1.
Geometry
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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
2.
Shape
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A shape is the form of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material composition. Psychologists have theorized that humans mentally break down images into simple geometric shapes called geons, examples of geons include cones and spheres. Some simple shapes can be put into broad categories, for instance, polygons are classified according to their number of edges as triangles, quadrilaterals, pentagons, etc. Each of these is divided into categories, triangles can be equilateral, isosceles, obtuse, acute, scalene, etc. while quadrilaterals can be rectangles, rhombi, trapezoids, squares. Other common shapes are points, lines, planes, and conic sections such as ellipses, circles, among the most common 3-dimensional shapes are polyhedra, which are shapes with flat faces, ellipsoids, which are egg-shaped or sphere-shaped objects, cylinders, and cones. If an object falls into one of these categories exactly or even approximately, thus, we say that the shape of a manhole cover is a disk, because it is approximately the same geometric object as an actual geometric disk. Similarity, Two objects are similar if one can be transformed into the other by a scaling, together with a sequence of rotations, translations. Isotopy, Two objects are isotopic if one can be transformed into the other by a sequence of deformations that do not tear the object or put holes in it. Sometimes, two similar or congruent objects may be regarded as having a different shape if a reflection is required to transform one into the other. For instance, the b and d are a reflection of each other, and hence they are congruent and similar. Sometimes, only the outline or external boundary of the object is considered to determine its shape, for instance, an hollow sphere may be considered to have the same shape as a solid sphere. Procrustes analysis is used in many sciences to determine whether or not two objects have the shape, or to measure the difference between two shapes. In advanced mathematics, quasi-isometry can be used as a criterion to state that two shapes are approximately the same. Simple shapes can often be classified into basic objects such as a point, a line, a curve, a plane. However, most shapes occurring in the world are complex. Some, such as plant structures and coastlines, may be so complicated as to defy traditional mathematical description – in which case they may be analyzed by differential geometry, or as fractals. In geometry, two subsets of a Euclidean space have the shape if one can be transformed to the other by a combination of translations, rotations. In other words, the shape of a set of points is all the information that is invariant to translations, rotations
3.
Rectangle
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In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as a quadrilateral, since equiangular means that all of its angles are equal. It can also be defined as a parallelogram containing a right angle, a rectangle with four sides of equal length is a square. The term oblong is occasionally used to refer to a non-square rectangle, a rectangle with vertices ABCD would be denoted as ABCD. The word rectangle comes from the Latin rectangulus, which is a combination of rectus and angulus, a crossed rectangle is a crossed quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals. It is a case of an antiparallelogram, and its angles are not right angles. Other geometries, such as spherical, elliptic, and hyperbolic, have so-called rectangles with sides equal in length. Rectangles are involved in many tiling problems, such as tiling the plane by rectangles or tiling a rectangle by polygons, a convex quadrilateral with successive sides a, b, c, d whose area is 12. A rectangle is a case of a parallelogram in which each pair of adjacent sides is perpendicular. A parallelogram is a case of a trapezium in which both pairs of opposite sides are parallel and equal in length. A trapezium is a quadrilateral which has at least one pair of parallel opposite sides. A convex quadrilateral is Simple, The boundary does not cross itself, star-shaped, The whole interior is visible from a single point, without crossing any edge. De Villiers defines a more generally as any quadrilateral with axes of symmetry through each pair of opposite sides. This definition includes both right-angled rectangles and crossed rectangles, quadrilaterals with two axes of symmetry, each through a pair of opposite sides, belong to the larger class of quadrilaterals with at least one axis of symmetry through a pair of opposite sides. These quadrilaterals comprise isosceles trapezia and crossed isosceles trapezia, a rectangle is cyclic, all corners lie on a single circle. It is equiangular, all its corner angles are equal and it is isogonal or vertex-transitive, all corners lie within the same symmetry orbit. It has two lines of symmetry and rotational symmetry of order 2. The dual polygon of a rectangle is a rhombus, as shown in the table below, the figure formed by joining, in order, the midpoints of the sides of a rectangle is a rhombus and vice versa
4.
Ratio
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In mathematics, a ratio is a relationship between two numbers indicating how many times the first number contains the second. For example, if a bowl of fruit contains eight oranges and six lemons, thus, a ratio can be a fraction as opposed to a whole number. Also, in example the ratio of lemons to oranges is 6,8. The numbers compared in a ratio can be any quantities of a kind, such as objects, persons, lengths. A ratio is written a to b or a, b, when the two quantities have the same units, as is often the case, their ratio is a dimensionless number. A rate is a quotient of variables having different units, but in many applications, the word ratio is often used instead for this more general notion as well. The numbers A and B are sometimes called terms with A being the antecedent, the proportion expressing the equality of the ratios A, B and C, D is written A, B = C, D or A, B, C, D. This latter form, when spoken or written in the English language, is expressed as A is to B as C is to D. A, B, C and D are called the terms of the proportion. A and D are called the extremes, and B and C are called the means, the equality of three or more proportions is called a continued proportion. Ratios are sometimes used three or more terms. The ratio of the dimensions of a two by four that is ten inches long is 2,4,10, a good concrete mix is sometimes quoted as 1,2,4 for the ratio of cement to sand to gravel. It is impossible to trace the origin of the concept of ratio because the ideas from which it developed would have been familiar to preliterate cultures. For example, the idea of one village being twice as large as another is so basic that it would have been understood in prehistoric society, however, it is possible to trace the origin of the word ratio to the Ancient Greek λόγος. Early translators rendered this into Latin as ratio, a more modern interpretation of Euclids meaning is more akin to computation or reckoning. Medieval writers used the word to indicate ratio and proportionalitas for the equality of ratios, Euclid collected the results appearing in the Elements from earlier sources. The Pythagoreans developed a theory of ratio and proportion as applied to numbers, the discovery of a theory of ratios that does not assume commensurability is probably due to Eudoxus of Cnidus. The exposition of the theory of proportions that appears in Book VII of The Elements reflects the earlier theory of ratios of commensurables, the existence of multiple theories seems unnecessarily complex to modern sensibility since ratios are, to a large extent, identified with quotients. This is a recent development however, as can be seen from the fact that modern geometry textbooks still use distinct terminology and notation for ratios
5.
Hyperrectangle
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In geometry, an n-orthotope is the generalization of a rectangle for higher dimensions, formally defined as the Cartesian product of intervals. A three-dimensional orthotope is also called a rectangular prism, rectangular cuboid. A special case of an n-orthotope, where all edges are equal length, is the n-cube, the dual polytope of an n-orthotope has been variously called a rectangular n-orthoplex, rhombic n-fusil, or n-lozenge. It is constructed by 2n points located in the center of the rectangular faces. An n-fusils Schläfli symbol can be represented by a sum of n line segments. A 1-fusil is a line segment and its plane cross selections in all pairs of axes are rhombi. Minimum bounding box Coxeter, Harold Scott MacDonald
6.
Aspect ratio (image)
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The aspect ratio of an image describes the proportional relationship between its width and its height. It is commonly expressed as two separated by a colon, as in 16,9.5 yards high. The most common ratios used today in the presentation of films in cinemas are 1.85,1 and 2.39,1. Two common videographic aspect ratios are 4,3, the video format of the 20th century. Other cinema and video aspect ratios exist, but are used infrequently, in still camera photography, the most common aspect ratios are 4,3,3,2, and more recently being found in consumer cameras 16,9. Other aspect ratios, such as 5,3,5,4, in motion picture formats, the physical size of the film area between the sprocket perforations determines the images size. The universal standard is a frame that is four perforations high, the film itself is 35 mm wide, but the area between the perforations is 24.89 mm×18.67 mm, leaving the de facto ratio of 4,3, or 1.33,1. A4,3 ratio mimics human eyesight visual angle of 155°h x 120°v, the motion picture industry convention assigns a value of 1.0 to the image’s height, an anamorphic frame is often incorrectly described as 2.40,1 or 2.40. After 1952, a number of aspect ratios were experimented with for anamorphic productions, a SMPTE specification for anamorphic projection from 1957 finally standardized the aperture to 2.35,1. An update in 1970 changed the ratio to 2.39,1 in order to make splices less noticeable. This aspect ratio of 2.39,1 was confirmed by the most recent revision from August 1993, in American cinemas, the common projection ratios are 1.85,1 and 2.39,1. Some European countries have 1.66,1 as the wide screen standard, the Academy ratio of 1.375,1 was used for all cinema films in the sound era until 1953. During that time, television, which had a aspect ratio of 1.33,1. Hollywood responded by creating a number of wide-screen formats, CinemaScope, Todd-AO. The flat 1.85,1 aspect ratio was introduced in May 1953, and became one of the most common cinema projection standards in the U. S. and elsewhere. The goal of these lenses and aspect ratios was to capture as much of the frame as possible, onto as large an area of the film as possible. In either case the image was squeezed horizontally to fit the frame size. Development of various film camera systems must ultimately cater to the placement of the frame in relation to the constraints of the perforations
7.
