1.
Number theory
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Number theory or, in older usage, arithmetic is a branch of pure mathematics devoted primarily to the study of the integers. It is sometimes called The Queen of Mathematics because of its place in the discipline. Number theorists study prime numbers as well as the properties of objects out of integers or defined as generalizations of the integers. Integers can be considered either in themselves or as solutions to equations, questions in number theory are often best understood through the study of analytical objects that encode properties of the integers, primes or other number-theoretic objects in some fashion. One may also study real numbers in relation to rational numbers, the older term for number theory is arithmetic. By the early century, it had been superseded by number theory. The use of the arithmetic for number theory regained some ground in the second half of the 20th century. In particular, arithmetical is preferred as an adjective to number-theoretic. The first historical find of a nature is a fragment of a table. The triples are too many and too large to have been obtained by brute force, the heading over the first column reads, The takiltum of the diagonal which has been subtracted such that the width. The tables layout suggests that it was constructed by means of what amounts, in language, to the identity 2 +1 =2. If some other method was used, the triples were first constructed and then reordered by c / a, presumably for use as a table. It is not known what these applications may have been, or whether there could have any, Babylonian astronomy, for example. It has been suggested instead that the table was a source of examples for school problems. While Babylonian number theory—or what survives of Babylonian mathematics that can be called thus—consists of this single, striking fragment, late Neoplatonic sources state that Pythagoras learned mathematics from the Babylonians. Much earlier sources state that Thales and Pythagoras traveled and studied in Egypt, Euclid IX 21—34 is very probably Pythagorean, it is very simple material, but it is all that is needed to prove that 2 is irrational. Pythagorean mystics gave great importance to the odd and the even, the discovery that 2 is irrational is credited to the early Pythagoreans. This forced a distinction between numbers, on the one hand, and lengths and proportions, on the other hand, the Pythagorean tradition spoke also of so-called polygonal or figurate numbers
2.
BASIC
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BASIC is a family of general-purpose, high-level programming languages whose design philosophy emphasizes ease of use. In 1964, John G. Kemeny and Thomas E. Kurtz designed the original BASIC language at Dartmouth College in the U. S. state of New Hampshire and they wanted to enable students in fields other than science and mathematics to use computers. At the time, nearly all use of computers required writing custom software, versions of BASIC became widespread on microcomputers in the mid-1970s and 1980s. Microcomputers usually shipped with BASIC, often in the machines firmware, having an easy-to-learn language on these early personal computers allowed small business owners, professionals, hobbyists, and consultants to develop custom software on computers they could afford. In the 2010s, BASIC remains popular in many computing dialects and in new languages influenced by BASIC, before the mid-1960s, the only computers were huge mainframe computers. Users submitted jobs on punched cards or similar media to specialist computer operators, the computer stored these, then used a batch processing system to run this queue of jobs one after another, allowing very high levels of utilization of these expensive machines. As the performance of computing hardware rose through the 1960s, multi-processing was developed and this allowed a mix of batch jobs to be run together, but the real revolution was the development of time-sharing. The original BASIC language was released on May 1,1964 by John G. Kemeny and Thomas E. Kurtz, the acronym BASIC comes from the name of an unpublished paper by Thomas Kurtz. BASIC was designed to allow students to write computer programs for the Dartmouth Time-Sharing System. It was intended specifically for technical users who did not have or want the mathematical background previously expected. Being able to use a computer to support teaching and research was quite novel at the time, the language was based on FORTRAN II, with some influences from ALGOL60 and with additions to make it suitable for timesharing. Wanting use of the language to become widespread, its designers made the available free of charge. They also made it available to schools in the Hanover area. In the following years, as dialects of BASIC appeared, Kemeny. A version was a part of the Pick operating system from 1973 onward. During this period a number of computer games were written in BASIC. A number of these were collected by DEC employee David H. Ahl and he later collected a number of these into book form,101 BASIC Computer Games, published in 1973. During the same period, Ahl was involved in the creation of a computer for education use
3.
