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102-478: The right to study and modify a computer program entails that the source code —the preferred format for making changes—be made available to users of that program. While this is often called "access to source code" or "public availability", the Free Software Foundation (FSF) recommends against thinking in those terms, because it might give the impression that users have an obligation (as opposed to

204-547: A Symphony No. 0 . In some universities, including Oxford and Cambridge, "week 0" or occasionally "noughth week" refers to the week before the first week of lectures in a term. In Australia, some universities refer to this as "O week", which serves as a pun on " orientation week ". As a parallel, the introductory weeks at university educations in Sweden are generally called nollning (zeroing). The United States Air Force starts basic training each Wednesday, and

306-441: A compiler or an assembler . The resulting executable is machine code ready for the computer. Alternatively, source code can be executed without conversion via an interpreter . An interpreter loads the source code into memory. It simultaneously translates and executes each statement . A method that combines compilation and interpretation is to first produce bytecode . Bytecode is an intermediate representation of source code that

408-426: A negative or positive liberty . Due to their restrictions on distribution, not everyone considers copyleft licenses to be free. Conversely, a permissive license may provide an incentive to create non-free software by reducing the cost of developing restricted software. Since this is incompatible with the spirit of software freedom, many people consider permissive licenses to be less free than copyleft licenses. There

510-462: A source-code editor that can alert the programmer to common errors. Modification often includes code refactoring (improving the structure without changing functionality) and restructuring (improving structure and functionality at the same time). Nearly every change to code will introduce new bugs or unexpected ripple effects , which require another round of fixes. Code reviews by other developers are often used to scrutinize new code added to

612-430: A trade secret . Proprietary, secret source code and algorithms are widely used for sensitive government applications such as criminal justice , which results in black box behavior with a lack of transparency into the algorithm's methodology. The result is avoidance of public scrutiny of issues such as bias. Access to the source code (not just the object code) is essential to modifying it. Understanding existing code

714-412: A zeroth derivative of a function is the function itself, obtained by differentiating zero times. Such usage corresponds to naming an element not properly belonging to the sequence but preceding it: the zeroth derivative is not really a derivative at all. However, just as the first derivative precedes the second derivative , so also does the zeroth derivative (or the original function itself) precede

816-412: A , b , i all integers). This usage follows from design choices embedded in many influential programming languages , including C , Java , and Lisp . In these three, sequence types (C arrays, Java arrays and lists, and Lisp lists and vectors) are indexed beginning with the zero subscript. Particularly in C, where arrays are closely tied to pointer arithmetic, this makes for a simpler implementation:

918-441: A Zeroth Law to his Three Laws of Robotics , essentially making them four laws. A standard roulette wheel contains the number 0 as well as 1-36. It appears in green, so is classed as neither a "red" nor "black" number for betting purposes. The card game Uno has number cards running from 0 to 9 along with special cards, within each coloured suit. The Four Essential Freedoms of Free Software are numbered starting from zero. This

1020-529: A bottleneck. This led to the introduction of high-level programming languages such as Fortran in the mid-1950s. These languages abstracted away the details of the hardware, instead being designed to express algorithms that could be understood more easily by humans. As instructions distinct from the underlying computer hardware , software is therefore relatively recent, dating to these early high-level programming languages such as Fortran , Lisp , and Cobol . The invention of high-level programming languages

1122-474: A closed interval without resorting to obtuse conventions like [1, 0] . Because of this property, zero-based indexing potentially reduces off-by-one and fencepost errors . On the other hand, the repeat count n is calculated in advance, making the use of counting from 0 to n − 1 (inclusive) less intuitive. Some authors prefer one-based indexing, as it corresponds more closely to how entities are indexed in other contexts. Another property of this convention

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1224-528: A copy of the free application itself. Fees are usually charged for distribution on compact discs and bootable USB drives, or for services of installing or maintaining the operation of free software. Development of large, commercially used free software is often funded by a combination of user donations, crowdfunding , corporate contributions, and tax money. The SELinux project at the United States National Security Agency

1326-409: A drop in revenue to the proprietary software industry by about $ 60 billion per year. Eric S. Raymond argued that the term free software is too ambiguous and intimidating for the business community. Raymond promoted the term open-source software as a friendlier alternative for the business and corporate world. Source code In computing , source code , or simply code or source ,

1428-632: A fee. The Free Software Foundation encourages selling free software. As the Foundation has written, "distributing free software is an opportunity to raise funds for development. Don't waste it!". For example, the FSF's own recommended license (the GNU GPL ) states that "[you] may charge any price or no price for each copy that you convey, and you may offer support or warranty protection for a fee." Microsoft CEO Steve Ballmer stated in 2001 that "open source