Display aspect ratio
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The aspect ratio of a computer display is the proportional relationship between its width and its height. It is expressed as two separated by a colon. Current common aspect ratios for displays are 5,4,4,3,16,10 and 16,9, computer displays with aspect ratio wider than 4,3 are also called widescreen. Widescreen computer displays are typically of the 16,9 or 16,10 aspect ratio, in 2008, the computer industry started to move over from 4,3 and 16,10 to 16,9. Until about 2003, most computer monitors had a 4,3 aspect ratio, between 2003 and 2006, monitors with 16,10 aspect ratio became commonly available, first in laptops and later also in standalone computer monitors. Reasons for this transition was productive uses for such monitors, i. e.16,10 became the most common aspect ratio for widescreen computer monitors until 2008. In 2008, the industry started to move from 4,3 and 16,10 to 16,9 as the standard aspect ratio for monitors. By 2010, virtually all computer monitor and laptop manufacturers had moved to the 16,9 aspect ratio. In 2011, non-widescreen displays with 4,3 aspect ratios still were being manufactured and he also predicted that by the end of 2011, production on all 4,3 or similar panels will be halted due to a lack of demand. In March 2011, the 16,9 resolution of 1920×1080 became the most commonly used resolution among Steam users, in April 2012, the 16,9 resolution of 1366×768 became the most commonly used resolution worldwide, the previous most common resolution was 1024×768. The third most common resolution at the time was 1280×800, since 2011, several monitors complying with the Digital Cinema Initiatives 4K standard have been produced. The standard specifies a resolution of 4096×2160 and a ratio of almost 1.9,1. Since 2005 most video games are made for the 16,9 aspect ratio and 16,9 computer displays therefore offer the best compatibility. 16,9 video games are letterboxed on a 16,10 or 4,3 display or have reduced field of view,4,3 monitors have the best compatibility with older games released prior to 2005 when that aspect ratio was the mainstream standard for computer displays. Movies usually are in 2.39,1,16,9 or 1.85,1,16,9 computer displays have the best compatibility. During the 2000s, DVDs and TV broadcasts shifted from 4,3 to 16,9, today the TV and DVDs are in 16,9 or 1.85,1 format,16,9 displays are optimal for their playback on a computer. 16,9 material on a 16,10 or 4,3 display will be letterboxed, in data processing or viewing 4,3 material such as older films, older TV broadcasts or older digital photographs, the widescreen will be letterboxed. Microsoft recommends a 16,9 display for computers running Windows 8
8.
Paper size
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Many paper size standards conventions have existed at different times and in different countries. Today, the A and B series of ISO216, which includes the commonly used A4 size, are the standard used by almost every country. In some areas under strong US influence, Letter is more prevalent, Paper sizes affect writing paper, stationery, cards, and some printed documents. The international standard for envelopes is the C series of ISO269, the international paper size standard is ISO216. It is based on the German DIN476 standard for paper sizes, ISO paper sizes are all based on a single aspect ratio of square root of 2, or approximately 1,1.4142. There are different series, as well as several extensions, the following international paper sizes are included in Cascading Style Sheets, A3, A4, A5, B4, B5. The base A0 size of paper is defined as having an area of 1 m2, rounded to the nearest millimetre, that is 841 by 1,189 millimetres. Successive paper sizes in the series A1, A2, A3 and this also effectively halves the area of each sheet. The most frequently used paper size is A4 measuring 210 by 297 millimetres, folded brochures of any size can be made by using sheets of the next larger size, e. g. A4 sheets are folded to make A5 brochures. The system allows scaling without compromising the aspect ratio from one size to another—as provided by office photocopiers, similarly, two sheets of A4 can be scaled down and fit exactly on 1 sheet without any cutoff or margins. The behavior of the ratio is easily proven, on a sheet of paper, let a be the long side and b be the short side, thus. When the sheet of paper is folded in half widthwise, let c be the length of the new short side, c = a/2. If we take the ratio of the folded paper we have, b c = b a 2 =2 a b =22 =2 Therefore. The advantages of basing a paper size upon an aspect ratio of √2 were first noted in 1786 by the German scientist and philosopher Georg Christoph Lichtenberg. The formats that became A2, A3, B3, B4 and B5 were developed in France on proposition of the mathematician Lazare Carnot, early in the 20th century, Dr Walter Porstmann turned Lichtenbergs idea into a proper system of different paper sizes. Porstmanns system was introduced as a DIN standard in Germany in 1922, even today, the paper sizes are called DIN A4 in everyday use in Germany and Austria. The DIN476 standard spread quickly to other countries, by 1977, A4 was the standard letter format in 88 of 148 countries. Today the standard has been adopted by all countries in the world except the United States, in Mexico, Costa Rica, Colombia, Venezuela, Chile, and the Philippines, the US letter format is still in common use, despite their official adoption of the ISO standard
9.
Standard ad size
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A web banner or banner ad is a form of advertising on the World Wide Web delivered by an ad server. This form of online advertising entails embedding an advertisement into a web page and it is intended to attract traffic to a website by linking to the website of the advertiser. In many cases, banners are delivered by an ad server. This payback system is often how the content provider is able to pay for the Internet access to supply the content in the first place, usually though, advertisers use ad networks to serve their advertisements, resulting in a revshare system and higher quality ad placement. Web banners differ in that the results for advertisement campaigns may be monitored real-time, behavior is often tracked through the use of a click tag. Many web surfers regard these advertisements as annoying because they distract from a web pages actual content or waste bandwidth, in some cases, web banners cover screen content that the user wishes to see. Newer web browsers often include software adblocker options to disable pop-ups or block images from selected websites, another way of avoiding banners is to use a proxy server that blocks them, such as Privoxy. Web browsers may also have extensions available that block banners, for example Adblock Plus for Mozilla Firefox, or AdThwart for Google Chrome, the pioneer of online advertising was Prodigy, a company owned by IBM and Sears at the time. Prodigy used online advertising first to promote Sears products in the 1980s, Prodigy was unable to capitalize on any of its first mover advantage in online advertising. The first clickable web ad was sold by Global Network Navigator in 1993 to Heller, Ehrman, White, & McAuliffe, GNN was the first commercially supported web publication and one of the very first commercial web sites ever. HotWired was the first web site to sell ads in large quantities to a wide range of major corporate advertisers. Andrew Anker was HotWireds first CEO, rick Boyce, a former media buyer with San Francisco advertising agency Hal Riney & Partners, spearheaded the sales effort for the company. HotWired coined the term banner ad and was the first company to provide click through rate reports to its customers, the first web banner sold by HotWired was paid for by AT&T Corp. and was put online on October 27,1994. Another source also credits Hotwired and October 1994, but has Coors Zima campaign as the first web banner, in May 1994, Ken McCarthy mentored Boyce in his transition from traditional to online advertising and first introduced the concept of a clickable/trackable ad. He stated that he believed only a direct response model—in which the return on investment of individual ads was measured—would prove sustainable over the long run for online advertising. In spite of this prediction, banner ads were valued and sold based on the number of impressions they generated, the first central ad server was released in July 1995 by Focalink Communications, which enabled the management, targeting, and tracking of online ads. A local ad server quickly followed from NetGravity in January 1996, the new online advertising model that emerged in the early years of the 21st century, introduced by GoTo. com, relies heavily on tracking ad response rather than impressions. The banner ad played a significant role in enabling the development of paid advertising on the Internet
10.
Pixel aspect ratio
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Pixel aspect ratio is a mathematical ratio that describes how the width of a pixel in a digital image compares to the height of that pixel. Most digital imaging systems display an image as a grid of tiny, Pixel Aspect Ratio describes this difference. Use of pixel aspect ratio mostly involves pictures pertaining to standard-definition television, most other imaging systems, including those that comply with SMPTE standards and practices, use square pixels. In digital images, there is a distinction with the Storage Aspect Ratio, if an image is displayed with square pixels, then these ratios agree, if not, then non-square, rectangular pixels are used, and these ratios disagree. The aspect ratio of the pixels themselves is known as the Pixel Aspect Ratio – for square pixels this is 1,1 – and these are related by the identity, SAR × PAR = DAR. For example, a 640 ×480 VGA image has a SAR of 640/480 =4,3, by contrast, a 720 ×576 D-1 PAL image has a SAR of 720/576 =5,4, but is displayed on a 4,3 display. In analog images such as there is no notion of pixel, nor notion of SAR or PAR. Today they arise also in transcoding between resolutions with different SARs, actual displays do not generally have non-square pixels, though digital sensors might, they are rather a mathematical abstraction used in resampling images to convert between resolutions. However, by a sampling rate, the effective horizontal resolution can be determined by the sampling theorem. Also, the resolution may be rounded, third, analog video signals are interlaced – each image is sent as two fields, each with half the lines. Thus pixels are either twice as tall as they would be without interlacing, Video is presented as a sequential series of images called video frames. Historically, video frames were created and recorded in analog form, as digital display technology, digital broadcast technology, and digital video compression evolved separately, it resulted in video frame differences that must be addressed using Pixel Aspect Ratio. Digital video frames are defined as a grid of pixels used to present each sequential image. The horizontal component is defined by pixels, and is known as a video line, the vertical component is defined by the number of lines, as in 480 lines. Such standards define an image as an array of well-defined horizontal Lines, well-defined vertical Line Duration, however, there is not a standard-definition television standard that properly defines image edges or explicitly demands a certain number of picture elements per line. As a result of becoming powerful enough to serve as video editing tools, video digital-to-analog converters. To convert analog video lines into a series of square pixels, the luma sampling rate for 480i pictures was 12 3⁄11 MHz and for 576i pictures was 14 3⁄4 MHz. The term pixel aspect ratio was first coined when ITU-R BT.601 specified that standard-definition television pictures are made of lines of exactly 720 non-square pixels and it designated 177,160 for 480i or 1035,1132 for 576i
11.
