83-526: Small Machine Algol Like Language ( SMALL ), is a computer programming language developed by Nevil Brownlee of the University of Auckland . The aim of the language was to enable writing ALGOL -like code that ran on a small machine. It also included the string data type for easier text manipulation. SMALL was used extensively from about 1980 to 1985 at Auckland University as a programming teaching aid, and for some internal projects. Originally, it
166-402: A compiler can make it crash when parsing some large source file, a simplification of the test case that results in only few lines from the original source file can be sufficient to reproduce the same crash. Trial-and-error/divide-and-conquer is needed: the programmer will try to remove some parts of the original test case and check if the problem still exists. When debugging the problem in a GUI,
249-498: A contributor. Ibn Muʿādh al-Jayyānī 's The book of unknown arcs of a sphere in the 11th century introduced the general law of sines. The plane law of sines was described in the 13th century by Nasīr al-Dīn al-Tūsī . In his On the Sector Figure , he stated the law of sines for plane and spherical triangles and provided proofs for this law. In the 9th century, Islamic mathematicians were familiar with negative numbers from
332-679: A crucial role in introducing Arabic mathematical ideas to the West. The translation of Arabic mathematical texts, along with Greek and Roman works, during the 14th to 17th century, played a pivotal role in shaping the intellectual landscape of the Renaissance . Arabic mathematics, particularly algebra, developed significantly during the medieval period . Muhammad ibn Musa al-Khwārizmī 's ( Arabic : محمد بن موسى الخوارزمي ; c. 780 – c. 850 ) work between AD 813 and 833 in Baghdad
415-574: A crucial role in shaping the trajectory of mathematics, with al-Khwārizmī 's algebraic innovations serving as a cornerstone. The dissemination of Arabic mathematics to the West during the Islamic Golden Age , facilitated by cultural exchanges and translations, left a lasting impact on Western mathematical thought. Mathematicians like Al-Battānī , Al-Khayyām , and Abū Kāmil , with their contributions to trigonometry , algebra , and geometry , extended their influence beyond their time. Despite
498-724: A distinction between magnitude and number . In the Greek view, magnitudes varied continuously and could be used for entities such as line segments, whereas numbers were discrete. Hence, irrationals could only be handled geometrically; and indeed Greek mathematics was mainly geometrical. Islamic mathematicians including Abū Kāmil Shujāʿ ibn Aslam and Ibn Tahir al-Baghdadi slowly removed the distinction between magnitude and number, allowing irrational quantities to appear as coefficients in equations and to be solutions of algebraic equations. They worked freely with irrationals as mathematical objects, but they did not examine closely their nature. In
581-540: A few simple readability transformations made code shorter and drastically reduced the time to understand it. Following a consistent programming style often helps readability. However, readability is more than just programming style. Many factors, having little or nothing to do with the ability of the computer to efficiently compile and execute the code, contribute to readability. Some of these factors include: The presentation aspects of this (such as indents, line breaks, color highlighting, and so on) are often handled by
664-406: A key role in translating and disseminating these works, thus making them accessible to a wider audience. Cremona is said to have translated into Latin "no fewer than 90 complete Arabic texts." European mathematicians, building on the foundations laid by Islamic scholars, further developed practical trigonometry for applications in navigation, cartography, and celestial navigation, thus pushing forward
747-402: A negative number is positive. If we subtract a negative number from a higher negative number, the remainder is their negative difference. The difference remains positive if we subtract a negative number from a lower negative number. If we subtract a negative number from a positive number, the remainder is their positive sum. If we subtract a positive number from an empty power ( martaba khāliyya ),
830-470: A result, loses efficiency and the ability for low-level manipulation). Debugging is a very important task in the software development process since having defects in a program can have significant consequences for its users. Some languages are more prone to some kinds of faults because their specification does not require compilers to perform as much checking as other languages. Use of a static code analysis tool can help detect some possible problems. Normally