1530-419: A freedom-respecting operating system , and to revive the spirit of cooperation once prevalent among hackers during the early days of computing. Free software differs from: For software under the purview of copyright to be free, it must carry a software license whereby the author grants users the aforementioned rights. Software that is not covered by copyright law, such as software in the public domain ,

1632-486: A lack of source code, there can exist additional obstacles keeping users from exercising freedom over a piece of software, such as software patents and digital rights management (more specifically, tivoization ). Free software can be a for-profit, commercial activity or not. Some free software is developed by volunteer computer programmers while other is developed by corporations; or even by both. Although both definitions refer to almost equivalent corpora of programs,

1734-410: A player's uniform number (in addition to the typical range of 1-99). The NFL voted to allow this from 2023 onwards. A chronological prequel of a series may be numbered as 0, such as Ring 0: Birthday or Zork Zero . The Swiss Federal Railways number certain classes of rolling stock from zero, for example, Re 460 000 to 118. In the realm of fiction, Isaac Asimov eventually added

1836-444: A project. The purpose of this phase is often to verify that the code meets style and maintainability standards and that it is a correct implementation of the software design . According to some estimates, code review dramatically reduce the number of bugs persisting after software testing is complete. Along with software testing that works by executing the code, static program analysis uses automated tools to detect problems with

1938-734: A remark distinguishing libre (freedom) software from gratis (zero price) software, the Free Software Foundation says: "Free software is a matter of liberty, not price. To understand the concept, you should think of 'free' as in ' free speech ', not as in 'free beer ' ". ( See Gratis versus libre . ) In the late 1990s, other groups published their own definitions that describe an almost identical set of software. The most notable are Debian Free Software Guidelines published in 1997, and The Open Source Definition , published in 1998. The BSD -based operating systems, such as FreeBSD , OpenBSD , and NetBSD , do not have their own formal definitions of free software. Users of these systems generally find

2040-563: A right) to give non-users a copy of the program. Although the term "free software" had already been used loosely in the past and other permissive software like the Berkeley Software Distribution released in 1978 existed, Richard Stallman is credited with tying it to the sense under discussion and starting the free software movement in 1983, when he launched the GNU Project : a collaborative effort to create

2142-496: A small set of licenses. The most popular of these licenses are: The Free Software Foundation and the Open Source Initiative both publish lists of licenses that they find to comply with their own definitions of free software and open-source software respectively: The FSF list is not prescriptive: free-software licenses can exist that the FSF has not heard about, or considered important enough to write about. So it

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2244-406: A specific platform, source code can be ported to a different machine and recompiled there. For the same source code, object code can vary significantly—not only based on the machine for which it is compiled, but also based on performance optimization from the compiler. Most programs do not contain all the resources needed to run them and rely on external libraries . Part of the compiler's function

2346-453: A widespread habit in this profession. Other programming languages, such as Fortran or COBOL , have array subscripts starting with one, because they were meant as high-level programming languages , and as such they had to have a correspondence to the usual ordinal numbers which predate the invention of the zero by a long time. Pascal allows the range of an array to be of any ordinal type (including enumerated types). APL allows setting

2448-399: A zero-based index due to the underlying address/offset logic mentioned above. To illustrate, suppose a is the memory address of the first element of an array, and i is the index of the desired element. To compute the address of the desired element, if the index numbers count from 1, the desired address is computed by this expression: where s is the size of each element. In contrast, if

2550-546: A zero-based indexing scheme, the first element is "element number zero"; likewise, the twelfth element is "element number eleven". Therefore, an analogy from the ordinal numbers to the quantity of objects numbered appears; the highest index of n objects will be n − 1 , and it refers to the n th element. For this reason, the first element is sometimes referred to as the zeroth element, in an attempt to avoid confusion. In mathematics , many sequences of numbers or of polynomials are indexed by nonnegative integers, for example,

2652-665: Is Christ Church on Harvard Square , whose address is 0 Garden Street. Formerly in Formula One , when a defending world champion did not compete in the following season, the number 1 was not assigned to any driver, but one driver of the world champion team would carry the number 0, and the other, number 2. This did happen both in 1993 and 1994 with Damon Hill carrying the number 0 in both seasons, as defending champion Nigel Mansell quit after 1992, and defending champion Alain Prost quit after 1993. However, in 2014

2754-405: Is a plain text computer program written in a programming language . A programmer writes the human readable source code to control the behavior of a computer . Since a computer, at base, only understands machine code , source code must be translated before a computer can execute it. The translation process can be implemented three ways. Source code can be converted into machine code by

