A computer program is a sequence or set of instructions in a programming language for a computer to execute . It is one component of software , which also includes documentation and other intangible components.
69-401: A validator is a computer program used to check the validity or syntactical correctness of a fragment of code or document. The term is commonly used in the context of validating HTML , CSS , and XML documents like RSS feeds, though it can be used for any defined format or language. Accessibility validators are automated tools that are designed to verify compliance of a web page or
138-509: A compiler written for the language. ( Assembly language programs are translated using an assembler .) The resulting file is called an executable . Alternatively, source code may execute within an interpreter written for the language. If the executable is requested for execution, then the operating system loads it into memory and starts a process . The central processing unit will soon switch to this process so it can fetch, decode, and then execute each machine instruction. If
207-462: A list of integers could be called integer_list . In object-oriented jargon, abstract datatypes are called classes . However, a class is only a definition; no memory is allocated. When memory is allocated to a class and bound to an identifier , it is called an object . Object-oriented imperative languages developed by combining the need for classes and the need for safe functional programming . A function, in an object-oriented language,
276-422: A programming language . Programming language features exist to provide building blocks to be combined to express programming ideals. Ideally, a programming language should: The programming style of a programming language to provide these building blocks may be categorized into programming paradigms . For example, different paradigms may differentiate: Each of these programming styles has contributed to
345-428: A store which consisted of memory to hold 1,000 numbers of 50 decimal digits each. Numbers from the store were transferred to the mill for processing. The engine was programmed using two sets of perforated cards. One set directed the operation and the other set inputted the variables. However, the thousands of cogged wheels and gears never fully worked together. Ada Lovelace worked for Charles Babbage to create
414-523: A web site with respect to one or more accessibility guidelines (such as WCAG , Section 508 or those associated with national laws such as the Stanca Act ). Computer program A computer program in its human-readable form is called source code . Source code needs another computer program to execute because computers can only execute their native machine instructions . Therefore, source code may be translated to machine instructions using
483-604: A description of the Analytical Engine (1843). The description contained Note G which completely detailed a method for calculating Bernoulli numbers using the Analytical Engine. This note is recognized by some historians as the world's first computer program . In 1936, Alan Turing introduced the Universal Turing machine , a theoretical device that can model every computation. It is a finite-state machine that has an infinitely long read/write tape. The machine can move
552-580: A language's basic syntax . The syntax of the language BASIC (1964) was intentionally limited to make the language easy to learn. For example, variables are not declared before being used. Also, variables are automatically initialized to zero. Here is an example computer program, in Basic, to average a list of numbers: Once the mechanics of basic computer programming are learned, more sophisticated and powerful languages are available to build large computer systems. Improvements in software development are
621-401: A mathematician is likely to write Bernoulli's formula as: This formula suggests setting B 1 = 1 / 2 when switching from the so-called 'archaic' enumeration which uses only the even indices 2, 4, 6... to the modern form (more on different conventions in the next paragraph). Most striking in this context is the fact that the falling factorial c has for k = 0
690-521: A profound influence on programming language design. Emerging from a committee of European and American programming language experts, it used standard mathematical notation and had a readable, structured design. Algol was first to define its syntax using the Backus–Naur form . This led to syntax-directed compilers. It added features like: Algol's direct descendants include Pascal , Modula-2 , Ada , Delphi and Oberon on one branch. On another branch
759-557: A result, the computer could be programmed quickly and perform calculations at very fast speeds. Presper Eckert and John Mauchly built the ENIAC. The two engineers introduced the stored-program concept in a three-page memo dated February 1944. Later, in September 1944, John von Neumann began working on the ENIAC project. On June 30, 1945, von Neumann published the First Draft of a Report on