913-405: A return to a more "spiritual and harmonious" lifestyle. Thus, the prevailing Orientalism in that period was one of the main reasons why Arabic mathematicians were often ignored for their contributions, as people outside the West were considered to be lacking the necessary rationality and scientific spirit to made significant contributions to math and science. The medieval Arab-Islamic world played
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#1732791772356996-430: A vehicle for the future development of the subject. Another important aspect of the introduction of algebraic ideas was that it allowed mathematics to be applied to itself in a way which had not happened before." Several other mathematicians during this time period expanded on the algebra of Al-Khwarizmi. Abu Kamil Shuja' wrote a book of algebra accompanied with geometrical illustrations and proofs. He also enumerated all
1079-469: A very different way both in its method employed and ultimate purpose, "the hallmark of Western science in its Greek origins as well as in its modern renaissance, is its conformity to rigorous standards". Thus, the perceived non-rigorous proof in Arabic mathematicians' book authorizes Bourbaki to exclude the Arabic period when he retraced the evolution of algebra. And instead, the history of classical algebra
1162-459: A visual environment. Different programming languages support different styles of programming (called programming paradigms ). The choice of language used is subject to many considerations, such as company policy, suitability to task, availability of third-party packages, or individual preference. Ideally, the programming language best suited for the task at hand will be selected. Trade-offs from this ideal involve finding enough programmers who know
1245-410: Is Entity-Relationship Modeling ( ER Modeling ). Implementation techniques include imperative languages ( object-oriented or procedural ), functional languages , and logic programming languages. It is very difficult to determine what are the most popular modern programming languages. Methods of measuring programming language popularity include: counting the number of job advertisements that mention
1328-607: Is directly executed by the central processing unit . Proficient programming usually requires expertise in several different subjects, including knowledge of the application domain , details of programming languages and generic code libraries , specialized algorithms, and formal logic . Auxiliary tasks accompanying and related to programming include analyzing requirements , testing , debugging (investigating and fixing problems), implementation of build systems , and management of derived artifacts , such as programs' machine code . While these are sometimes considered programming, often
1411-580: Is essentially European", and just some technical innovations to the Greek heritage rather than open up a completely new branch of mathematics. In the French philosopher Ernest Renan 's work, Arabic math is merely "a reflection of Greece , combined with Persian and Indian influences". And according to Duhem , "Arabic science only reproduced the teachings received from Greek science". Besides being considered as merely some insignificant additions or reflections to
1494-450: Is important to understand just how significant this new idea was. It was a revolutionary move away from the Greek concept of mathematics which was essentially geometry. Algebra was a unifying theory which allowed rational numbers , irrational numbers , geometrical magnitudes, etc., to all be treated as "algebraic objects". It gave mathematics a whole new development path so much broader in concept to that which had existed before, and provided
1577-464: Is little more than a different notation for a machine language, two machines with different instruction sets also have different assembly languages. High-level languages made the process of developing a program simpler and more understandable, and less bound to the underlying hardware . The first compiler related tool, the A-0 System , was developed in 1952 by Grace Hopper , who also coined
1660-561: Is now known that his work is based on older Indian or Greek sources. He revised Ptolemy 's Geography and wrote on astronomy and astrology. However, C.A. Nallino suggests that al-Khwarizmi's original work was not based on Ptolemy but on a derivative world map, presumably in Syriac or Arabic . The spherical law of sines was discovered in the 10th century: it has been attributed variously to Abu-Mahmud Khojandi , Nasir al-Din al-Tusi and Abu Nasr Mansur , with Abu al-Wafa' Buzjani as
1743-496: Is similar to learning a foreign language . Mathematics in medieval Islam Mathematics during the Golden Age of Islam , especially during the 9th and 10th centuries, was built upon syntheses of Greek mathematics ( Euclid , Archimedes , Apollonius ) and Indian mathematics ( Aryabhata , Brahmagupta ). Important developments of the period include extension of the place-value system to include decimal fractions ,