Fortran
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Fortran is a general-purpose, imperative programming language that is especially suited to numeric computation and scientific computing. It is a language for high-performance computing and is used for programs that benchmark. Fortran encompasses a lineage of versions, each of which evolved to add extensions to the language while usually retaining compatibility with prior versions, the names of earlier versions of the language through FORTRAN77 were conventionally spelled in all-capitals. The capitalization has been dropped in referring to newer versions beginning with Fortran 90, the official language standards now refer to the language as Fortran rather than all-caps FORTRAN. In late 1953, John W. Backus submitted a proposal to his superiors at IBM to develop a practical alternative to assembly language for programming their IBM704 mainframe computer. Backus historic FORTRAN team consisted of programmers Richard Goldberg, Sheldon F. Best, Harlan Herrick, Peter Sheridan, Roy Nutt, Robert Nelson, Irving Ziller, Lois Haibt, and David Sayre. Its concepts included easier entry of equations into a computer, a developed by J. Halcombe Laning and demonstrated in the Laning. A draft specification for The IBM Mathematical Formula Translating System was completed by mid-1954, the first manual for FORTRAN appeared in October 1956, with the first FORTRAN compiler delivered in April 1957. John Backus said during a 1979 interview with Think, the IBM employee magazine, the language was widely adopted by scientists for writing numerically intensive programs, which encouraged compiler writers to produce compilers that could generate faster and more efficient code. The inclusion of a complex data type in the language made Fortran especially suited to technical applications such as electrical engineering. By 1960, versions of FORTRAN were available for the IBM709,650,1620, significantly, the increasing popularity of FORTRAN spurred competing computer manufacturers to provide FORTRAN compilers for their machines, so that by 1963 over 40 FORTRAN compilers existed. For these reasons, FORTRAN is considered to be the first widely used programming language supported across a variety of computer architectures, the arithmetic IF statement was similar to a three-way branch instruction on the IBM704. However, the 704 branch instructions all contained only one destination address, an optimizing compiler like FORTRAN would most likely select the more compact and usually faster Transfers instead of the Compare. Also the Compare considered −0 and +0 to be different values while the Transfer Zero, the FREQUENCY statement in FORTRAN was used originally to give branch probabilities for the three branch cases of the arithmetic IF statement. The Monte Carlo technique is documented in Backus et al, many years later, the FREQUENCY statement had no effect on the code, and was treated as a comment statement, since the compilers no longer did this kind of compile-time simulation. A similar fate has befallen compiler hints in other programming languages. The first FORTRAN compiler reported diagnostic information by halting the program when an error was found and that code could be looked up by the programmer in a error messages table in the operators manual, providing them with a brief description of the problem. Before the development of disk files, text editors and terminals, programs were most often entered on a keyboard onto 80-column punched cards
4.
Computing
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Computing is any goal-oriented activity requiring, benefiting from, or creating a mathematical sequence of steps known as an algorithm — e. g. through computers. The field of computing includes computer engineering, software engineering, computer science, information systems, the ACM Computing Curricula 2005 defined computing as follows, In a general way, we can define computing to mean any goal-oriented activity requiring, benefiting from, or creating computers. For example, an information systems specialist will view computing somewhat differently from a software engineer, regardless of the context, doing computing well can be complicated and difficult. Because society needs people to do computing well, we must think of computing not only as a profession, the fundamental question underlying all computing is What can be automated. The term computing is also synonymous with counting and calculating, in earlier times, it was used in reference to the action performed by mechanical computing machines, and before that, to human computers. Computing is intimately tied to the representation of numbers, but long before abstractions like the number arose, there were mathematical concepts to serve the purposes of civilization. These concepts include one-to-one correspondence, comparison to a standard, the earliest known tool for use in computation was the abacus, and it was thought to have been invented in Babylon circa 2400 BC. Its original style of usage was by lines drawn in sand with pebbles, abaci, of a more modern design, are still used as calculation tools today. This was the first known computer and most advanced system of calculation known to date - preceding Greek methods by 2,000 years. The first recorded idea of using electronics for computing was the 1931 paper The Use of Thyratrons for High Speed Automatic Counting of Physical Phenomena by C. E. Wynn-Williams. Claude Shannons 1938 paper A Symbolic Analysis of Relay and Switching Circuits then introduced the idea of using electronics for Boolean algebraic operations, a computer is a machine that manipulates data according to a set of instructions called a computer program. The program has a form that the computer can use directly to execute the instructions. The same program in its source code form, enables a programmer to study. Because the instructions can be carried out in different types of computers, the execution process carries out the instructions in a computer program. Instructions express the computations performed by the computer and they trigger sequences of simple actions on the executing machine. Those actions produce effects according to the semantics of the instructions, computer software or just software, is a collection of computer programs and related data that provides the instructions for telling a computer what to do and how to do it. Software refers to one or more programs and data held in the storage of the computer for some purposes. In other words, software is a set of programs, procedures, algorithms, program software performs the function of the program it implements, either by directly providing instructions to the computer hardware or by serving as input to another piece of software
5.