Photolithography
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Photolithography, also termed optical lithography or UV lithography, is a process used in microfabrication to pattern parts of a thin film or the bulk of a substrate. It uses light to transfer a geometric pattern from a photomask to a light-sensitive chemical photoresist, or simply resist, on the substrate. A series of chemical treatments then engraves the exposure pattern into, or enables deposition of a new material in the desired pattern upon. For example, in integrated circuits, a modern CMOS wafer will go through the photolithographic cycle up to 50 times. This procedure is comparable to a high precision version of the used to make printed circuit boards. Subsequent stages in the process have more in common with etching than with lithographic printing and its main disadvantages are that it requires a flat substrate to start with, it is not very effective at creating shapes that are not flat, and it can require extremely clean operating conditions. The root words photo, litho, and graphy all have Greek origins, with the light, stone. As suggested by the name compounded from them, photolithography is a method in which light plays an essential role. In the 1820s, Nicephore Niepce invented a process that used Bitumen of Judea. In 1954, Louis Plambeck Jr. developed the Dycryl polymeric letterpress plate, a single iteration of photolithography combines several steps in sequence. Modern cleanrooms use automated, robotic wafer track systems to coordinate the process, the procedure described here omits some advanced treatments, such as thinning agents or edge-bead removal. Other solutions made with trichloroethylene, acetone or methanol can also be used to clean, the wafer is initially heated to a temperature sufficient to drive off any moisture that may be present on the wafer surface,150 °C for ten minutes is sufficient. Wafers that have been in storage must be cleaned to remove contamination. A liquid or gaseous adhesion promoter, such as Bisamine, is applied to promote adhesion of the photoresist to the wafer. The surface layer of silicon dioxide on the wafer reacts with HMDS to form tri-methylated silicon-dioxide, in order to ensure the development of the image, it is best covered and placed over a hot plate and let it dry while stabilizing the temperature at 120 °C. The wafer is covered with photoresist by spin coating, a viscous, liquid solution of photoresist is dispensed onto the wafer, and the wafer is spun rapidly to produce a uniformly thick layer. The spin coating typically runs at 1200 to 4800 rpm for 30 to 60 seconds, the spin coating process results in a uniform thin layer, usually with uniformity of within 5 to 10 nanometres. Thus, the top layer of resist is quickly ejected from the edge while the bottom layer still creeps slowly radially along the wafer
12.
Tire code
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Automobile tires are described by an alphanumeric tire code or tyre code, which is generally molded into the sidewall of the tire. This code specifies the dimensions of the tire, and some of its key limitations, such as load-bearing ability, sometimes the inner sidewall contains information not included on the outer sidewall, and vice versa. The code has grown in complexity over the years, as is evident from the mix of S. I. and English units, new automotive tires frequently have ratings for traction, treadwear, and temperature resistance. Most tires sizes are given using the ISO Metric sizing system, however, some pickup trucks and SUVs use the Light Truck Numeric or Light Truck High Flotation system. The European Tyre and Rim Technical Organisation and the Tire and Rim Association are two organizations that influence national tire standards, the objective of the ETRTO include aligning national tire and rim standards in Europe. The Tire and Rim Association, formerly known as The Tire and Rim Association of America, in the United States, the Office of Vehicle Safety Compliance, a component of the Department of Transportation, is one of the agencies tasked to enforce the Federal Motor Vehicle Safety Standard. Canada has published tire regulations, such as the Motor Vehicle Tire Safety Regulations SOR 95-148, in practice, the standards of the two organizations have evolved together and are fairly interchangeable, but not fully, since the Load Index will be different for the same size tire. 3-digit number, The nominal section width of the tire in millimeters, the tire surface that touches the road usually has smaller width. /, Slash character for character separation, 2- or 3-digit number, The aspect ratio of the sidewall height as a percentage of the nominal section width of the tire. If the information is omitted, it is assumed to be 82%, if the number is larger than 200, then this is the diameter of the entire tire in millimeters. There is the exception of metric-diameter tires, such as the use of the 390 size. Few tires are made to this size currently, the number may be longer where a half-inch size is used, for example many heavy transport trucks now use 22. 5-inch tires. 2- or 3-digit number, Load index, see table below, Some light-truck tires are approved for dual use, that is they can be run in pairs next to each other. If so, separate load indexes will be specified for single, in the example shown in the light-truck tire illustration, the tire has a load index of 114 if used as a single tire, and a load index of 111 if used in a dual pair. Tires without this designation are unsafe for dual usage, 1- or 2-digit/letter combo, Speed rating, see table below Additional marks, See subheading below. 2-digit number, The diameter of the tire in inches, 3- or 4-digit number, The section width of the tire in inches. If the tire diameter is not given, section widths ending in zero indicate the ratio of 92%. These aspect ratios often vary from todays tire manufacturer specification, construction of the fabric of the tire, B, bias belt D, diagonal R, radial 2-digit number, Diameter in inches of the wheel rim that this tire is designed to fit
13.
Aspect ratio (wing)
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In aeronautics, the aspect ratio of a wing is the ratio of its span to its mean chord. It is equal to the square of the wingspan divided by the wing area, thus, a long, narrow wing has a high aspect ratio, whereas a short, wide wing has a low aspect ratio. A large wingspan is working on a cylinder of air. The aft-leaning component of change in velocity is proportional to the induced drag. The interaction between undisturbed air outside the cylinder of air, and the downward-moving cylinder of air occurs at the wingtips. Also, longer wings may have some torsion for a given load, maneuverability, a low aspect-ratio wing will have a higher roll angular acceleration than one of high aspect ratio, because a high-aspect-ratio wing has a higher moment of inertia to overcome. In a steady roll, the longer wing gives a roll moment because of the longer moment arm of the aileron. Low aspect ratio wings are used on fighter aircraft, not only for the higher roll rates. Parasitic drag, While high aspect wings create less induced drag and this is because, for an equal wing area, the average chord is smaller. However, this variation is small when compared to the variation in induced drag with changing wingspan. For example, the drag coefficient c d of a NACA23012 airfoil is inversely proportional to chord length to the power 0.129. A20 percent increase in length would decrease the section drag coefficient by 2.38 percent. Practicality, low aspect ratios have a greater internal volume, since the maximum thickness is greater. This limits the Airbus A380 to 80m wide with a ratio of 7.8, while the Boeing 787 or Airbus A350 have an aspect ratio of 9.5. Aircraft which approach or exceed the speed of sound sometimes incorporate variable-sweep wings and these wings give a high aspect ratio when unswept and a low aspect ratio at maximum sweep. In subsonic flow, steeply swept and narrow wings are inefficient compared to a high-aspect-ratio wing, thus a long span, valuable at low speeds, causes excessive drag at transonic and supersonic speeds. By varying the sweep the wing can be optimised for the current flight speed, however the extra weight and complexity of a moveable wing mean that it is not often used. The aspect ratios of birds and bats wings vary considerably, birds that fly long distances or spend long periods soaring such as albatrosses and eagles often have wings of high aspect ratio
14.
Optical lens
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A lens is a transmissive optical device that focuses or disperses a light beam by means of refraction. A simple lens consists of a piece of transparent material, while a compound lens consists of several simple lenses. Lenses are made from such as glass or plastic, and are ground. A lens can focus light to form an image, unlike a prism, devices that similarly focus or disperse waves and radiation other than visible light are also called lenses, such as microwave lenses, electron lenses, acoustic lenses, or explosive lenses. The word lens comes from the Latin name of the lentil, the genus of the lentil plant is Lens, and the most commonly eaten species is Lens culinaris. The lentil plant also gives its name to a geometric figure, the variant spelling lense is sometimes seen. While it is listed as a spelling in some dictionaries. The oldest lens artifact is the Nimrud lens, dating back 2700 years to ancient Assyria, david Brewster proposed that it may have been used as a magnifying glass, or as a burning-glass to start fires by concentrating sunlight. Another early reference to magnification dates back to ancient Egyptian hieroglyphs in the 8th century BC, the earliest written records of lenses date to Ancient Greece, with Aristophanes play The Clouds mentioning a burning-glass. Some scholars argue that the evidence indicates that there was widespread use of lenses in antiquity. Such lenses were used by artisans for fine work, and for authenticating seal impressions, both Pliny and Seneca the Younger described the magnifying effect of a glass globe filled with water. The Viking lenses were capable of concentrating enough sunlight to ignite fires, between the 11th and 13th century reading stones were invented. Often used by monks to assist in illuminating manuscripts, these were primitive plano-convex lenses initially made by cutting a sphere in half. As the stones were experimented with, it was understood that shallower lenses magnified more effectively. Lenses came into use in Europe with the invention of spectacles. Spectacle makers created improved types of lenses for the correction of vision based more on knowledge gained from observing the effects of the lenses. With the invention of the telescope and microscope there was a deal of experimentation with lens shapes in the 17th. Opticians tried to construct lenses of varying forms of curvature, wrongly assuming errors arose from defects in the figure of their surfaces
15.