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#17327917723561826-542: Is still strong in corporate data centers often on large mainframe computers , Fortran in engineering applications, scripting languages in Web development, and C in embedded software . Many applications use a mix of several languages in their construction and use. New languages are generally designed around the syntax of a prior language with new functionality added, (for example C++ adds object-orientation to C, and Java adds memory management and bytecode to C++, but as
1909-476: Is to look upstream towards al-Khayyām and al-Ṭūsī; and downstream towards Newton, Leibniz, Cramer, Bézout and the Bernoulli brothers". Numerous problems that appear in "La Géométrie" (Geometry) have foundations that date back to al-Khayyām. Abū Kāmil (Arabic: أبو كامل شجاع بن أسلم بن محمد بن شجاع , also known as Al-ḥāsib al-miṣrī—lit. "The Egyptian Calculator") (c. 850 – c. 930), was studied algebra following
1992-419: Is used. The transition to symbolic algebra, where only symbols are used, can be seen in the work of Ibn al-Banna' al-Marrakushi and Abū al-Ḥasan ibn ʿAlī al-Qalaṣādī . On the work done by Al-Khwarizmi, J. J. O'Connor and Edmund F. Robertson said: "Perhaps one of the most significant advances made by Arabic mathematics began at this time with the work of al-Khwarizmi, namely the beginnings of algebra. It
2075-434: Is usually easier to code in "high-level" languages than in "low-level" ones. Programming languages are essential for software development. They are the building blocks for all software, from the simplest applications to the most sophisticated ones. Allen Downey , in his book How To Think Like A Computer Scientist , writes: Many computer languages provide a mechanism to call functions provided by shared libraries . Provided
2158-515: Is written as the work of the Renaissance and the origin of algebraic geometry is traced back to Descartes, while Arabic mathematicians' contributions are ignored deliberately. In Rashed's words: "To justify the exclusion of science written in Arabic from the history of science, one invokes its absence of rigor, its calculatory appearance and its practical aims. Furthermore, strictly dependent on Greek science and, lastly, incapable of introducing experimental norms, scientists of that time were relegated to
2241-580: The Algebra of al-Khwārizmī. Khayyám obtained the solutions of these equations by finding the intersection points of two conic sections . This method had been used by the Greeks, but they did not generalize the method to cover all equations with positive roots . Sharaf al-Dīn al-Ṭūsī (? in Tus, Iran – 1213/4) developed a novel approach to the investigation of cubic equations—an approach which entailed finding
2324-649: The Arabic word meaning completion or "reunion of broken parts", flourished during the Islamic golden age . Muhammad ibn Musa al-Khwarizmi , a Persian scholar in the House of Wisdom in Baghdad was the founder of algebra, is along with the Greek mathematician Diophantus , known as the father of algebra. In his book The Compendious Book on Calculation by Completion and Balancing , Al-Khwarizmi deals with ways to solve for
2407-504: The Jacquard loom could produce entirely different weaves by changing the "program" – a series of pasteboard cards with holes punched in them. Code-breaking algorithms have also existed for centuries. In the 9th century, the Arab mathematician Al-Kindi described a cryptographic algorithm for deciphering encrypted code, in A Manuscript on Deciphering Cryptographic Messages . He gave
2490-603: The Middle East is that of Qusta ibn Luqa (10th century), an Arab mathematician from Baalbek , Lebanon . He justified the technique by a formal, Euclidean-style geometric proof . Within the tradition of Golden Age Muslim mathematics, double false position was known as hisāb al-khaṭāʾayn ("reckoning by two errors"). It was used for centuries to solve practical problems such as commercial and juridical questions (estate partitions according to rules of Quranic inheritance ), as well as purely recreational problems. The algorithm
2573-594: The medieval era , driven by the practical applications of Al-Khwārizmī 's methods. This dissemination was influenced not only by economic and political factors but also by cultural exchanges, exemplified by events such as the Crusades and the translation movement. The Islamic Golden Age , spanning from the 8th to the 14th century, marked a period of considerable advancements in various scientific disciplines, attracting scholars from medieval Europe seeking access to this knowledge. Trade routes and cultural interactions played