Maple (software)
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Maple is a symbolic and numeric computing environment, and is also a multi-paradigm programming language. Developed by Maplesoft, Maple also covers aspects of technical computing, including visualization, data analysis, matrix computation. A toolbox, MapleSim, adds functionality for physical modeling. Users can enter mathematics in traditional mathematical notation, custom user interfaces can also be created. There is support for numeric computations, to arbitrary precision, as well as symbolic computation and visualization, examples of symbolic computations are given below. Maple incorporates a dynamically typed programming language which resembles Pascal. The language permits variables of lexical scope, there are also interfaces to other languages. There is also an interface to Excel, Maple supports MathML2.0, a W3C format for representing and interpreting mathematical expressions, including their display in Web pages. Maple is based on a kernel, written in C. Most functionality is provided by libraries, which come from a variety of sources, most of the libraries are written in the Maple language, these have viewable source code. Many numerical computations are performed by the NAG Numerical Libraries, ATLAS libraries, different functionality in Maple requires numerical data in different formats. Symbolic expressions are stored in memory as directed acyclic graphs, the standard interface and calculator interface are written in Java. The first concept of Maple arose from a meeting in November 1980 at the University of Waterloo, researchers at the university wished to purchase a computer powerful enough to run Macsyma. Instead, it was decided that they would develop their own computer system that would be able to run on lower cost computers. The first limited version appearing in December 1980 with Maple demonstrated first at conferences beginning in 1982, the name is a reference to Maples Canadian heritage. By the end of 1983, over 50 universities had copies of Maple installed on their machines, in 1984, the research group arranged with Watcom Products Inc to license and distribute the first commercially available version, Maple 3.3. In 1988 Waterloo Maple Inc. was founded, the company’s original goal was to manage the distribution of the software. In 1989, the first graphical user interface for Maple was developed and included with version 4.3 for the Macintosh, x11 and Windows versions of the new interface followed in 1990 with Maple V
6.
Mathcad
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Mathcad is computer software primarily intended for the verification, validation, documentation and re-use of engineering calculations. First introduced in 1986 on DOS, it was the first to introduce live editing of typeset mathematical notation, Mathcad today includes some of the capabilities of a computer algebra system, but remains oriented towards ease of use and simultaneous documentation of numerical engineering applications. Mathcad is part of a product development system developed by PTC. It integrates with PTC’s other solutions that aid product development, including Creo Elements/Pro, Windchill, the Mathcad interface allows users to combine a variety of different elements into the form of a worksheet, which is naturally readable. Because the mathematics are core to the program, the math is inherently live and this allows for simple manipulation of input variables, assumptions, and expressions, which in turn update in real-time. The examples below serve to outline the scope of Mathcad’s capabilities, as of March 2017, there are two actively maintained Mathcad releases and a free express version available to consumers, Mathcad 15.0 was originally released in June,2010. Its first maintenance release was released in November,2010, Mathcad 15.0 is the next progressive release of the traditional product line, sharing the same worksheet file structure and extension as its predecessor, Mathcad 14.0. Mathcad Prime 4.0, PTC’s latest generation product, was introduced in March 2017 and this is PTCs latest release, after 2 years. This Freemium pilot marks a new marketing approach for PTC, review and markup of engineering notes can now be done directly by team members without them all requiring a full Mathcad Prime license. Mathcad is currently a Windows-only application, current releases of Mathcad 15.0 and Mathcad Prime 4.0 are supported on 32-bit and 64-bit versions of Windows XP, Windows Vista, and Windows 7. Currently releases are 32-bit applications only, however, while users do utilize emulation to establish other platform operability, Mathcad’s last officially supported, natively installed Mac OS release was on January 8,1998. This version can still be run on Macintosh computers that support the Mac OS X Classic Environment or SheepShaver, after PTC’s purchase of Mathcad in 2006, changes were made to the Mathcad support policy. That change specified non-maintenance bearing licenses were no longer able to receive updates, including bug fixes, though disagreeable to some long-time, pre-PTC-acquisition customers, this is PTC’s standard policy for all its other products. The Mathcad Business Unit within PTC recently updated their support policy, for Mathcad 15.0 and future versions of Mathcad, the first year of maintenance entitlements and support will be included in the purchase or upgrade price.1 and PTC Mathcad 15.0 M040
7.