Square
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In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles. It can also be defined as a rectangle in which two adjacent sides have equal length, a square with vertices ABCD would be denoted ◻ ABCD. e. A rhombus with equal diagonals a convex quadrilateral with sides a, b, c, d whose area is A =12 =12. Opposite sides of a square are both parallel and equal in length, all four angles of a square are equal. All four sides of a square are equal, the diagonals of a square are equal. The square is the n=2 case of the families of n-hypercubes and n-orthoplexes, a truncated square, t, is an octagon. An alternated square, h, is a digon, the perimeter of a square whose four sides have length ℓ is P =4 ℓ and the area A is A = ℓ2. In classical times, the power was described in terms of the area of a square. This led to the use of the square to mean raising to the second power. The area can also be calculated using the diagonal d according to A = d 22. In terms of the circumradius R, the area of a square is A =2 R2, since the area of the circle is π R2, in terms of the inradius r, the area of the square is A =4 r 2. Because it is a polygon, a square is the quadrilateral of least perimeter enclosing a given area. Dually, a square is the quadrilateral containing the largest area within a given perimeter. Indeed, if A and P are the area and perimeter enclosed by a quadrilateral, then the isoperimetric inequality holds,16 A ≤ P2 with equality if. The diagonals of a square are 2 times the length of a side of the square and this value, known as the square root of 2 or Pythagoras constant, was the first number proven to be irrational. A square can also be defined as a parallelogram with equal diagonals that bisect the angles, if a figure is both a rectangle and a rhombus, then it is a square. If a circle is circumscribed around a square, the area of the circle is π /2 times the area of the square, if a circle is inscribed in the square, the area of the circle is π /4 times the area of the square. A square has an area than any other quadrilateral with the same perimeter
16.
VGA
–
Video Graphics Array is the display hardware first introduced with the IBM PS/2 line of computers in 1987. Today, the VGA analog interface is used for high definition video, including resolutions of 1080p, while the transmission bandwidth of VGA is high enough to support even higher resolution playback, there can be picture quality degradation depending on cable quality and length. How discernible this degradation is depends on the individuals eyesight and the display, though it is noticeable when switching to. The VGA supports both All Points Addressable graphics modes, and alphanumeric text modes, the other color modes defaulted to standard EGA or CGA compatible palettes, but could still be redefined if desired using VGA-specific programming. g. 400×600 or 360×480, and thin pixels,16 colors and the 70 Hz refresh rate with e. g. 736×410 mode. Its single-chip implementation allowed the VGA to be placed directly on a PC′s motherboard with a minimum of difficulty, since it only required video memory, timing crystals, the original VGA specifications are as follows,256 KB Video RAM 16-color and 256-color paletted display modes. 262, 144-color global palette Selectable 25.175 MHz or 28.322 MHz master pixel clock Usual line rate fixed at 31, compatibility is almost full at BIOS level, but even at register level, a very high value of compatibility is reached. The formula for the VGA horizontal frequency is thus ×525 kHz =4500 ÷143 kHz ≈31.4685 kHz, All other frequencies used by the VGA card are derived from this value by integer multiplication or division. Since the exactness of quartz oscillators is limited, real cards will have higher or lower frequency. All derived VGA timings can be varied widely by software that bypasses the VGA firmware interface and communicates directly with the VGA hardware, the use of other timings may in fact damage such monitors and thus was usually avoided by software publishers. For the most common VGA mode, the timings are. The figures shown in this image may be inaccurate and not match the above table exactly. The same general layout applies, merely at a lower frequency and these timings are the same in the higher frequency mode, but all pixel counts are correspondingly multiplied by 9/8ths – thus,720 active pixels,900 total per line, and a 54 pixel back porch. The monitor is triggered into synchronising at the higher frame rate by use of a positive-polarity VSync pulse. Depending on manufacturer, the details of active period and front/back porch widths, particularly in the horizontal domain. 640×400 @70 Hz is traditionally the video used for booting most VGA-compatible x86 personal computers which show a graphical boot screen. 640×480 @60 Hz is the default Windows graphics mode, up to Windows 2000 and it remains an option in XP and later versions via the boot menu low resolution video option and per-application compatibility mode settings. 320×200 @70 Hz is the most common mode for VGA-era PC games, using exactly the same timings as the 640×400 mode, the actual timings vary slightly from the defined standard
17.
XGA
–
The graphics display resolution is the width and height dimensions of an electronic visual display device, such as a computer monitor, in pixels. Certain combinations of width and height are standardized and typically given a name, a higher display resolution in a display of the same size means that displayed content appears sharper. The favored aspect ratio of mass market display industry products has changed gradually from 4,3, then to 16,10, and then to 16,9, the 4,3 aspect ratio generally reflects older products, especially the era of the cathode ray tube. The 16,10 aspect ratio had its largest use in the 1995–2010 period, and the 16,9 aspect ratio tends to reflect post-2010 mass market computer monitor, laptop, and entertainment products displays. The 4,3 aspect ratio was common in older cathode ray tube displays. The 16,10 ratio allowed some compromise between showing older 4,3 aspect ratio broadcast TV shows, but also allowing better viewing of widescreen movies. However, around the year 2005, entertainment industry displays gradually moved from 16,10 to the 16,9 aspect ratio, by about 2007, virtually all mass market entertainment displays were 16,9. In 2011, 1920×1080 was the resolution in the most heavily marketed entertainment market displays. The next standard, 3840×2160 emerged on the market in 2013, also in 2013, displays with 2560×1080 appeared, which closely approximate the common CinemaScope movie standard aspect ratio of 2. 35–2.40. In 2014,21,9 screens with pixel dimensions of 3440×1440 became available as well, in this time frame, with the notable exception of Apple, almost all desktop, laptop, and display manufacturers gradually moved to promoting only 16,9 aspect ratio displays. By 2011, the 16,10 aspect ratio had virtually disappeared from the Windows laptop display market, one consequence of this transition was that the highest available resolutions moved generally downward. NHD is a resolution of 640×360 pixels, which is exactly one ninth of a Full HD frame. Pixel doubling nHD frames will form one 720p frame and pixel tripling nHD frames will form one 1080p frame, one drawback of this resolution is that the vertical resolution is not an even multiple of 16, which is a common macroblock size for video codecs. Video frames encoded with 16×16 pixel macroblocks would be padded to 640×368, H.264 codecs have this padding and cropping ability built-in as standard. The same is true for qHD and 1080p but the amount of padding is more for lower resolutions such as nHD. To avoid storing the eight lines of padded pixels, some prefer to encode video at 624×352. When such video streams are encoded from HD frames or played back on HD displays in full screen mode they are scaled by non-integer scale factors. True nHD frames on the hand has integer scale factors
18.
ISO 216
–
ISO216 specifies international standard paper sizes used in most countries in the world today, although not in Canada or the United States. The standard defines the A and B series of sizes, including A4. Two supplementary standards, ISO217 and ISO269, define related paper sizes, all ISO216, ISO217 and ISO269 paper sizes have the same aspect ratio,1, √2, at least to within the rounding to whole numbers of millimetres. This ratio has the property that when cut or folded in half widthwise. Each ISO paper size is one half of the area of the larger size. In 1786, the German scientist Georg Christoph Lichtenberg described the advantages of basing a paper size on a ratio of 2 in a letter to Johann Beckmann. The formats that became ISO paper sizes A2, A3, B3, B4 and they were listed in a 1798 law on taxation of publications that was based in part on page sizes. The main advantage of this system is its scaling, the ISO system of paper sizes exploit these properties of the 2 aspect ratio. In each series of sizes, the largest size is numbered 0, a folded brochure can be made by using a sheet of the next larger size (for example, an A4 sheet is folded in half to make a brochure with size A5 pages. An office photocopier or printer can be designed to reduce a page from A4 to A5 or to enlarge a page from A4 to A3, similarly, two sheets of A4 can be scaled down to fit one A4 sheet without excess empty paper. This system also simplifies calculating the weight paper, under ISO536, papers grammage is defined as a sheets weight in grams per area in square metres. Since an A0 sheet has an area of 1 m2, its weight in grams is the same as its grammage, one can derive the grammage of other sizes by arithmetic division in g/m2. A standard A4 sheet made from 80 g/m2 paper weighs 5 g, thus the weight, and the associated postage rate, can be easily approximated by counting the number of sheets used.414 aspect ratio, rounded to the nearest millimetre. A0 is defined so that it has an area of 1 square metre before rounding. Successive paper sizes in the series are defined by halving the length of the paper size. The most frequently used of this series is the size A4 which is 210 mm ×297 mm, for comparison, the letter paper size commonly used in North America is approximately 6 mm wider and 18 mm shorter than A4. The geometric rationale behind the root of 2 is to maintain the aspect ratio of each subsequent rectangle after cutting or folding an A series sheet in half. The formula that gives the border of the paper size An in metres and without rounding off is the geometric sequence
19.