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2656-433: The positive roots of first and second-degree (linear and quadratic) polynomial equations . He introduces the method of reduction , and unlike Diophantus, also gives general solutions for the equations he deals with. Al-Khwarizmi's algebra was rhetorical, which means that the equations were written out in full sentences. This was unlike the algebraic work of Diophantus, which was syncopated, meaning that some symbolism
2739-439: The source code editor , but the content aspects reflect the programmer's talent and skills. Various visual programming languages have also been developed with the intent to resolve readability concerns by adopting non-traditional approaches to code structure and display. Integrated development environments (IDEs) aim to integrate all such help. Techniques like Code refactoring can enhance readability. The academic field and
2822-486: The 10th century, Abū al-Wafā' al-Būzjānī considered debts as negative numbers in A Book on What Is Necessary from the Science of Arithmetic for Scribes and Businessmen . By the 12th century, al-Karaji's successors were to state the general rules of signs and use them to solve polynomial divisions . As al-Samaw'al writes: the product of a negative number— al-nāqiṣ —by a positive number— al-zāʾid —is negative, and by
2905-642: The 14th to 17th century, the translation of Arabic mathematical texts, along with Greek and Roman ones, played a crucial role in shaping the intellectual landscape of the Renaissance. Figures like Fibonacci , who studied in North Africa and the Middle East, helped introduce and popularize Arabic numerals and mathematical concepts in Europe. The study of algebra , the name of which is derived from
2988-568: The 9th century, a programmable music sequencer was invented by the Persian Banu Musa brothers, who described an automated mechanical flute player in the Book of Ingenious Devices . In 1206, the Arab engineer Al-Jazari invented a programmable drum machine where a musical mechanical automaton could be made to play different rhythms and drum patterns, via pegs and cams . In 1801,
3071-526: The AE in 1837. In the 1880s, Herman Hollerith invented the concept of storing data in machine-readable form. Later a control panel (plug board) added to his 1906 Type I Tabulator allowed it to be programmed for different jobs, and by the late 1940s, unit record equipment such as the IBM 602 and IBM 604 , were programmed by control panels in a similar way, as were the first electronic computers . However, with
3154-616: The Islamic world found its way to Europe through various channels. Meanwhile, the Crusades connected Western Europeans with the Islamic world. While the primary purpose of the Crusades was military, there was also cultural exchange and exposure to Islamic knowledge, including mathematics. European scholars who traveled to the Holy Land and other parts of the Islamic world gained access to Arabic manuscripts and mathematical treatises. During
3237-678: The Islamic world made substantial contributions to mathematics , astronomy , medicine , and other sciences . As a result, the intellectual achievements of Islamic scholars attracted the attention of scholars in medieval Europe who sought to access this wealth of knowledge. Trade routes, such as the Silk Road , facilitated the movement of goods, ideas, and knowledge between the East and West. Cities like Baghdad , Cairo , and Cordoba became centers of learning and attracted scholars from different cultural backgrounds.Therefore, mathematical knowledge from
3320-539: The West was driven by its practical applications, the expansion of mathematical concepts by his successors, and the translation and adaptation of these ideas into the Western context. This spread was a complex process involving economics, politics, and cultural exchange, greatly influencing Western mathematics. The period known as the Islamic Golden Age (8th to 14th century) was characterized by significant advancements in various fields, including mathematics . Scholars in
3403-496: The West was facilitated by several factors. The practicality and general applicability of al-Khwārizmī's methods were significant. They were designed to convert numerical or geometrical problems into equations in normal form, leading to canonical solution formulae. His work and that of his successors like al-Karaji laid the foundation for advances in various mathematical fields, including number theory , numerical analysis , and rational Diophantine analysis . Al-Khwārizmī's algebra