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#1732851951570828-442: A shortcut for 1 × 2 × ... × k was not introduced until 100 years later. The integral symbol on the left hand side goes back to Gottfried Wilhelm Leibniz in 1675 who used it as a long letter S for "summa" (sum). The letter n on the left hand side is not an index of summation but gives the upper limit of the range of summation which is to be understood as 1, 2, ..., n . Putting things together, for positive c , today
897-406: A suggestion of Abraham de Moivre . Bernoulli's formula is sometimes called Faulhaber's formula after Johann Faulhaber who found remarkable ways to calculate sum of powers but never stated Bernoulli's formula. According to Knuth a rigorous proof of Faulhaber's formula was first published by Carl Jacobi in 1834. Knuth's in-depth study of Faulhaber's formula concludes (the nonstandard notation on
966-456: Is an asymptotic series . It contains the trigamma function ψ 1 . From the generating functions above, one can obtain the following integral formula for the even Bernoulli numbers: The Bernoulli numbers can be expressed in terms of the Riemann zeta function : Here the argument of the zeta function is 0 or negative. As ζ ( k ) {\displaystyle \zeta (k)}
1035-418: Is assigned to a class. An assigned function is then referred to as a method , member function , or operation . Object-oriented programming is executing operations on objects . Object-oriented languages support a syntax to model subset/superset relationships. In set theory , an element of a subset inherits all the attributes contained in the superset. For example, a student is a person. Therefore,
1104-457: Is disputed whether Lovelace or Babbage developed the algorithm . As a result, the Bernoulli numbers have the distinction of being the subject of the first published complex computer program . The superscript ± used in this article distinguishes the two sign conventions for Bernoulli numbers. Only the n = 1 term is affected: In the formulas below, one can switch from one sign convention to
1173-585: Is negative if n is divisible by 4 and positive otherwise. The Bernoulli numbers are special values of the Bernoulli polynomials B n ( x ) {\displaystyle B_{n}(x)} , with B n − = B n ( 0 ) {\displaystyle B_{n}^{-{}}=B_{n}(0)} and B n + = B n ( 1 ) {\displaystyle B_{n}^{+}=B_{n}(1)} . The Bernoulli numbers were discovered around
1242-548: Is to alter the electrical resistivity and conductivity of a semiconductor junction . First, naturally occurring silicate minerals are converted into polysilicon rods using the Siemens process . The Czochralski process then converts the rods into a monocrystalline silicon , boule crystal . The crystal is then thinly sliced to form a wafer substrate . The planar process of photolithography then integrates unipolar transistors, capacitors , diodes , and resistors onto
1311-731: Is useful to be able to compute the Bernoulli numbers B 0 through B p − 3 modulo p , where p is a prime; for example to test whether Vandiver's conjecture holds for p , or even just to determine whether p is an irregular prime . It is not feasible to carry out such a computation using the above recursive formulae, since at least (a constant multiple of) p arithmetic operations would be required. Fortunately, faster methods have been developed which require only O ( p (log p ) ) operations (see big O notation ). David Harvey describes an algorithm for computing Bernoulli numbers by computing B n modulo p for many small primes p , and then reconstructing B n via
1380-482: Is zero for negative even integers (the trivial zeroes ), if n>1 is odd, ζ ( 1 − n ) {\displaystyle \zeta (1-n)} is zero. By means of the zeta functional equation and the gamma reflection formula the following relation can be obtained: Now the argument of the zeta function is positive. It then follows from ζ → 1 ( n → ∞ ) and Stirling's formula that In some applications it
1449-551: The new statement. A module's other file is the source file . Here is a C++ source file for the GRADE class in a simple school application: Here is a C++ header file for the PERSON class in a simple school application: Bernoulli number In mathematics , the Bernoulli numbers B n are a sequence of rational numbers which occur frequently in analysis . The Bernoulli numbers appear in (and can be defined by)
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#17328519515701518-767: The Chinese remainder theorem . Harvey writes that the asymptotic time complexity of this algorithm is O ( n log( n ) ) and claims that this implementation is significantly faster than implementations based on other methods. Using this implementation Harvey computed B n for n = 10 . Harvey's implementation has been included in SageMath since version 3.1. Prior to that, Bernd Kellner computed B n to full precision for n = 10 in December 2002 and Oleksandr Pavlyk for n = 10 with Mathematica in April 2008. Arguably