2856-617: Is an example of a federally funded free-software project. Proprietary software, on the other hand, tends to use a different business model, where a customer of the proprietary application pays a fee for a license to legally access and use it. This license may grant the customer the ability to configure some or no parts of the software themselves. Often some level of support is included in the purchase of proprietary software, but additional support services (especially for enterprise applications) are usually available for an additional fee. Some proprietary software vendors will also customize software for

2958-442: Is an overarching term that can refer to a code's correct and efficient behavior, its reusability and portability , or the ease of modification. It is usually more cost-effective to build quality into the product from the beginning rather than try to add it later in the development process. Higher quality code will reduce lifetime cost to both suppliers and customers as it is more reliable and easier to maintain . Maintainability

3060-464: Is biased by counting more vulnerabilities for the free software systems, since their source code is accessible and their community is more forthcoming about what problems exist as a part of full disclosure , and proprietary software systems can have undisclosed societal drawbacks, such as disenfranchising less fortunate would-be users of free programs. As users can analyse and trace the source code, many more people with no commercial constraints can inspect

3162-451: Is debate over the security of free software in comparison to proprietary software, with a major issue being security through obscurity . A popular quantitative test in computer security is to use relative counting of known unpatched security flaws. Generally, users of this method advise avoiding products that lack fixes for known security flaws, at least until a fix is available. Free software advocates strongly believe that this methodology

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3264-570: Is followed by AD 1 . However, there is a year zero in astronomical year numbering (where it coincides with the Julian year 1 BC) and in ISO ;8601:2004 (where it coincides with the Gregorian year 1 BC), as well as in all Buddhist and Hindu calendars . In many countries, the ground floor in buildings is considered as floor number 0 rather than as the "1st floor",

3366-569: Is free as long as the source code is also in the public domain, or otherwise available without restrictions. Proprietary software uses restrictive software licences or EULAs and usually does not provide users with the source code. Users are thus legally or technically prevented from changing the software, and this results in reliance on the publisher to provide updates, help, and support. ( See also vendor lock-in and abandonware ). Users often may not reverse engineer , modify, or redistribute proprietary software. Beyond copyright law, contracts and

3468-452: Is frequently cited as a contributing factor to the maturation of their programming skills. Some people consider source code an expressive artistic medium . Source code often contains comments —blocks of text marked for the compiler to ignore. This content is not part of the program logic, but is instead intended to help readers understand the program. Companies often keep the source code confidential in order to hide algorithms considered

3570-448: Is in the use of modular arithmetic as implemented in modern computers. Usually, the modulo function maps any integer modulo N to one of the numbers 0, 1, 2, ..., N − 1 , where N ≥ 1 . Because of this, many formulas in algorithms (such as that for calculating hash table indices) can be elegantly expressed in code using the modulo operation when array indices start at zero. Pointer operations can also be expressed more elegantly on

3672-496: Is necessary to understand how it works and before modifying it. The rate of understanding depends both on the code base as well as the skill of the programmer. Experienced programmers have an easier time understanding what the code does at a high level. Software visualization is sometimes used to speed up this process. Many software programmers use an integrated development environment (IDE) to improve their productivity. IDEs typically have several features built in, including

3774-468: Is not available to commercial companies. The way the license is written, if you use any open-source software, you have to make the rest of your software open source." This misunderstanding is based on a requirement of copyleft licenses (like the GPL) that if one distributes modified versions of software, they must release the source and use the same license. This requirement does not extend to other software from

3876-493: Is not endorsed by the FSF and does not use Linux-libre, it is also a popular distribution available without kernel blobs by default since 2011. The Linux community uses the term "blob" to refer to all nonfree firmware in a kernel whereas OpenBSD uses the term to refer to device drivers. The FSF does not consider OpenBSD to be blob free under the Linux community's definition of blob. Selling software under any free-software licence

3978-400: Is not unusual to number from zero rather than one. If an array is used to represent a cycle, it is convenient to obtain the index with a modulo function , which can result in zero. With zero-based numbering, a range can be expressed as the half-open interval , [0, n ) , as opposed to the closed interval, [1, n ] . Empty ranges, which often occur in algorithms, are tricky to express with

4080-455: Is numbered R0. It was built after the ring road around Antwerp , but Brussels (being the capital city) was deemed deserving of a more basic number. Similarly the (unfinished) orbital motorway around Budapest in Hungary is called M0 . Zero is sometimes used in street addresses , especially in schemes where even numbers are one side of the street and odd numbers on the other. A case in point