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3486-654: The age of discovery and scientific revolution. The practical applications of trigonometry for navigation and astronomy became increasingly important during the Age of Exploration. Al-Battānī is one of the islamic mathematicians who made great contributions to the development of trigonometry. He "innovated new trigonometric functions, created a table of cotangents, and made some formulas in spherical trigonometry." These discoveries, together with his astronomical works which are praised for their accuracy, greatly advanced astronomical calculations and instruments. Al-Khayyām (1048–1131)
3569-452: The algebra. His algebra was initially focused on linear and quadratic equations and the elementary arithmetic of binomials and trinomials. This approach, which involved solving equations using radicals and related algebraic calculations, influenced mathematical thinking long after his death. Al-Khwārizmī's proof of the rule for solving quadratic equations of the form (ax^2 + bx = c), commonly referred to as "squares plus roots equal numbers,"
3652-478: The arithmetization of algebra , influencing mathematical thought for an extended period. Successors like Al-Karaji expanded on his work, contributing to advancements in various mathematical domains. The practicality and broad applicability of these mathematical methods facilitated the dissemination of Arabic mathematics to the West, contributing substantially to the evolution of Western mathematics. Arabic mathematical knowledge spread through various channels during
3735-474: The author of Algebra , al-Khwārizmī. His Book of Algebra (Kitāb fī al-jabr wa al-muqābala) is "essentially a commentary on and elaboration of al-Khwārizmī's work; in part for that reason and in part for its own merit, the book enjoyed widespread popularity in the Muslim world". It contains 69 problems, which is more than al-Khwārizmī who had 40 in his book. Abū Kāmil's Algebra plays a significant role in shaping
3818-414: The circumstances. The first step in most formal software development processes is requirements analysis , followed by testing to determine value modeling, implementation, and failure elimination (debugging). There exist a lot of different approaches for each of those tasks. One approach popular for requirements analysis is Use Case analysis. Many programmers use forms of Agile software development where
3901-495: The code, making it easy to target varying machine instruction sets via compilation declarations and heuristics . Compilers harnessed the power of computers to make programming easier by allowing programmers to specify calculations by entering a formula using infix notation . Programs were mostly entered using punched cards or paper tape . By the late 1960s, data storage devices and computer terminals became inexpensive enough that programs could be created by typing directly into
3984-467: The computers. Text editors were also developed that allowed changes and corrections to be made much more easily than with punched cards . Whatever the approach to development may be, the final program must satisfy some fundamental properties. The following properties are among the most important: Using automated tests and fitness functions can help to maintain some of the aforementioned attributes. In computer programming, readability refers to
4067-548: The concept of the stored-program computer introduced in 1949, both programs and data were stored and manipulated in the same way in computer memory . Machine code was the language of early programs, written in the instruction set of the particular machine, often in binary notation. Assembly languages were soon developed that let the programmer specify instructions in a text format (e.g., ADD X, TOTAL), with abbreviations for each operation code and meaningful names for specifying addresses. However, because an assembly language
4150-401: The development of Algebra and algebraic geometry, Western historians in the 18th and early 19th century still regarded it as a fact that Classical science and math were unique phenomena of the West. Even though some math contributions from Arab mathematicians are occasionally acknowledged, they are considered to be "outside history or only integrated in so far as it contributed to science, which
4233-537: The early language), and to formalise file manipulation operations. Computer programming Computer programming or coding is the composition of sequences of instructions, called programs , that computers can follow to perform tasks. It involves designing and implementing algorithms , step-by-step specifications of procedures, by writing code in one or more programming languages . Programmers typically use high-level programming languages that are more easily intelligible to humans than machine code , which