1587-604: The IBM System/360 (1964) had a CPU made from circuit boards containing discrete components on ceramic substrates . The Intel 4004 (1971) was a 4- bit microprocessor designed to run the Busicom calculator. Five months after its release, Intel released the Intel 8008 , an 8-bit microprocessor. Bill Pentz led a team at Sacramento State to build the first microcomputer using the Intel 8008:
1656-480: The Sac State 8008 (1972). Its purpose was to store patient medical records. The computer supported a disk operating system to run a Memorex , 3- megabyte , hard disk drive . It had a color display and keyboard that was packaged in a single console. The disk operating system was programmed using IBM's Basic Assembly Language (BAL) . The medical records application was programmed using a BASIC interpreter. However,
1725-1126: The Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of m -th powers of the first n positive integers, in the Euler–Maclaurin formula , and in expressions for certain values of the Riemann zeta function . The values of the first 20 Bernoulli numbers are given in the adjacent table. Two conventions are used in the literature, denoted here by B n − {\displaystyle B_{n}^{-{}}} and B n + {\displaystyle B_{n}^{+{}}} ; they differ only for n = 1 , where B 1 − = − 1 / 2 {\displaystyle B_{1}^{-{}}=-1/2} and B 1 + = + 1 / 2 {\displaystyle B_{1}^{+{}}=+1/2} . For every odd n > 1 , B n = 0 . For every even n > 0 , B n
1794-550: The circuits . At its core, it was a series of Pascalines wired together. Its 40 units weighed 30 tons, occupied 1,800 square feet (167 m ), and consumed $ 650 per hour ( in 1940s currency ) in electricity when idle. It had 20 base-10 accumulators . Programming the ENIAC took up to two months. Three function tables were on wheels and needed to be rolled to fixed function panels. Function tables were connected to function panels by plugging heavy black cables into plugboards . Each function table had 728 rotating knobs. Programming
1863-404: The programming environment to advance from a computer terminal (until the 1990s) to a graphical user interface (GUI) computer. Computer terminals limited programmers to a single shell running in a command-line environment . During the 1970s, full-screen source code editing became possible through a text-based user interface . Regardless of the technology available, the goal is to program in
1932-993: The Bernoulli numbers, usually giving some reference in the older literature. One of them is (for m ≥ 1 {\displaystyle m\geq 1} ): The exponential generating functions are where the substitution is t → − t {\displaystyle t\to -t} . The two generating functions only differ by t . If we let F ( t ) = ∑ i = 1 ∞ f i t i {\displaystyle F(t)=\sum _{i=1}^{\infty }f_{i}t^{i}} and G ( t ) = 1 / ( 1 + F ( t ) ) = ∑ i = 0 ∞ g i t i {\displaystyle G(t)=1/(1+F(t))=\sum _{i=0}^{\infty }g_{i}t^{i}} then Then g 0 = 1 {\displaystyle g_{0}=1} and for m > 0 {\displaystyle m>0}
2001-494: The EDVAC , which equated the structures of the computer with the structures of the human brain. The design became known as the von Neumann architecture . The architecture was simultaneously deployed in the constructions of the EDVAC and EDSAC computers in 1949. The IBM System/360 (1964) was a family of computers, each having the same instruction set architecture . The Model 20 was
2070-433: The ENIAC also involved setting some of the 3,000 switches. Debugging a program took a week. It ran from 1947 until 1955 at Aberdeen Proving Ground , calculating hydrogen bomb parameters, predicting weather patterns, and producing firing tables to aim artillery guns. Instead of plugging in cords and turning switches, a stored-program computer loads its instructions into memory just like it loads its data into memory. As
2139-532: The LHS is explained further on): In the above Knuth meant B 1 − {\displaystyle B_{1}^{-}} ; instead using B 1 + {\displaystyle B_{1}^{+}} the formula avoids subtraction: The Bernoulli numbers OEIS : A164555 (n)/ OEIS : A027642 (n) were introduced by Jakob Bernoulli in the book Ars Conjectandi published posthumously in 1713 page 97. The main formula can be seen in
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2208-640: The cheaper Intel 8088 . IBM embraced the Intel 8088 when they entered the personal computer market (1981). As consumer demand for personal computers increased, so did Intel's microprocessor development. The succession of development is known as the x86 series . The x86 assembly language is a family of backward-compatible machine instructions . Machine instructions created in earlier microprocessors were retained throughout microprocessor upgrades. This enabled consumers to purchase new computers without having to purchase new application software . The major categories of instructions are: VLSI circuits enabled