Wolfram Mathematica
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Wolfram Mathematica is a mathematical symbolic computation program, sometimes termed a computer algebra system or program, used in many scientific, engineering, mathematical, and computing fields. It was conceived by Stephen Wolfram and is developed by Wolfram Research of Champaign, the Wolfram Language is the programming language used in Mathematica. The kernel interprets expressions and returns result expressions, all content and formatting can be generated algorithmically or edited interactively. Standard word processing capabilities are supported, including real-time multi-lingual spell-checking, documents can be structured using a hierarchy of cells, which allow for outlining and sectioning of a document and support automatic numbering index creation. Documents can be presented in an environment for presentations. Notebooks and their contents are represented as Mathematica expressions that can be created, modified or analyzed by Mathematica programs or converted to other formats, the front end includes development tools such as a debugger, input completion, and automatic syntax highlighting. Among the alternative front ends is the Wolfram Workbench, an Eclipse based integrated development environment and it provides project-based code development tools for Mathematica, including revision management, debugging, profiling, and testing. There is a plugin for IntelliJ IDEA based IDEs to work with Wolfram Language code which in addition to syntax highlighting can analyse and auto-complete local variables, the Mathematica Kernel also includes a command line front end. Other interfaces include JMath, based on GNU readline and MASH which runs self-contained Mathematica programs from the UNIX command line, version 5.2 added automatic multi-threading when computations are performed on multi-core computers. This release included CPU specific optimized libraries, in addition Mathematica is supported by third party specialist acceleration hardware such as ClearSpeed. Support for CUDA and OpenCL GPU hardware was added in 2010, also, since version 8 it can generate C code, which is automatically compiled by a system C compiler, such as GCC or Microsoft Visual Studio. A free-of-charge version, Wolfram CDF Player, is provided for running Mathematica programs that have saved in the Computable Document Format. It can also view standard Mathematica files, but not run them and it includes plugins for common web browsers on Windows and Macintosh. WebMathematica allows a web browser to act as a front end to a remote Mathematica server and it is designed to allow a user written application to be remotely accessed via a browser on any platform. It may not be used to full access to Mathematica. Due to bandwidth limitations interactive 3D graphics is not fully supported within a web browser, Wolfram Language code can be converted to C code or to an automatically generated DLL. Wolfram Language code can be run on a Wolfram cloud service as a web-app or as an API either on Wolfram-hosted servers or in an installation of the Wolfram Enterprise Private Cloud. Communication with other applications occurs through a protocol called Wolfram Symbolic Transfer Protocol and it allows communication between the Wolfram Mathematica kernel and front-end, and also provides a general interface between the kernel and other applications
8.