135 film
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135 is photographic film in a film format used for still photography. It is a film with a film gauge of 35 mm. Its engineering standard for the film is controlled by ISO1007, the term 135 was introduced by Kodak in 1934 as a designation for the cassette for 35 mm film, specifically for still photography. It quickly grew in popularity, surpassing 120 film by the late 1960s to become the most popular photographic film size, despite competition from formats such as 828,126,110, and APS, it remains so today. 135 camera film always comes perforated with Kodak Standard perforations, the size of the 135 film frame has been adopted by many high-end digital single-lens reflex and digital mirrorless cameras, commonly referred to as full frame. Even though the format is smaller than historical medium format and large format film, it is much larger than image sensors in most compact cameras. Individual rolls of 135 film are enclosed in single-spool, light-tight, the film is clipped or taped to a spool and exits via a slot lined with flocking. The end of the film is cut on one side to form a leader and it has the same dimensions and perforation pitch as 35 mm movie print film. Most cameras require the film to be rewound before the camera is opened, disposable cameras use the same technique so that the user does not have to rewind. Since the 1980s film cassettes have been marked with a DX encoding pattern, conforming cameras detect this, different films are sensitive to light at different degrees, this film speed is standardised by ISO.135 film has been made in several emulsion types and sensitivities. Since the introduction of digital cameras the most usual films have colour emulsions of ISO 100/21° to ISO 800/30°, films of lower sensitivity and higher sensitivity are for more specialist purposes. There are colour and monochrome films, negative and positive, monochrome film is usually panchromatic, orthochromatic has fallen out of use. Film designed to be sensitive to infrared radiation can be obtained, more exotic emulsions have been available in 135 than other roll-film sizes. The term 135 format usually refers to a 36×24 mm film format, the 36×24 mm format is common to digital image sensors, where it is typically referred to as full frame format. On 135 film, the dimension of the 36×24 mm frame runs parallel to the length of the film. The perforation size and pitch are according to the standard specification KS-1870, for each frame, the film advances 8 perforations. This is specified as 38.00 mm and this allows for 2 mm gaps between frames. Each camera model has a different location for the sprocket which advances the film, therefore, each camera model’s frame will vary in position relative to the perforations
20.
IPhone
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IPhone is a line of smartphones designed and marketed by Apple Inc. They run Apples iOS mobile operating system, the first generation iPhone was released on June 29,2007, the most recent iPhone model is the iPhone 7, which was unveiled at a special event on September 7,2016. The user interface is built around the devices multi-touch screen, including a virtual keyboard, the iPhone has Wi-Fi and can connect to cellular networks. Other functionality, such as games, reference works, and social networking. As of January 2017, Apples App Store contained more than 2.2 million applications available for the iPhone, Apple has released ten generations of iPhone models, each accompanied by one of the ten major releases of the iOS operating system. The iPhone 5 featured a taller, 4-inch display and Apples newly introduced Lightning connector, in 2013, Apple released the 5S with improved hardware and a fingerprint reader, and the lower-cost 5C, a version of the 5 with colored plastic casings instead of metal. They were followed by the larger iPhone 6, with models featuring 4.7 and 5. 5-inch displays.5 mm headphone jack found on previous phones. The iPhones commercial success has been credited with reshaping the smartphone industry, the original iPhone was one of the first phones to use a design featuring a slate format with a touchscreen interface. Almost all modern smartphones have replicated this style of design, in the US, the iPhone holds the largest share of the smartphone market. As of late 2015, the iPhone had a 43. 6% market share, followed by Samsung, LG, Apple CEO Steve Jobs steered the original focus away from a tablet and towards a phone. Apple created the device during a collaboration with Cingular Wireless at the time—at an estimated development cost of US$150 million over thirty months. Apple rejected the design by committee approach that had yielded the Motorola ROKR E1, among other deficiencies, the ROKR E1s firmware limited storage to only 100 iTunes songs to avoid competing with Apples iPod nano. Jobs unveiled the iPhone to the public on January 9,2007, the passionate reaction to the launch of the iPhone resulted in sections of the media dubbing it the Jesus phone. Following this successful release in the US, the first generation iPhone was made available in the UK, France, and Germany in November 2007, on July 11,2008, Apple released the iPhone 3G in twenty-two countries, including the original six. Apple released the iPhone 3G in upwards of eighty countries and territories. Apple announced the iPhone 3GS on June 8,2009, along with plans to release it later in June, July, many would-be users objected to the iPhones cost, and 40% of users had household incomes over US$100,000. The back of the original first generation iPhone was made of aluminum with a black plastic accent, the iPhone 3G and 3GS feature a full plastic back to increase the strength of the GSM signal. The iPhone 3G was available in an 8 GB black model, the iPhone 3GS was available in both colors, regardless of storage capacity
21.
IPhone 5
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The iPhone 5 is a smartphone that was designed and marketed by Apple Inc. It is the generation of the iPhone, succeeding the iPhone 4S and preceding the iPhone 5S. Formally unveiled as part of an event on September 12,2012. It was the first iPhone to be developed under the guidance of Tim Cook. The iPhone 5 featured major changes in comparison to its predecessor. This was the second Apple phone to include its new Sony made 8MP camera, Apple began taking pre-orders on September 14,2012, and over two million were received within 24 hours. The iPhone 5 was officially discontinued by Apple on September 10,2013 with the announcement of its successors, the iPhone 5S, rumors about the iPhone 5 began shortly after the announcement of the iPhone 4S, though detailed leaks did not emerge until June 2012. On July 30,2012, reports pinpointed the dates on which the iPhone 5 would be unveiled and released, on September 4,2012, Apple announced they would be hosting an event at the Yerba Buena Center for the Arts in San Francisco on September 12,2012. A shadow of the numeral 5 was featured in the invitations sent to the media, at the unveiling, Apple announced the iPhone 5 and also introduced new iPod Nano and iPod Touch models. They also stated that pre-orders would be accepted starting September 14,2012, over two million pre-orders were received within 24 hours. Initial demand for the new phone exceeded the set by its predecessor. On November 30,2012, Apple added a version of the iPhone 5 to their online US store. The iPhone 5 was officially discontinued by Apple on September 10,2013 with the announcement of its successors, the iPhone 5S and the iPhone 5C. While the 5C shared almost the same hardware as the iPhone 5. The introduction of the 5C deviated from Apples previous market strategy, where the previous iPhone model would remain in production, but sold at a lower price point below the new model. On April 28,2014, Apple initiated an out of warranty recall program to replace any failing power buttons of iPhone 5 models which were manufactured prior to March 2013 at no cost. Following the release of the iPhone 5, Samsung announced that it was filing a lawsuit against Apple for infringing eight of its patents, the case was scheduled to begin in 2014. In a statement, Samsung said it had little choice but to take the necessary to protect our innovations
22.
Widescreen
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Widescreen images are a variety of aspect ratios used in film, television and computer screens. In film, a film is any film image with a width-to-height aspect ratio greater than the standard 1.37,1 Academy aspect ratio provided by 35mm film. For television, the screen ratio for broadcasts was 4,3. Between the 1990s and early 2000s,16,9 TV displays came into wide use and they are typically used in conjunction with high-definition television receivers, or Standard-Definition DVD players and other digital television sources. With computer displays, aspect ratios wider than 4,3 are also called widescreen, widescreen computer displays were previously of 16,10 aspect ratio, but now are usually 16,9. Widescreen was first used in the film of the Corbett-Fitzsimmons Fight in 1897 and this was not only the longest film that had been released to date at 100 minutes, it was also the first widescreen film being shot on 63mm Eastman stock with five perforations per frame. Widescreen was first widely used in the late 1920s in some films and newsreels. Claude Autant-Lara released a film Pour construire un feu in the early Henri Chretien widescreen process, in 1927, The American aka The Flag Maker was released. In 1926, Spoor and Berggren had released a Natural Vision film of Niagara Falls, the Natural Vision widescreen process used 63. 5mm film and had a 2,1 aspect ratio. On May 26,1929, Fox Film Corporation released Fox Grandeur News, which premiered at Graumans Chinese Theatre in Hollywood on October 2,1930, all of which were also made in the 70mm Fox Grandeur process. On November 13,1930, United Artists released The Bat Whispers directed by Roland West in a 70mm widescreen process known as Magnafilm, warner Brothers released Song of the Flame and Kismet in a widescreen process they called Vitascope. In 1930, after experimenting with the system called Fanthom Screen for The Trail of 98, MGM filmed The Great Meadow in Realife—however, its unclear if it was ever released in that widescreen process due to declining interest of the movie-going public. S. However, a few producers and directors, among them Alfred Hitchcock, have been reluctant to use the anamorphic widescreen size featured in such formats as Cinemascope. Hitchcock alternatively used VistaVision, a widescreen process developed by Paramount Pictures. Masked widescreen was introduced in April 1953, a detriment is that the film grain size is thus increased because only part of the image is being expanded to full height. Films are designed to be shown in cinemas in masked widescreen format, in such an instance, a photographer will compose for widescreen, but protect the full image from things such as microphones and other filming equipment. Standardized flat wide screen ratios are 1.66,1,1.75,1,1.85,1,1.85,1 has become the predominant aspect ratio for the format. 35mm anamorphic – This type of widescreen is used for CinemaScope, Panavision, the film is essentially shot squeezed, so that the actors appear vertically elongated on the actual film
23.