2277-419: The computer was an evolutionary dead-end because it was extremely expensive. Also, it was built at a public university lab for a specific purpose. Nonetheless, the project contributed to the development of the Intel 8080 (1974) instruction set . In 1978, the modern software development environment began when Intel upgraded the Intel 8080 to the Intel 8086 . Intel simplified the Intel 8086 to manufacture
2346-537: The configuration, an execute button was pressed. This process was then repeated. Computer programs also were automatically inputted via paper tape , punched cards or magnetic-tape . After the medium was loaded, the starting address was set via switches, and the execute button was pressed. A major milestone in software development was the invention of the Very Large Scale Integration (VLSI) circuit (1964). Following World War II , tube-based technology
2415-434: The descendants include C , C++ and Java . BASIC (1964) stands for "Beginner's All-Purpose Symbolic Instruction Code". It was developed at Dartmouth College for all of their students to learn. If a student did not go on to a more powerful language, the student would still remember Basic. A Basic interpreter was installed in the microcomputers manufactured in the late 1970s. As the microcomputer industry grew, so did
2484-662: The early history of the computation of sums of integer powers, which have been of interest to mathematicians since antiquity. Methods to calculate the sum of the first n positive integers, the sum of the squares and of the cubes of the first n positive integers were known, but there were no real 'formulas', only descriptions given entirely in words. Among the great mathematicians of antiquity to consider this problem were Pythagoras (c. 572–497 BCE, Greece), Archimedes (287–212 BCE, Italy), Aryabhata (b. 476, India), Abu Bakr al-Karaji (d. 1019, Persia) and Abu Ali al-Hasan ibn al-Hasan ibn al-Haytham (965–1039, Iraq). During
2553-460: The first Fortran standard in 1966. In 1978, Fortran 77 became the standard until 1991. Fortran 90 supports: COBOL (1959) stands for "COmmon Business Oriented Language". Fortran manipulated symbols. It was soon realized that symbols did not need to be numbers, so strings were introduced. The US Department of Defense influenced COBOL's development, with Grace Hopper being a major contributor. The statements were English-like and verbose. The goal
2622-475: The language BCPL was replaced with B , and AT&T Bell Labs called the next version "C". Its purpose was to write the UNIX operating system . C is a relatively small language, making it easy to write compilers. Its growth mirrored the hardware growth in the 1980s. Its growth also was because it has the facilities of assembly language , but uses a high-level syntax . It added advanced features like: C allows
2691-400: The language. Basic pioneered the interactive session . It offered operating system commands within its environment: However, the Basic syntax was too simple for large programs. Recent dialects added structure and object-oriented extensions. Microsoft's Visual Basic is still widely used and produces a graphical user interface . C programming language (1973) got its name because
2760-430: The last term; it should be − 3 20 n 2 {\displaystyle -{\tfrac {3}{20}}n^{2}} instead of − 1 12 n 2 {\displaystyle -{\tfrac {1}{12}}n^{2}} . Many characterizations of the Bernoulli numbers have been found in the last 300 years, and each could be used to introduce these numbers. Here only four of
2829-523: The late sixteenth and early seventeenth centuries mathematicians made significant progress. In the West Thomas Harriot (1560–1621) of England, Johann Faulhaber (1580–1635) of Germany, Pierre de Fermat (1601–1665) and fellow French mathematician Blaise Pascal (1623–1662) all played important roles. Thomas Harriot seems to have been the first to derive and write formulas for sums of powers using symbolic notation, but even he calculated only up to
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2898-401: The m term in the series for G ( t ) {\displaystyle G(t)} is: If then we find that showing that the values of i ! g i {\displaystyle i!g_{i}} obey the recursive formula for the Bernoulli numbers B i − {\displaystyle B_{i}^{-}} . The (ordinary) generating function
2967-485: The matrix was to burn out the unneeded connections. There were so many connections, firmware programmers wrote a computer program on another chip to oversee the burning. The technology became known as Programmable ROM . In 1971, Intel installed the computer program onto the chip and named it the Intel 4004 microprocessor . The terms microprocessor and central processing unit (CPU) are now used interchangeably. However, CPUs predate microprocessors. For example,
3036-566: The most important application of the Bernoulli numbers in mathematics is their use in the Euler–Maclaurin formula . Assuming that f is a sufficiently often differentiable function the Euler–Maclaurin formula can be written as This formulation assumes the convention B 1 = − 1 / 2 . Using the convention B 1 = + 1 / 2 the formula becomes Here f ( 0 ) = f {\displaystyle f^{(0)}=f} (i.e.