Division by zero
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In mathematics, division by zero is division where the divisor is zero. Such a division can be expressed as a/0 where a is the dividend. In ordinary arithmetic, the expression has no meaning, as there is no number which, multiplied by 0, gives a, and so division by zero is undefined. Since any number multiplied by zero is zero, the expression 0/0 also has no defined value, in computing, a program error may result from an attempt to divide by zero. When division is explained at the elementary level, it is often considered as splitting a set of objects into equal parts. As an example, consider having ten cookies, and these cookies are to be distributed equally to five people at a table, each person would receive 105 =2 cookies. Similarly, if there are ten cookies, and only one person at the table, so, for dividing by zero, what is the number of cookies that each person receives when 10 cookies are evenly distributed amongst 0 people at a table. Certain words can be pinpointed in the question to highlight the problem, the problem with this question is the when. There is no way to evenly distribute 10 cookies to nobody, in mathematical jargon, a set of 10 items cannot be partitioned into 0 subsets. So 100, at least in elementary arithmetic, is said to be either meaningless, similar problems occur if one has 0 cookies and 0 people, but this time the problem is in the phrase the number. A partition is possible, but since the partition has 0 parts, vacuously every set in our partition has a number of elements, be it 0,2,5. If there are, say,5 cookies and 2 people, in any integer partition of a 5-set into 2 parts, one of the parts of the partition will have more elements than the other. But the problem with 5 cookies and 2 people can be solved by cutting one cookie in half, the problem with 5 cookies and 0 people cannot be solved in any way that preserves the meaning of divides. Another way of looking at division by zero is that division can always be checked using multiplication. Considering the 10/0 example above, setting x = 10/0, if x equals ten divided by zero, then x times zero equals ten, but there is no x that, when multiplied by zero, gives ten. If instead of x=10/0 we have x=0/0, then every x satisfies the question what number x, multiplied by zero, the Brahmasphutasiddhanta of Brahmagupta is the earliest known text to treat zero as a number in its own right and to define operations involving zero. The author could not explain division by zero in his texts, according to Brahmagupta, A positive or negative number when divided by zero is a fraction with the zero as denominator. Zero divided by a negative or positive number is zero or is expressed as a fraction with zero as numerator
9.
AMPL
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A Mathematical Programming Language is an algebraic modeling language to describe and solve high-complexity problems for large-scale mathematical computing. It was developed by Robert Fourer, David Gay, and Brian Kernighan at Bell Laboratories, AMPL supports dozens of solvers, both open source and commercial software, including CBC, CPLEX, FortMP, Gurobi, MINOS, IPOPT, SNOPT, KNITRO, and LGO. Problems are passed to solvers as nl files, AMPL is used by more than 100 corporate clients, and by government agencies and academic institutions. One advantage of AMPL is the similarity of its syntax to the notation of optimization problems. This allows for a concise and readable definition of problems in the domain of optimization. Many modern solvers available on the NEOS Server accept AMPL input, according to the NEOS statistics AMPL is the most popular format for representing mathematical programming problems. AMPL features a mix of declarative and imperative programming styles, to support re-use and simplify construction of large-scale optimization problems, AMPL allows separation of model and data. AMPL is available for many popular 32- and 64-bit operating systems including Linux, Mac OS X, some Unix, the translator is proprietary software maintained by AMPL Optimization LLC. However, several services exist, providing free modeling and solving facilities using AMPL. A free student version with limited functionality and a free full-featured version for academic courses are also available, AMPL can be used from within Microsoft Excel via the SolverStudio Excel add-in. The AMPL Solver Library, which allows reading nl files and provides the differentiation, is open-source. It is used in many solvers to implement AMPL connection and this table present significant steps in AMPL history. A transportation problem from George Dantzig is used to provide a sample AMPL model and this problem finds the least cost shipping schedule that meets requirements at markets and supplies at factories. Here is a partial list of solvers supported by AMPL, sol Official website Prof. Fourers home page at Northwestern University
10.