4182-411: Is permissible, as is commercial use. This is true for licenses with or without copyleft . Since free software may be freely redistributed, it is generally available at little or no fee. Free software business models are usually based on adding value such as customization, accompanying hardware, support, training, integration, or certification. Exceptions exist however, where the user is charged to obtain

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4284-521: Is possible for a license to be free and not in the FSF list. The OSI list only lists licenses that have been submitted, considered and approved. All open-source licenses must meet the Open Source Definition in order to be officially recognized as open source software. Free software, on the other hand, is a more informal classification that does not rely on official recognition. Nevertheless, software licensed under licenses that do not meet

4386-470: Is quickly interpreted. The first programmable computers, which appeared at the end of the 1940s, were programmed in machine language (simple instructions that could be directly executed by the processor). Machine language was difficult to debug and was not portable between different computer systems. Initially, hardware resources were scarce and expensive, while human resources were cheaper. As programs grew more complex, programmer productivity became

4488-408: Is said to have zero-order intentionality if it shows "no intention of anything at all". This would include a situation where the organism's genetically predetermined phenotype results in a fitness benefit to itself, because it did not "intend" to express its genes. In the similar sense, a computer may be considered from this perspective a zero-order intentional entity, as it does not "intend" to express

4590-612: Is summarized at the Debian web site. It is rare that a license announced as being in-compliance with the FSF guidelines does not also meet the Open Source Definition , although the reverse is not necessarily true (for example, the NASA Open Source Agreement is an OSI-approved license, but non-free according to FSF). There are different categories of free software. Proponents of permissive and copyleft licenses disagree on whether software freedom should be viewed as

4692-631: Is that many software engineering courses do not emphasize it. Development engineers who know that they will not be responsible for maintaining the software do not have an incentive to build in maintainability. The situation varies worldwide, but in the United States before 1974, software and its source code was not copyrightable and therefore always public domain software . In 1974, the US Commission on New Technological Uses of Copyrighted Works (CONTU) decided that "computer programs, to

4794-477: Is the quality of software enabling it to be easily modified without breaking existing functionality. Following coding conventions such as using clear function and variable names that correspond to their purpose makes maintenance easier. Use of conditional loop statements only if the code could execute more than once, and eliminating code that will never execute can also increase understandability. Many software development organizations neglect maintainability during

4896-418: Is to link these files in such a way that the program can be executed by the hardware. Software developers often use configuration management to track changes to source code files ( version control ). The configuration management system also keeps track of which object code file corresponds to which version of the source code file. The number of lines of source code is often used as a metric when evaluating

4998-756: The Apache web server; and the Sendmail mail transport agent. Other influential examples include the Emacs text editor; the GIMP raster drawing and image editor; the X Window System graphical-display system; the LibreOffice office suite; and the TeX and LaTeX typesetting systems. From the 1950s up until the early 1970s, it was normal for computer users to have the software freedoms associated with free software, which

5100-519: The Bernoulli numbers and the Bell numbers . In both mechanics and statistics , the zeroth moment is defined, representing total mass in the case of physical density , or total probability, i.e. one, for a probability distribution . The zeroth law of thermodynamics was formulated after the first, second, and third laws, but considered more fundamental, thus its name. In biology, an organism

5202-830: The GNU operating system began in January 1984, and the Free Software Foundation (FSF) was founded in October 1985. He developed a free software definition and the concept of " copyleft ", designed to ensure software freedom for all. Some non-software industries are beginning to use techniques similar to those used in free software development for their research and development process; scientists, for example, are looking towards more open development processes, and hardware such as microchips are beginning to be developed with specifications released under copyleft licenses ( see

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5304-741: The Linux kernel and other device drivers motivated some developers in Ireland to launch gNewSense , a Linux-based distribution with all the binary blobs removed. The project received support from the Free Software Foundation and stimulated the creation, headed by the Free Software Foundation Latin America , of the Linux-libre kernel. As of October 2012, Trisquel is the most popular FSF endorsed Linux distribution ranked by Distrowatch (over 12 months). While Debian

5406-660: The OpenCores project, for instance ). Creative Commons and the free-culture movement have also been largely influenced by the free software movement. In 1983, Richard Stallman , longtime member of the hacker community at the MIT Artificial Intelligence Laboratory , announced the GNU Project, saying that he had become frustrated with the effects of the change in culture of the computer industry and its users. Software development for

5508-665: The United Space Alliance , which manages the computer systems for the International Space Station (ISS), regarding their May 2013 decision to migrate ISS computer systems from Windows to Linux The economic viability of free software has been recognized by large corporations such as IBM , Red Hat , and Sun Microsystems . Many companies whose core business is not in the IT sector choose free software for their Internet information and sales sites, due to