3105-553: The most useful ones are mentioned: For the proof of the equivalence of the four approaches. The Bernoulli numbers obey the sum formulas where m = 0 , 1 , 2... {\displaystyle m=0,1,2...} and δ denotes the Kronecker delta . The first of these is sometimes written as the formula (for m > 1) ( B + 1 ) m − B m = 0 , {\displaystyle (B+1)^{m}-B_{m}=0,} where
3174-517: The other with the relation B n + = ( − 1 ) n B n − {\displaystyle B_{n}^{+}=(-1)^{n}B_{n}^{-}} , or for integer n = 2 or greater, simply ignore it. Since B n = 0 for all odd n > 1 , and many formulas only involve even-index Bernoulli numbers, a few authors write " B n " instead of B 2 n . This article does not follow that notation. The Bernoulli numbers are rooted in
3243-413: The power is expanded formally using the binomial theorem and B k {\displaystyle B^{k}} is replaced by B k {\displaystyle B_{k}} . Solving for B m ∓ {\displaystyle B_{m}^{\mp {}}} gives the recursive formulas In 1893 Louis Saalschütz listed a total of 38 explicit formulas for
3312-443: The programmer to control which region of memory data is to be stored. Global variables and static variables require the fewest clock cycles to store. The stack is automatically used for the standard variable declarations . Heap memory is returned to a pointer variable from the malloc() function. In the 1970s, software engineers needed language support to break large projects down into modules . One obvious feature
3381-486: The result of improvements in computer hardware . At each stage in hardware's history, the task of computer programming changed dramatically. In 1837, Jacquard's loom inspired Charles Babbage to attempt to build the Analytical Engine . The names of the components of the calculating device were borrowed from the textile industry. In the textile industry, yarn was brought from the store to be milled. The device had
3450-500: The same time by the Swiss mathematician Jacob Bernoulli , after whom they are named, and independently by Japanese mathematician Seki Takakazu . Seki's discovery was posthumously published in 1712 in his work Katsuyō Sanpō ; Bernoulli's, also posthumously, in his Ars Conjectandi of 1713. Ada Lovelace 's note G on the Analytical Engine from 1842 describes an algorithm for generating Bernoulli numbers with Babbage 's machine; it
3519-487: The second half of the corresponding facsimile. The constant coefficients denoted A , B , C and D by Bernoulli are mapped to the notation which is now prevalent as A = B 2 , B = B 4 , C = B 6 , D = B 8 . The expression c · c −1· c −2· c −3 means c ·( c −1)·( c −2)·( c −3) – the small dots are used as grouping symbols. Using today's terminology these expressions are falling factorial powers c . The factorial notation k ! as
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#17328519515703588-438: The set of students is a subset of the set of persons. As a result, students inherit all the attributes common to all persons. Additionally, students have unique attributes that other people do not have. Object-oriented languages model subset/superset relationships using inheritance . Object-oriented programming became the dominant language paradigm by the late 1990s. C++ (1985) was originally called "C with Classes". It
3657-467: The smallest and least expensive. Customers could upgrade and retain the same application software . The Model 195 was the most premium. Each System/360 model featured multiprogramming —having multiple processes in memory at once. When one process was waiting for input/output , another could compute. IBM planned for each model to be programmed using PL/1 . A committee was formed that included COBOL , Fortran and ALGOL programmers. The purpose
3726-418: The source code is requested for execution, then the operating system loads the corresponding interpreter into memory and starts a process. The interpreter then loads the source code into memory to translate and execute each statement . Running the source code is slower than running an executable . Moreover, the interpreter must be installed on the computer. The "Hello, World!" program is used to illustrate
3795-423: The sum of the fourth powers. Johann Faulhaber gave formulas for sums of powers up to the 17th power in his 1631 Academia Algebrae , far higher than anyone before him, but he did not give a general formula. Blaise Pascal in 1654 proved Pascal's identity relating the sums of the p th powers of the first n positive integers for p = 0, 1, 2, ..., k . The Swiss mathematician Jakob Bernoulli (1654–1705)
3864-430: The synthesis of different programming languages . A programming language is a set of keywords , symbols , identifiers , and rules by which programmers can communicate instructions to the computer. They follow a set of rules called a syntax . Programming languages get their basis from formal languages . The purpose of defining a solution in terms of its formal language is to generate an algorithm to solve
3933-447: The tape back and forth, changing its contents as it performs an algorithm . The machine starts in the initial state, goes through a sequence of steps, and halts when it encounters the halt state. All present-day computers are Turing complete . The Electronic Numerical Integrator And Computer (ENIAC) was built between July 1943 and Fall 1945. It was a Turing complete , general-purpose computer that used 17,468 vacuum tubes to create