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#17327917723564316-510: The ease with which a human reader can comprehend the purpose, control flow , and operation of source code . It affects the aspects of quality above, including portability, usability and most importantly maintainability. Readability is important because programmers spend the majority of their time reading, trying to understand, reusing, and modifying existing source code, rather than writing new source code. Unreadable code often leads to bugs, inefficiencies, and duplicated code . A study found that
4399-527: The engineering practice of computer programming are concerned with discovering and implementing the most efficient algorithms for a given class of problems. For this purpose, algorithms are classified into orders using Big O notation , which expresses resource use—such as execution time or memory consumption—in terms of the size of an input. Expert programmers are familiar with a variety of well-established algorithms and their respective complexities and use this knowledge to choose algorithms that are best suited to
4482-504: The equation would have no solutions, one solution or two solutions, depending on whether the height of the curve at that point was less than, equal to, or greater than a . His surviving works give no indication of how he discovered his formulae for the maxima of these curves. Various conjectures have been proposed to account for his discovery of them. The earliest implicit traces of mathematical induction can be found in Euclid 's proof that
4565-400: The first description of cryptanalysis by frequency analysis , the earliest code-breaking algorithm. The first computer program is generally dated to 1843 when mathematician Ada Lovelace published an algorithm to calculate a sequence of Bernoulli numbers , intended to be carried out by Charles Babbage 's Analytical Engine . However, Charles Babbage himself had written a program for
4648-404: The first step in debugging is to attempt to reproduce the problem. This can be a non-trivial task, for example as with parallel processes or some unusual software bugs. Also, specific user environment and usage history can make it difficult to reproduce the problem. After the bug is reproduced, the input of the program may need to be simplified to make it easier to debug. For example, when a bug in
4731-432: The foundational contributions of Arab mathematicians, Western historians in the 18th and early 19th centuries, influenced by Orientalist views, sometimes marginalized these achievements. The East lacking rationality and scientific spirit perpetuated a biased perspective, hindering the recognition of the significant role played by Arabic mathematics in the development of algebra and other mathematical disciplines. Reevaluating
4814-464: The functions in a library follow the appropriate run-time conventions (e.g., method of passing arguments ), then these functions may be written in any other language. Computer programmers are those who write computer software. Their jobs usually involve: Although programming has been presented in the media as a somewhat mathematical subject, some research shows that good programmers have strong skills in natural human languages, and that learning to code
4897-441: The great tradition of Greek classical science, math works from Arabic mathematicians are also blamed for lacking rigor and too focused on practical applications and calculations, and this is why Western historians argued they could never reach the level of Greek mathematicians. As Tannery wrote, Arabic math "in no way superseded the level attained by Diophantus". On the other hand, they perceived that Western mathematicians went into
4980-404: The language to build a team, the availability of compilers for that language, and the efficiency with which programs written in a given language execute. Languages form an approximate spectrum from "low-level" to "high-level"; "low-level" languages are typically more machine-oriented and faster to execute, whereas "high-level" languages are more abstract and easier to use but execute less quickly. It
5063-465: The language, the number of books sold and courses teaching the language (this overestimates the importance of newer languages), and estimates of the number of existing lines of code written in the language (this underestimates the number of users of business languages such as COBOL). Some languages are very popular for particular kinds of applications, while some languages are regularly used to write many different kinds of applications. For example, COBOL
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#17327917723565146-532: The number of primes is infinite (c. 300 BCE). The first explicit formulation of the principle of induction was given by Pascal in his Traité du triangle arithmétique (1665). In between, implicit proof by induction for arithmetic sequences was introduced by al-Karaji (c. 1000) and continued by al-Samaw'al , who used it for special cases of the binomial theorem and properties of Pascal's triangle . The Greeks had discovered irrational numbers , but were not happy with them and only able to cope by drawing