Computer display
–
A computer monitor or a computer display is an electronic visual display for computers. A monitor usually comprises the display device, circuitry, casing, the display device in modern monitors is typically a thin film transistor liquid crystal display or a flat panel LED display, while older monitors used a cathode ray tubes. It can be connected to the computer via VGA, DVI, HDMI, DisplayPort, Thunderbolt, LVDS or other proprietary connectors, originally, computer monitors were used for data processing while television receivers were used for entertainment. From the 1980s onwards, computers have used for both data processing and entertainment, while televisions have implemented some computer functionality. The common aspect ratio of televisions, and computer monitors, has changed from 4,3 to 16,10, to 16,9. Early electronic computers were fitted with a panel of light bulbs where the state of each particular bulb would indicate the state of a particular register bit inside the computer. This allowed the operating the computer to monitor the internal state of the machine. As early monitors were only capable of displaying a limited amount of information. Instead, a printer was the primary output device, while the monitor was limited to keeping track of the programs operation. Multiple technologies have used for computer monitors. Until the 21st century most used cathode ray tubes but they have largely superseded by LCD monitors. The first computer monitors used cathode ray tubes, high-resolution CRT displays were developed for specialized military, industrial and scientific applications but they were far too costly for general use. Either computer could be connected to the terminals of an ordinary color TV set or used with a purpose-made CRT color monitor for optimum resolution. In 1984 IBM introduced the Enhanced Graphics Adapter which was capable of producing 16 colors and had a resolution of 640 x 350, by the end of the 1980s color CRT monitors that could clearly display 1024 x 768 pixels were widely available and increasingly affordable. During the following decade maximum display resolutions gradually increased and prices continued to fall, CRT technology remained dominant in the PC monitor market into the new millennium partly because it was cheaper to produce and offered viewing angles close to 180 degrees. CRTs still offer some image quality advantages over LCDs but improvements to the latter have made much less obvious. There are multiple technologies that have used to implement liquid crystal displays. Commonly, the laptop would be offered with an assortment of display options at increasing price points, monochrome
24.
WXGA (graphics)
–
The graphics display resolution is the width and height dimensions of an electronic visual display device, such as a computer monitor, in pixels. Certain combinations of width and height are standardized and typically given a name, a higher display resolution in a display of the same size means that displayed content appears sharper. The favored aspect ratio of mass market display industry products has changed gradually from 4,3, then to 16,10, and then to 16,9, the 4,3 aspect ratio generally reflects older products, especially the era of the cathode ray tube. The 16,10 aspect ratio had its largest use in the 1995–2010 period, and the 16,9 aspect ratio tends to reflect post-2010 mass market computer monitor, laptop, and entertainment products displays. The 4,3 aspect ratio was common in older cathode ray tube displays. The 16,10 ratio allowed some compromise between showing older 4,3 aspect ratio broadcast TV shows, but also allowing better viewing of widescreen movies. However, around the year 2005, entertainment industry displays gradually moved from 16,10 to the 16,9 aspect ratio, by about 2007, virtually all mass market entertainment displays were 16,9. In 2011, 1920×1080 was the resolution in the most heavily marketed entertainment market displays. The next standard, 3840×2160 emerged on the market in 2013, also in 2013, displays with 2560×1080 appeared, which closely approximate the common CinemaScope movie standard aspect ratio of 2. 35–2.40. In 2014,21,9 screens with pixel dimensions of 3440×1440 became available as well, in this time frame, with the notable exception of Apple, almost all desktop, laptop, and display manufacturers gradually moved to promoting only 16,9 aspect ratio displays. By 2011, the 16,10 aspect ratio had virtually disappeared from the Windows laptop display market, one consequence of this transition was that the highest available resolutions moved generally downward. NHD is a resolution of 640×360 pixels, which is exactly one ninth of a Full HD frame. Pixel doubling nHD frames will form one 720p frame and pixel tripling nHD frames will form one 1080p frame, one drawback of this resolution is that the vertical resolution is not an even multiple of 16, which is a common macroblock size for video codecs. Video frames encoded with 16×16 pixel macroblocks would be padded to 640×368, H.264 codecs have this padding and cropping ability built-in as standard. The same is true for qHD and 1080p but the amount of padding is more for lower resolutions such as nHD. To avoid storing the eight lines of padded pixels, some prefer to encode video at 624×352. When such video streams are encoded from HD frames or played back on HD displays in full screen mode they are scaled by non-integer scale factors. True nHD frames on the hand has integer scale factors
25.
Golden ratio
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In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The figure on the right illustrates the geometric relationship, expressed algebraically, for quantities a and b with a > b >0, a + b a = a b = def φ, where the Greek letter phi represents the golden ratio. Its value is, φ =1 +52 =1.6180339887 …, A001622 The golden ratio is also called the golden mean or golden section. Other names include extreme and mean ratio, medial section, divine proportion, divine section, golden proportion, golden cut, the golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other plant parts. The golden ratio has also used to analyze the proportions of natural objects as well as man-made systems such as financial markets. Two quantities a and b are said to be in the golden ratio φ if a + b a = a b = φ, one method for finding the value of φ is to start with the left fraction. Through simplifying the fraction and substituting in b/a = 1/φ, a + b a =1 + b a =1 +1 φ, multiplying by φ gives φ +1 = φ2 which can be rearranged to φ2 − φ −1 =0. First, the line segment A B ¯ is about doubled and then the semicircle with the radius A S ¯ around the point S is drawn, now the semicircle is drawn with the radius A B ¯ around the point B. The arising intersection point E corresponds 2 φ, next up, the perpendicular on the line segment A E ¯ from the point D will be establish. The subsequent parallel F S ¯ to the line segment C M ¯, produces, as it were and it is well recognizable, this triangle and the triangle M S C are similar to each other. The hypotenuse F S ¯ has due to the cathetuses S D ¯ =1 and D F ¯ =2 according the Pythagorean theorem, finally, the circle arc is drawn with the radius 5 around the point F. The golden ratio has been claimed to have held a fascination for at least 2,400 years. But the fascination with the Golden Ratio is not confined just to mathematicians, biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics. Ancient Greek mathematicians first studied what we now call the golden ratio because of its frequent appearance in geometry, the division of a line into extreme and mean ratio is important in the geometry of regular pentagrams and pentagons. Euclid explains a construction for cutting a line in extreme and mean ratio, throughout the Elements, several propositions and their proofs employ the golden ratio. The golden ratio is explored in Luca Paciolis book De divina proportione, since the 20th century, the golden ratio has been represented by the Greek letter φ or less commonly by τ. Timeline according to Priya Hemenway, Phidias made the Parthenon statues that seem to embody the golden ratio, plato, in his Timaeus, describes five possible regular solids, some of which are related to the golden ratio
26.
Super 16 mm
–
16 mm film is a historically popular and economical gauge of film. 16 mm refers to the width of the film, with other film gauges including 8 and 35 mm. It is generally used for film making or for low budget motion pictures. It also existed as an amateur or home movie making format for several decades, alongside 8 mm film. In 1923, Eastman Kodak released the first 16 mm outfit consisting of a camera, projector, tripod, screen, rCA-Victor introduced a 16 mm sound movie projector in 1932 and developed an optical sound-on-film 16 mm camera, released in 1935. Eastman Kodak introduced 16 mm film in 1923 as a less expensive alternative to 35 mm film. During the 1920s, the format was often referred to as sub-standard by the professional industry, Kodak hired Willard Beech Cook from his 28 mm Pathescope of America company to create the new 16 mm Kodascope Library. In addition to making movies, people could buy or rent films from the library. Intended for amateur use,16 mm film was one of the first formats to use acetate safety film as a film base, Kodak never used nitrate film for the format because of the high flammability of the nitrate base. 35 mm nitrate was discontinued in 1952, the silent 16 mm format was initially aimed at the home enthusiast, but by the 1930s it had begun to make inroads into the educational market. The addition of sound tracks and, most notably, Kodachrome in 1935. Used extensively in WW2, there was an expansion of 16 mm professional filmmaking in the post-war years. Films for government, business, medical and industrial clients created a network of 16 mm professional filmmakers. The advent of television production also enhanced the use of 16 mm film, initially for its advantage of cost and portability over 35 mm. At first used as a format, the 16 mm format was also used to create television programming shot outside the confines of the more rigid television studio production sets. The home movie market gradually switched to the less expensive 8 mm film. 16 mm has been used for television production with light cameras in many countries before portable video cameras appeared. Replacing analog video devices, digital video has made significant inroads in television production use, nevertheless,16 mm is still in use in its Super 16 ratio for low cost productions
27.