GameMaker Studio
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GameMaker, Studio is a proprietary game creation system created by Mark Overmars in the Delphi programming language. GameMaker was designed to allow novice computer programmers to be able to make computer games without much programming knowledge by use of these actions, originally titled Animo, the program was first released in 1999, and began as a program for creating 2D animations. The name was changed to GameMaker, lacking a space to avoid intellectual property conflicts with the 1991 software Game-Maker. GameMaker primarily runs games that use 2D graphics, allowing the use of limited 3D graphics, GameMaker is designed to allow its users to easily develop video games without having to learn a complex programming language such as C++ or Java through its proprietary drag and drop system. These icons represent actions that would occur in a game, such as movement, basic drawing and it is also possible to create custom action libraries using the Library Maker. Game Maker Language is the primary interpreted scripting language used in GameMaker and it is used to further enhance and control the design of a game through more conventional programming, as opposed to the drag and drop system. GameMaker accommodates redistribution on multiple platforms, the program builds for these platforms, Windows, Windows 8, Mac OS X, Ubuntu, HTML5, Android, iOS, Windows Phone 8, Tizen, Xbox One, and Playstation. However, a Windows desktop computer with system requirements equal to that of the game produced is required in order to develop the games along with an internet connection. Several versions of the software made reverse engineering easy by packing resource data to the end of the executable with no encryption or internal obfuscation, a decompiler was released specifically for decompiling games distributed with the early iOS runner. Obfuscation programs were developed and released to deter hackers from extracting the game resources from executable files built with the program. YoYoGames later issued a cease and desist to the hackers warning against further infringement of their intellectual property posing as a financial threat to the company. The latest version of the software, GM, Studio, makes it harder to decompile games given its compiled nature and this was due to a fault in their digital rights management software implementation which they use as a method of combating infringing copies of the software. YoYoGames publicly stated they would remove the DRM at a point in time. List of game engines Game engine Unity Construct Undertale Official website
11.
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
12.
ALGOL 68
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ALGOL68 was designed by the IFIP Working Group 2.1. On December 20,1968, the language was adopted by Working Group 2.1. ALGOL68 was defined using a grammar formalism invented by Adriaan van Wijngaarden. ALGOL68 has been criticized, most prominently by some members of its design such as C. A. R. In 1970, ALGOL 68-R became the first working compiler for ALGOL68, in the 1973 revision, certain features – such as proceduring, gommas and formal bounds – were omitted. The language of the unrevised report, steve Bourne, who was on the Algol 68 revision committee, took some of its ideas to his Bourne shell and to C. The complete history of the project can be found in C. H, lindseys A History of ALGOL68. For a full-length treatment of the language, see Programming Algol 68 Made Easy by Dr. Sian Mountbatten, or Learning Algol 68 Genie by Dr. Marcel van der Veer which includes the Revised Report. A Shorter History of Algol 68 ALGOL68 - 3rd generation ALGOL Mar.1968, Draft Report on the Algorithmic Language ALGOL68 - Edited by, A. van Wijngaarden, B. J. Mailloux, J. E. L. Oct.1968, Penultimate Draft Report on the Algorithmic Language ALGOL68 – Chapters 1-9 Chapters 10-12 – Edited by, A. van Wijngaarden, B. J. Mailloux, J. E. L. Dec.1968, Report on the Algorithmic Language ALGOL68 – Offprint from Numerische Mathematik,14, 79-218, – Edited by, A. van Wijngaarden, B. J. Mailloux, J. E. L. WG2.1 members active in the design of ALGOL68. 1973, Revised Report on the Algorithmic Language Algol 68 - Springer-Verlag 1976 - Edited by, A. van Wijngaarden, B. J. Mailloux, J. E. L. 1968, On December 20,1968, the Final Report was adopted by the Working Group, translations of the standard were made for Russian, German, French and Bulgarian, and then later Japanese and Chinese. The standard was made available in Braille. 1984, TC97 considered Algol 68 for standardisation as New Work Item TC97/N1642,1988, Subsequently ALGOL68 became one of the GOST standards in Russia. r0. The basic language construct is the unit, a unit may be a formula, an enclosed clause, a routine text or one of several technically needed constructs. The technical term enclosed clause unifies some of the inherently bracketing constructs known as block, do statement, when keywords are used, generally the reversed character sequence of the introducing keyword is used for terminating the enclosure, e. g