5229-526: The point at which a cubic polynomial obtains its maximum value. For example, to solve the equation x 3 + a = b x {\displaystyle \ x^{3}+a=bx} , with a and b positive, he would note that the maximum point of the curve y = b x − x 3 {\displaystyle \ y=bx-x^{3}} occurs at x = b 3 {\displaystyle x=\textstyle {\sqrt {\frac {b}{3}}}} , and that
5312-536: The possible solutions to some of his problems. Abu al-Jud , Omar Khayyam , along with Sharaf al-Dīn al-Tūsī , found several solutions of the cubic equation . Omar Khayyam found the general geometric solution of a cubic equation. Omar Khayyam (c. 1038/48 in Iran – 1123/24) wrote the Treatise on Demonstration of Problems of Algebra containing the systematic solution of cubic or third-order equations , going beyond
5395-476: The programmer can try to skip some user interaction from the original problem description and check if the remaining actions are sufficient for bugs to appear. Scripting and breakpointing are also part of this process. Debugging is often done with IDEs . Standalone debuggers like GDB are also used, and these often provide less of a visual environment, usually using a command line . Some text editors such as Emacs allow GDB to be invoked through them, to provide
5478-563: The remainder is the same negative, and if we subtract a negative number from an empty power, the remainder is the same positive number. Between the 9th and 10th centuries, the Egyptian mathematician Abu Kamil wrote a now-lost treatise on the use of double false position, known as the Book of the Two Errors ( Kitāb al-khaṭāʾayn ). The oldest surviving writing on double false position from
5561-517: The role of conscientious guardians of the Hellenistic museum." In 18th century Germany and France , the prevailing Orientalist view was "East and West oppose each other not as geographical but as historical positivities", which labeled " Rationalism " as the essence of the West, while the "Call of the Orient " movement emerged in the 19th century was interpreted as "against Rationalism" and
5644-399: The systematised study of algebra and advances in geometry and trigonometry . The medieval Islamic world underwent significant developments in mathematics. Muhammad ibn Musa al-Khwārizmī played a key role in this transformation, introducing algebra as a distinct field in the 9th century. Al-Khwārizmī 's approach, departing from earlier arithmetical traditions, laid the groundwork for
5727-411: The term software development is used for this larger overall process – with the terms programming , implementation , and coding reserved for the writing and editing of code per se. Sometimes software development is known as software engineering , especially when it employs formal methods or follows an engineering design process . Programmable devices have existed for centuries. As early as
5810-409: The term 'compiler'. FORTRAN , the first widely used high-level language to have a functional implementation, came out in 1957, and many other languages were soon developed—in particular, COBOL aimed at commercial data processing, and Lisp for computer research. These compiled languages allow the programmer to write programs in terms that are syntactically richer, and more capable of abstracting
5893-459: The trajectory of Western mathematics, particularly in its impact on the works of the Italian mathematician Leonardo of Pisa, widely recognized as Fibonacci. In his Liber Abaci (1202), Fibonacci extensively incorporated ideas from Arabic mathematicians, using approximately 29 problems from Book of Algebra with scarce modification. Despite the fundamental works Arabic mathematicians have done on
5976-516: The twelfth century, Latin translations of Al-Khwarizmi 's Arithmetic on the Indian numerals introduced the decimal positional number system to the Western world . His Compendious Book on Calculation by Completion and Balancing presented the first systematic solution of linear and quadratic equations . In Renaissance Europe, he was considered the original inventor of algebra, although it
6059-560: The various stages of formal software development are more integrated together into short cycles that take a few weeks rather than years. There are many approaches to the Software development process. Popular modeling techniques include Object-Oriented Analysis and Design ( OOAD ) and Model-Driven Architecture ( MDA ). The Unified Modeling Language ( UML ) is a notation used for both the OOAD and MDA. A similar technique used for database design