Domino (mathematics)
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In mathematics, a domino is a polyomino of order 2, that is, a polygon in the plane made of two equal-sized squares connected edge-to-edge. When rotations and reflections are not considered to be distinct shapes, since it has reflection symmetry, it is also the only one-sided domino. When rotations are also considered distinct, there are two fixed dominoes, The second one can be created by rotating the one above by 90°, a domino tiling is a covering of another polyomino with dominoes. In a wider sense, the term domino is often understood to mean a tile of any shape. Dominoes, a set of domino-shaped gaming pieces Tatami, Japanese domino-shaped floor mats
28.
Major axis
–
In geometry, the major axis of an ellipse is its longest diameter, a line segment that runs through the center and both foci, with ends at the widest points of the perimeter. The semi-major axis is one half of the axis, and thus runs from the centre, through a focus. Essentially, it is the radius of an orbit at the two most distant points. For the special case of a circle, the axis is the radius. One can think of the axis as an ellipses long radius. The semi-major axis of a hyperbola is, depending on the convention, thus it is the distance from the center to either vertex of the hyperbola. A parabola can be obtained as the limit of a sequence of ellipses where one focus is fixed as the other is allowed to move arbitrarily far away in one direction. Thus a and b tend to infinity, a faster than b, the semi-minor axis is a line segment associated with most conic sections that is at right angles with the semi-major axis and has one end at the center of the conic section. It is one of the axes of symmetry for the curve, in an ellipse, the one, in a hyperbola. The semi-major axis is the value of the maximum and minimum distances r max and r min of the ellipse from a focus — that is. In astronomy these extreme points are called apsis, the semi-minor axis of an ellipse is the geometric mean of these distances, b = r max r min. The eccentricity of an ellipse is defined as e =1 − b 2 a 2 so r min = a, r max = a. Now consider the equation in polar coordinates, with one focus at the origin, the mean value of r = ℓ / and r = ℓ /, for θ = π and θ =0 is a = ℓ1 − e 2. In an ellipse, the axis is the geometric mean of the distance from the center to either focus. The semi-minor axis of an ellipse runs from the center of the ellipse to the edge of the ellipse, the semi-minor axis is half of the minor axis. The minor axis is the longest line segment perpendicular to the axis that connects two points on the ellipses edge. The semi-minor axis b is related to the axis a through the eccentricity e. A parabola can be obtained as the limit of a sequence of ellipses where one focus is fixed as the other is allowed to move arbitrarily far away in one direction
29.
Minor axis
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In geometry, the major axis of an ellipse is its longest diameter, a line segment that runs through the center and both foci, with ends at the widest points of the perimeter. The semi-major axis is one half of the axis, and thus runs from the centre, through a focus. Essentially, it is the radius of an orbit at the two most distant points. For the special case of a circle, the axis is the radius. One can think of the axis as an ellipses long radius. The semi-major axis of a hyperbola is, depending on the convention, thus it is the distance from the center to either vertex of the hyperbola. A parabola can be obtained as the limit of a sequence of ellipses where one focus is fixed as the other is allowed to move arbitrarily far away in one direction. Thus a and b tend to infinity, a faster than b, the semi-minor axis is a line segment associated with most conic sections that is at right angles with the semi-major axis and has one end at the center of the conic section. It is one of the axes of symmetry for the curve, in an ellipse, the one, in a hyperbola. The semi-major axis is the value of the maximum and minimum distances r max and r min of the ellipse from a focus — that is. In astronomy these extreme points are called apsis, the semi-minor axis of an ellipse is the geometric mean of these distances, b = r max r min. The eccentricity of an ellipse is defined as e =1 − b 2 a 2 so r min = a, r max = a. Now consider the equation in polar coordinates, with one focus at the origin, the mean value of r = ℓ / and r = ℓ /, for θ = π and θ =0 is a = ℓ1 − e 2. In an ellipse, the axis is the geometric mean of the distance from the center to either focus. The semi-minor axis of an ellipse runs from the center of the ellipse to the edge of the ellipse, the semi-minor axis is half of the minor axis. The minor axis is the longest line segment perpendicular to the axis that connects two points on the ellipses edge. The semi-minor axis b is related to the axis a through the eccentricity e. A parabola can be obtained as the limit of a sequence of ellipses where one focus is fixed as the other is allowed to move arbitrarily far away in one direction
30.
Compact set
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In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed and bounded. Examples include a closed interval, a rectangle, or a set of points. This notion is defined for general topological spaces than Euclidean space in various ways. One such generalization is that a space is compact if any infinite sequence of points sampled from the space must frequently get arbitrarily close to some point of the space. An equivalent definition is that every sequence of points must have an infinite subsequence that converges to some point of the space, the Heine-Borel theorem states that a subset of Euclidean space is compact in this sequential sense if and only if it is closed and bounded. Thus, if one chooses a number of points in the closed unit interval some of those points must get arbitrarily close to some real number in that space. For instance, some of the numbers 1/2, 4/5, 1/3, 5/6, 1/4, 6/7, the same set of points would not accumulate to any point of the open unit interval, so the open unit interval is not compact. Euclidean space itself is not compact since it is not bounded, in particular, the sequence of points 0, 1, 2, 3, … has no subsequence that converges to any given real number. Apart from closed and bounded subsets of Euclidean space, typical examples of compact spaces include spaces consisting not of geometrical points, the term compact was introduced into mathematics by Maurice Fréchet in 1904 as a distillation of this concept. Various equivalent notions of compactness, including sequential compactness and limit point compactness, in general topological spaces, however, different notions of compactness are not necessarily equivalent. This more subtle notion, introduced by Pavel Alexandrov and Pavel Urysohn in 1929, the term compact set is sometimes a synonym for compact space, but usually refers to a compact subspace of a topological space. In the 19th century, several disparate mathematical properties were understood that would later be seen as consequences of compactness. On the one hand, Bernard Bolzano had been aware that any bounded sequence of points has a subsequence that must eventually get close to some other point. The process could then be repeated by dividing the resulting smaller interval into smaller and smaller parts until it closes down on the limit point. The full significance of Bolzanos theorem, and its method of proof, in the 1880s, it became clear that results similar to the Bolzano–Weierstrass theorem could be formulated for spaces of functions rather than just numbers or geometrical points. The idea of regarding functions as points of a generalized space dates back to the investigations of Giulio Ascoli. The uniform limit of this sequence then played precisely the same role as Bolzanos limit point and this ultimately led to the notion of a compact operator as an offshoot of the general notion of a compact space. It was Maurice Fréchet who, in 1906, had distilled the essence of the Bolzano–Weierstrass property, in 1870, Eduard Heine showed that a continuous function defined on a closed and bounded interval was in fact uniformly continuous
31.
Storage Aspect Ratio
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The aspect ratio of an image describes the proportional relationship between its width and its height. It is commonly expressed as two separated by a colon, as in 16,9.5 yards high. The most common ratios used today in the presentation of films in cinemas are 1.85,1 and 2.39,1. Two common videographic aspect ratios are 4,3, the video format of the 20th century. Other cinema and video aspect ratios exist, but are used infrequently, in still camera photography, the most common aspect ratios are 4,3,3,2, and more recently being found in consumer cameras 16,9. Other aspect ratios, such as 5,3,5,4, in motion picture formats, the physical size of the film area between the sprocket perforations determines the images size. The universal standard is a frame that is four perforations high, the film itself is 35 mm wide, but the area between the perforations is 24.89 mm×18.67 mm, leaving the de facto ratio of 4,3, or 1.33,1. A4,3 ratio mimics human eyesight visual angle of 155°h x 120°v, the motion picture industry convention assigns a value of 1.0 to the image’s height, an anamorphic frame is often incorrectly described as 2.40,1 or 2.40. After 1952, a number of aspect ratios were experimented with for anamorphic productions, a SMPTE specification for anamorphic projection from 1957 finally standardized the aperture to 2.35,1. An update in 1970 changed the ratio to 2.39,1 in order to make splices less noticeable. This aspect ratio of 2.39,1 was confirmed by the most recent revision from August 1993, in American cinemas, the common projection ratios are 1.85,1 and 2.39,1. Some European countries have 1.66,1 as the wide screen standard, the Academy ratio of 1.375,1 was used for all cinema films in the sound era until 1953. During that time, television, which had a aspect ratio of 1.33,1. Hollywood responded by creating a number of wide-screen formats, CinemaScope, Todd-AO. The flat 1.85,1 aspect ratio was introduced in May 1953, and became one of the most common cinema projection standards in the U. S. and elsewhere. The goal of these lenses and aspect ratios was to capture as much of the frame as possible, onto as large an area of the film as possible. In either case the image was squeezed horizontally to fit the frame size. Development of various film camera systems must ultimately cater to the placement of the frame in relation to the constraints of the perforations
32.
