x =0, y =0, z =0, two other solutions are x =3, y =6, z =1, and x =8, y =9, z =2. In fact, this set of solutions describes a plane in three-dimensional space. The solution set of a set of equations or inequalities is the set of all its solutions, a solution being a tuple of values, one for each unknown
7.
Parameter
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A parameter, generally, is any characteristic that can help in defining or classifying a particular system. That is, a parameter is an element of a system that is useful, or critical, parameter has more specific meanings within various disciplines, including mathematics, computing and computer programming, engineering, statistics, logic and linguistics. Mathematical functions have one or more arguments that are designated in the definition by variables, a function definition can also contain parameters, but unlike variables, parameters are not listed among the arguments that the function takes. When parameters are present, the definition actually defines a family of functions. A parameter could be incorporated into the name to indicate its dependence on the parameter. For instance, one may define the base b of a logarithm by log b = log log where b is a parameter that indicates which logarithmic function is being used. It is not an argument of the function, and will, for instance, in some informal situations it is a matter of convention whether some or all of the symbols in a function definition are called parameters. However, changing the status of symbols between parameter and variable changes the function as a mathematical object, for instance, the notation for the falling factorial power n k _ = n ⋯, defines a polynomial function of n, but is not a polynomial function of k. Indeed, in the case, it is only defined for non-negative integer arguments. Sometimes it is useful to all functions with certain parameters as parametric family. Examples from probability theory are given further below, a variable is one of the many things a parameter is not. The dependent variable, the speed of the car, depends on the independent variable, change the lever arms of the linkage. Will still depend on the pedal position and you have changed a parameter A parametric equaliser is an audio filter that allows the frequency of maximum cut or boost to be set by one control, and the size of the cut or boost by another. These settings, the level of the peak or trough, are two of the parameters of a frequency response curve, and in a two-control equaliser they completely describe the curve. More elaborate parametric equalisers may allow other parameters to be varied and these parameters each describe some aspect of the response curve seen as a whole, over all frequencies. A graphic equaliser provides individual level controls for various frequency bands, if asked to imagine the graph of the relationship y = ax2, one typically visualizes a range of values of x, but only one value of a. Of course a different value of a can be used, generating a different relation between x and y, thus a is a parameter, it is less variable than the variable x or y, but it is not an explicit constant like the exponent 2. More precisely, changing the parameter a gives a different problem, in calculating income based on wage and hours worked, it is typically assumed that the number of hours worked is easily changed, but the wage is more static
8.
Kromme
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In mathematics, a curve is, generally speaking, an object similar to a line but that need not be straight. Thus, a curve is a generalization of a line, in that curvature is not necessarily zero, various disciplines within mathematics have given the term different meanings depending on the area of study, so the precise meaning depends on context. However, many of these meanings are special instances of the definition which follows, a curve is a topological space which is locally homeomorphic to a line. In everyday language, this means that a curve is a set of points which, near each of its points, looks like a line, a simple example of a curve is the parabola, shown to the right. A large number of curves have been studied in multiple mathematical fields. A closed curve is a curve that forms a path whose starting point is also its ending point—that is, closely related meanings include the graph of a function and a two-dimensional graph. Interest in curves began long before they were the subject of mathematical study and this can be seen in numerous examples of their decorative use in art and on everyday objects dating back to prehistoric times. Curves, or at least their graphical representations, are simple to create, historically, the term line was used in place of the more modern term curve. Hence the phrases straight line and right line were used to distinguish what are called lines from curved lines. For example, in Book I of Euclids Elements, a line is defined as a breadthless length, Euclids idea of a line is perhaps clarified by the statement The extremities of a line are points. Later commentators further classified according to various schemes. For example, Composite lines Incomposite lines Determinate Indeterminate The Greek geometers had studied many kinds of curves. One reason was their interest in solving problems that could not be solved using standard compass. These curves include, The conic sections, deeply studied by Apollonius of Perga The cissoid of Diocles, studied by Diocles, the conchoid of Nicomedes, studied by Nicomedes as a method to both double the cube and to trisect an angle. The Archimedean spiral, studied by Archimedes as a method to trisect an angle, the spiric sections, sections of tori studied by Perseus as sections of cones had been studied by Apollonius. A fundamental advance in the theory of curves was the advent of analytic geometry in the seventeenth century and this enabled a curve to be described using an equation rather than an elaborate geometrical construction. Previously, curves had been described as geometrical or mechanical according to how they were, or supposedly could be, conic sections were applied in astronomy by Kepler. Newton also worked on an example in the calculus of variations
9.
Meetkunde
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Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer, Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, contemporary geometry has many subfields, Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to problems in geometry. It has applications in physics, including in general relativity, topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, the earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space
10.
Identiteit (wiskunde)
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In other words, A = B is an identity if A and B define the same functions. This means that an identity is an equality between functions that are differently defined, for example,2 = a2 + 2ab + b2 and cos2 + sin2 =1 are identities. Identities are sometimes indicated by the triple bar symbol ≡ instead of =, geometrically, these are identities involving certain functions of one or more angles. They are distinct from triangle identities, which are identities involving both angles and side lengths of a triangle, only the former are covered in this article. These identities are useful whenever expressions involving trigonometric functions need to be simplified, for example, the latter equation is true when θ =0, false when θ =2. The following identities hold for all integer exponents, provided that the base is non-zero and this contrasts with addition and multiplication, which are. For example,2 +3 =3 +2 =5 and 2 ·3 =3 ·2 =6, but 23 =8, whereas 32 =9. For example, +4 =2 + =9 and ·4 =2 · =24, but 23 to the 4 is 84 or 4,096, whereas 2 to the 34 is 281 or 2,417,851,639,229,258,349,412,352. Without parentheses to modify the order of calculation, by convention the order is top-down, not bottom-up, several important formulas, sometimes called logarithmic identities or log laws, relate logarithms to one another. The logarithm of a product is the sum of the logarithms of the numbers being multiplied, the logarithm of the p-th power of a number is p times the logarithm of the number itself, the logarithm of a p-th root is the logarithm of the number divided by p. The following table lists these identities with examples, each of the identities can be derived after substitution of the logarithm definitions x = blogb, and/or y = blogb, in the left hand sides. The logarithm logb can be computed from the logarithms of x and b with respect to an arbitrary base k using the following formula, typical scientific calculators calculate the logarithms to bases 10 and e. Logarithms with respect to any base b can be determined using either of these two logarithms by the formula, log b = log 10 log 10 = log e log e . Given a number x and its logarithm logb to a base b. The hyperbolic functions satisfy many identities, all of similar in form to the trigonometric identities. The Gudermannian function gives a relationship between the circular functions and the hyperbolic ones that does not involve complex numbers. Accounting identity List of mathematical identities Encyclopedia of Equation Online encyclopedia of mathematical identities A Collection of Algebraic Identities