6142-827: The western world through Spain and Sicily during the translation movement. "The Moors (western Mohammedans from that part of North Africa once known as Mauritania) crossed over into Spain early in the seventh century, bringing with them the cultural resources of the Arab world". In the 13th century, King Alfonso X of Castile established the Toledo School of Translators , in the Kingdom of Castile , where scholars translated numerous scientific and philosophical works from Arabic into Latin . The translations included Islamic contributions to trigonometry , which helps European mathematicians and astronomers in their studies. European scholars such as Gerard of Cremona (1114–1187) played
6225-513: The works of Indian mathematicians, but the recognition and use of negative numbers during this period remained timid. Al-Khwarizmi did not use negative numbers or negative coefficients. But within fifty years, Abu Kamil illustrated the rules of signs for expanding the multiplication ( a ± b ) ( c ± d ) {\displaystyle (a\pm b)(c\pm d)} . Al-Karaji wrote in his book al-Fakhrī that "negative quantities must be counted as terms". In
6308-435: Was a Persian mathematician, astronomer, and poet, known for his work on algebra and geometry, particularly his investigations into the solutions of cubic equations. He was "the first in history to elaborate a geometrical theory of equations with degrees ≤ 3", and has great influence on the work of Descartes, a French mathematician who is often regarded as the founder of analytical geometry. Indeed, "to read Descartes ' Géométrie
6391-611: Was a monumental achievement in the history of algebra. This breakthrough laid the groundwork for the systematic approach to solving quadratic equations, which became a fundamental aspect of algebra as it developed in the Western world. Al-Khwārizmī's method, which involved completing the square, not only provided a practical solution for equations of this type but also introduced an abstract and generalized approach to mathematical problems. His work, encapsulated in his seminal text "Al-Kitab al-Mukhtasar fi Hisab al-Jabr wal-Muqabala" (The Compendious Book on Calculation by Completion and Balancing),
6474-505: Was a turning point. He introduced the term "algebra" in the title of his book, " Kitab al-jabr wa al-muqabala ," marking it as a distinct discipline. He regarded his work as "a short work on Calculation by (the rules of) Completion and Reduction, confining it to what is easiest and most useful in arithmetic". Later, people commented his work was not just a theoretical treatise but also practical, aimed at solving problems in areas like commerce and land measurement. Al-Khwārizmī 's approach
6557-425: Was an autonomous discipline with its historical perspective, eventually leading to the "arithmetization of algebra". His successors expanded on his work, adapting it to new theoretical and technical challenges and reorienting it towards a more arithmetical direction for abstract algebraic calculation. Arabic mathematics, epitomized by al-Khwārizmī's work, was crucial in shaping the mathematical landscape. Its spread to
6640-424: Was groundbreaking in that it did not arise from any previous "arithmetical" tradition, including that of Diophantus . He developed a new vocabulary for algebra, distinguishing between purely algebraic terms and those shared with arithmetic. Al-Khwārizmī noticed that the representation of numbers is crucial in daily life. Thus, he wanted to find or summarize a way to simplify the mathematical operation, so-called later,
6723-454: Was often memorized with the aid of mnemonics , such as a verse attributed to Ibn al-Yasamin and balance-scale diagrams explained by al-Hassar and Ibn al-Banna , who were each mathematicians of Moroccan origin. The influence of medieval Arab-Islamic mathematics to the rest of the world is wide and profound, in both the realm of science and mathematics. The knowledge of the Arabs went into
6806-488: Was translated into Latin in the 12th century. This translation played a pivotal role in the transmission of algebraic knowledge to Europe, significantly influencing mathematicians during the Renaissance and shaping the evolution of modern mathematics. Al-Khwārizmī's contributions, especially his proof for quadratic equations, are a testament to the rich mathematical heritage of the Islamic world and its enduring impact on Western mathematics. The spread of Arabic mathematics to
6889-570: Was written in Fortran IV, to run on a Burroughs Corporation B6700 mainframe computer . Subsequently, it was rewritten in SMALL, and ported to a Digital Equipment Corporation (DEC) PDP-10 mainframe (on the operating system TOPS-10 ) and an IBM S360 mainframe (on the operating system VM Conversational Monitor System (VM/CMS)). About 1985, SMALL had some object-oriented programming features added to handle structures (that were missing from
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