Equidimensional
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Equidimensional is an adjective applied to objects that have nearly the same size or spread in multiple directions. As a mathematical concept, it may be applied to objects that extend across any number of dimensions, more specifically, its also used to characterize the shape of three-dimensional solids. The word equidimensional is sometimes used by geologists to describe the shape of three-dimensional objects, in that case it is a synonym for equant. Deviations from equidimensional are used to classify the shape of objects like rocks or particles. Perhaps this is an intuitively reasonable setting in general for the point at which somethings dimensions become significantly unequal. The relationship between the four categories is illustrated in the figure at right, which one to plot long. Perfectly equidimensional spheres plot in the right corner. Objects with equal short and intermediate axes lie on the bound, while objects with equal long. The dotted gray and black lines correspond to integer values ranging from 2 up to 10. The point of intersection for all four classes on this plot occurs when the axes a, b, c have ratios of R2, R,1. Make axis b any shorter and the object becomes prolate, make axis b any longer and it becomes oblate. Bring a and c closer to b and the object becomes equidimensional, separate a and c further from b and it becomes bladed. For example, the envelope for some humans might plot near the black dot in the upper left of the figure. Aspect ratio between long and short Equant as a used in astronomy Oblate spheroid Prolate spheroid shape analysis Theodor Zingg PhD thesis, http
33.
Squeeze mapping
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In linear algebra, a squeeze mapping is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is not a rotation or shear mapping. For a fixed real number a, the mapping ↦ is the squeeze mapping with parameter a. Since is a hyperbola, if u = ax and v = y/a, then uv = xy, for this reason it is natural to think of the squeeze mapping as a hyperbolic rotation, as did Émile Borel in 1914, by analogy with circular rotations which preserve circles. The squeeze mapping sets the stage for development of the concept of logarithms, the problem of finding the area bounded by a hyperbola is one of quadrature. The solution, found by Grégoire de Saint-Vincent and Alphonse Antonio de Sarasa in 1647, required the natural logarithm function, some insight into logarithms comes through hyperbolic sectors that are permuted by squeeze mappings while preserving their area. The area of a sector is taken as a measure of a hyperbolic angle associated with the sector. Both circular and hyperbolic angle generate invariant measures but with respect to different transformation groups, the hyperbolic functions, which take hyperbolic angle as argument, perform the role that circular functions play with the circular angle argument. If r and s are real numbers, the composition of their squeeze mappings is the squeeze mapping of their product. Therefore, the collection of squeeze mappings forms a group isomorphic to the multiplicative group of positive real numbers. An additive view of this arises from consideration of hyperbolic sectors. This is equivalent to preserving the form xy via the change of basis x = u + v, y = u − v, and corresponds geometrically to preserving hyperbolae. The perspective of the group of squeeze mappings as hyperbolic rotation is analogous to interpreting the group SO preserving quadratic form x2 + y2) as being circular rotations, in the language of Möbius transforms, the squeeze transformations are the hyperbolic elements in the classification of elements. These applications are somewhat bland compared to two physical and a philosophical application, in fluid dynamics one of the fundamental motions of an incompressible flow involves bifurcation of a flow running up against an immovable wall. The same model gives fluid convergence when time is run backward, indeed, the area of any hyperbolic sector is invariant under squeezing. For another approach to a flow with hyperbolic streamlines, see the potential flow. In 1989 Ottino described the linear isochoric two-dimensional flow as v 1 = G x 2 v 2 = K G x 1 where K lies in the interval and we consider a region of flow forming an angle of π/2 and delimited on the left and bottom by symmetry planes. Stocker and Hosoi then recall Moffatts consideration of flow in a corner between rigid boundaries, induced by a disturbance at a large distance. According to Stocker and Hosoi, For a free fluid in a square corner and that hyperbolic coordinates are indeed the natural choice to describe these flows
34.
International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker
35.
Fraction (mathematics)
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A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A common, vulgar, or simple fraction consists of an integer numerator displayed above a line, numerators and denominators are also used in fractions that are not common, including compound fractions, complex fractions, and mixed numerals. The numerator represents a number of parts, and the denominator. For example, in the fraction 3/4, the numerator,3, tells us that the fraction represents 3 equal parts, the picture to the right illustrates 34 or ¾ of a cake. Fractional numbers can also be written without using explicit numerators or denominators, by using decimals, percent signs, an integer such as the number 7 can be thought of as having an implicit denominator of one,7 equals 7/1. Other uses for fractions are to represent ratios and to represent division, thus the fraction ¾ is also used to represent the ratio 3,4 and the division 3 ÷4. The test for a number being a number is that it can be written in that form. In a fraction, the number of parts being described is the numerator. Informally, they may be distinguished by placement alone but in formal contexts they are separated by a fraction bar. The fraction bar may be horizontal, oblique, or diagonal and these marks are respectively known as the horizontal bar, the slash or stroke, the division slash, and the fraction slash. In typography, horizontal fractions are known as en or nut fractions and diagonal fractions as em fractions. The denominators of English fractions are expressed as ordinal numbers. When the denominator is 1, it may be expressed in terms of wholes but is commonly ignored. When the numerator is one, it may be omitted, a fraction may be expressed as a single composition, in which case it is hyphenated, or as a number of fractions with a numerator of one, in which case they are not. Fractions should always be hyphenated when used as adjectives, alternatively, a fraction may be described by reading it out as the numerator over the denominator, with the denominator expressed as a cardinal number. The term over is used even in the case of solidus fractions, Fractions with large denominators that are not powers of ten are often rendered in this fashion while those with denominators divisible by ten are typically read in the normal ordinal fashion. A simple fraction is a number written as a/b or a b
36.
Division (mathematics)
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Division is one of the four basic operations of arithmetic, the others being addition, subtraction, and multiplication. The division of two numbers is the process of calculating the number of times one number is contained within one another. For example, in the picture on the right, the 20 apples are divided into groups of five apples, Division can also be thought of as the process of evaluating a fraction, and fractional notation is commonly used to represent division. Division is the inverse of multiplication, if a × b = c, then a = c ÷ b, as long as b is not zero. Division by zero is undefined for the numbers and most other contexts, because if b =0, then a cannot be deduced from b and c. In some contexts, division by zero can be defined although to a limited extent, in division, the dividend is divided by the divisor to get a quotient. In the above example,20 is the dividend, five is the divisor, in some cases, the divisor may not be contained fully by the dividend, for example,10 ÷3 leaves a remainder of one, as 10 is not a multiple of three. Sometimes this remainder is added to the quotient as a fractional part, but in the context of integer division, where numbers have no fractional part, the remainder is kept separately or discarded. Besides dividing apples, division can be applied to other physical, Division has been defined in several contexts, such as for the real and complex numbers and for more abstract contexts such as for vector spaces and fields. Division is the most mentally difficult of the four operations of arithmetic. Teaching the objective concept of dividing integers introduces students to the arithmetic of fractions, unlike addition, subtraction, and multiplication, the set of all integers is not closed under division. Dividing two integers may result in a remainder, to complete the division of the remainder, the number system is extended to include fractions or rational numbers as they are more generally called. When students advance to algebra, the theory of division intuited from arithmetic naturally extends to algebraic division of variables, polynomials. Division is often shown in algebra and science by placing the dividend over the divisor with a line, also called a fraction bar. For example, a divided by b is written a b This can be read out loud as a divided by b, a fraction is a division expression where both dividend and divisor are integers, and there is no implication that the division must be evaluated further. A second way to show division is to use the obelus, common in arithmetic, in this manner, ISO 80000-2-9.6 states it should not be used. The obelus is also used alone to represent the operation itself. In some non-English-speaking cultures, a divided by b is written a, b and this notation was introduced in 1631 by William Oughtred in his Clavis Mathematicae and later popularized by Gottfried Wilhelm Leibniz
37.
Divisor
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In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some other integer to produce n. In this case one says also that n is a multiple of m, an integer n is divisible by another integer m if m is a divisor of n, this implies dividing n by m leaves no remainder. Under this definition, the statement m ∣0 holds for every m, as before, but with the additional constraint k ≠0. Under this definition, the statement m ∣0 does not hold for m ≠0, in the remainder of this article, which definition is applied is indicated where this is significant. Divisors can be negative as well as positive, although sometimes the term is restricted to positive divisors. For example, there are six divisors of 4, they are 1,2,4, −1, −2, and −4,1 and −1 divide every integer. Every integer is a divisor of itself, every integer is a divisor of 0. Integers divisible by 2 are called even, and numbers not divisible by 2 are called odd,1, −1, n and −n are known as the trivial divisors of n. A divisor of n that is not a divisor is known as a non-trivial divisor. A non-zero integer with at least one divisor is known as a composite number, while the units −1 and 1. There are divisibility rules which allow one to recognize certain divisors of a number from the numbers digits, the generalization can be said to be the concept of divisibility in any integral domain. 7 is a divisor of 42 because 7 ×6 =42 and it can also be said that 42 is divisible by 7,42 is a multiple of 7,7 divides 42, or 7 is a factor of 42. The non-trivial divisors of 6 are 2, −2,3, the positive divisors of 42 are 1,2,3,6,7,14,21,42. 5 ∣0, because 5 ×0 =0, if a ∣ b and b ∣ a, then a = b or a = − b. If a ∣ b and a ∣ c, then a ∣ holds, however, if a ∣ b and c ∣ b, then ∣ b does not always hold. If a ∣ b c, and gcd =1, then a ∣ c, if p is a prime number and p ∣ a b then p ∣ a or p ∣ b. A positive divisor of n which is different from n is called a proper divisor or a part of n. A number that does not evenly divide n but leaves a remainder is called an aliquant part of n, an integer n >1 whose only proper divisor is 1 is called a prime number