4002-553: The underlining problem. An algorithm is a sequence of simple instructions that solve a problem. The evolution of programming languages began when the EDSAC (1949) used the first stored computer program in its von Neumann architecture . Programming the EDSAC was in the first generation of programming language . Imperative languages specify a sequential algorithm using declarations , expressions , and statements : FORTRAN (1958)
4071-415: The value 1 / c + 1 . Thus Bernoulli's formula can be written if B 1 = 1/2 , recapturing the value Bernoulli gave to the coefficient at that position. The formula for ∑ k = 1 n k 9 {\displaystyle \textstyle \sum _{k=1}^{n}k^{9}} in the first half of the quotation by Bernoulli above contains an error at
4140-448: The wafer to build a matrix of metal–oxide–semiconductor (MOS) transistors. The MOS transistor is the primary component in integrated circuit chips . Originally, integrated circuit chips had their function set during manufacturing. During the 1960s, controlling the electrical flow migrated to programming a matrix of read-only memory (ROM). The matrix resembled a two-dimensional array of fuses. The process to embed instructions onto
4209-407: The zeroth-order derivative of f {\displaystyle f} is just f {\displaystyle f} ). Moreover, let f ( − 1 ) {\displaystyle f^{(-1)}} denote an antiderivative of f {\displaystyle f} . By the fundamental theorem of calculus , Thus the last formula can be further simplified to
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#17328519515704278-427: Was designed to expand C's capabilities by adding the object-oriented facilities of the language Simula . An object-oriented module is composed of two files. The definitions file is called the header file . Here is a C++ header file for the GRADE class in a simple school application: A constructor operation is a function with the same name as the class name. It is executed when the calling operation executes
4347-523: Was published posthumously in Ars Conjectandi in 1713. Seki Takakazu independently discovered the Bernoulli numbers and his result was published a year earlier, also posthumously, in 1712. However, Seki did not present his method as a formula based on a sequence of constants. Bernoulli's formula for sums of powers is the most useful and generalizable formulation to date. The coefficients in Bernoulli's formula are now called Bernoulli numbers, following
4416-436: Was replaced with point-contact transistors (1947) and bipolar junction transistors (late 1950s) mounted on a circuit board . During the 1960s , the aerospace industry replaced the circuit board with an integrated circuit chip . Robert Noyce , co-founder of Fairchild Semiconductor (1957) and Intel (1968), achieved a technological improvement to refine the production of field-effect transistors (1963). The goal
4485-418: Was the first to realize the existence of a single sequence of constants B 0 , B 1 , B 2 ,... which provide a uniform formula for all sums of powers. The joy Bernoulli experienced when he hit upon the pattern needed to compute quickly and easily the coefficients of his formula for the sum of the c th powers for any positive integer c can be seen from his comment. He wrote: Bernoulli's result
4554-405: Was to decompose large projects physically into separate files . A less obvious feature was to decompose large projects logically into abstract data types . At the time, languages supported concrete (scalar) datatypes like integer numbers, floating-point numbers, and strings of characters . Abstract datatypes are structures of concrete datatypes, with a new name assigned. For example,
4623-433: Was to design a language so managers could read the programs. However, the lack of structured statements hindered this goal. COBOL's development was tightly controlled, so dialects did not emerge to require ANSI standards. As a consequence, it was not changed for 15 years until 1974. The 1990s version did make consequential changes, like object-oriented programming . ALGOL (1960) stands for "ALGOrithmic Language". It had
4692-425: Was to develop a language that was comprehensive, easy to use, extendible, and would replace Cobol and Fortran. The result was a large and complex language that took a long time to compile . Computers manufactured until the 1970s had front-panel switches for manual programming. The computer program was written on paper for reference. An instruction was represented by a configuration of on/off settings. After setting
4761-423: Was unveiled as "The IBM Mathematical FORmula TRANslating system". It was designed for scientific calculations, without string handling facilities. Along with declarations , expressions , and statements , it supported: It succeeded because: However, non-IBM vendors also wrote Fortran compilers, but with a syntax that would likely fail IBM's compiler. The American National Standards Institute (ANSI) developed
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