5610-581: The first derivative . Martin Richards , creator of the BCPL language (a precursor of C ), designed arrays initiating at 0 as the natural position to start accessing the array contents in the language, since the value of a pointer p used as an address accesses the position p + 0 in memory. BCPL was first compiled for the IBM ;7094 ; the language introduced no run-time indirection lookups , so

5712-409: The source code was distributed to use these programs. Software was also shared and distributed as printed source code ( Type-in program ) in computer magazines (like Creative Computing , SoftSide , Compute! , Byte , etc.) and books, like the bestseller BASIC Computer Games . By the early 1970s, the picture changed: software costs were dramatically increasing, a growing software industry

5814-557: The Free Software Definition cannot rightly be considered free software. Apart from these two organizations, the Debian project is seen by some to provide useful advice on whether particular licenses comply with their Debian Free Software Guidelines . Debian does not publish a list of approved licenses, so its judgments have to be tracked by checking what software they have allowed into their software archives. That

5916-426: The Free Software Foundation recommends using the term "free software" rather than " open-source software " (an alternative, yet similar, concept coined in 1998), because the goals and messaging are quite dissimilar. According to the Free Software Foundation, "Open source" and its associated campaign mostly focus on the technicalities of the public development model and marketing free software to businesses, while taking

6018-631: The GNU operating system began in January 1984, and the Free Software Foundation (FSF) was founded in October 1985. An article outlining the project and its goals was published in March 1985 titled the GNU Manifesto . The manifesto included significant explanation of the GNU philosophy, Free Software Definition and " copyleft " ideas. The Linux kernel , started by Linus Torvalds , was released as freely modifiable source code in 1991. The first licence

6120-581: The Internet. Users can easily download and install those applications via a package manager that comes included with most Linux distributions . The Free Software Directory maintains a large database of free-software packages. Some of the best-known examples include Linux-libre , Linux-based operating systems, the GNU Compiler Collection and C library ; the MySQL relational database;

6222-435: The array. Superficially, the fictitious element doesn't scale well to multidimensional arrays. Indexing multidimensional arrays from zero makes a naive (contiguous) conversion to a linear address space (systematically varying one index after the other) look simpler than when indexing from one. For instance, when mapping the three-dimensional array A[ P ][ N ][ M ] to a linear array L[ M⋅N⋅P ] , both with M ⋅ N ⋅ P elements,

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6324-463: The canon of his works, and he wrote gilt nicht ("doesn't count") on the score and a circle with a crossbar, intending it to mean "invalid". But posthumously, this work came to be known as Symphony No. 0 in D minor , even though it was actually written after Symphony No. 1 in C minor . There is an even earlier Symphony in F minor of Bruckner's, which is sometimes called No. 00 . The Russian composer Alfred Schnittke also wrote

6426-583: The code and find bugs and loopholes than a corporation would find practicable. According to Richard Stallman, user access to the source code makes deploying free software with undesirable hidden spyware functionality far more difficult than for proprietary software. Some quantitative studies have been done on the subject. In 2006, OpenBSD started the first campaign against the use of binary blobs in kernels . Blobs are usually freely distributable device drivers for hardware from vendors that do not reveal driver source code to users or developers. This restricts

6528-470: The code of the programs it runs. In biological or medical experiments, the first day of an experiment is often numbered as day 0. Patient zero (or index case ) is the initial patient in the population sample of an epidemiological investigation. The year zero does not exist in the widely used Gregorian calendar or in its predecessor, the Julian calendar . Under those systems, the year 1 BC

6630-426: The development phase, even though it will increase long-term costs. Technical debt is incurred when programmers, often out of laziness or urgency to meet a deadline, choose quick and dirty solutions rather than build maintainability into their code. A common cause is underestimates in software development effort estimation , leading to insufficient resources allocated to development. A challenge with maintainability

6732-473: The elements, gives to access the same element, which arguably looks more complicated. Of course, r ′ = r + 1, since [ z = z ′ – 1], [ y = y ′ – 1], and [ x = x ′ – 1]. A simple and everyday-life example is positional notation , which the invention of the zero made possible. In positional notation, tens, hundreds, thousands and all other digits start with zero, only units start at one. This situation can lead to some confusion in terminology. In

6834-490: The ethical issue of user rights very lightly or even antagonistically. Stallman has also stated that considering the practical advantages of free software is like considering the practical advantages of not being handcuffed, in that it is not necessary for an individual to consider practical reasons in order to realize that being handcuffed is undesirable in itself. The FSF also notes that "Open Source" has exactly one specific meaning in common English, namely that "you can look at

6936-406: The extent that they embody an author's original creation, are proper subject matter of copyright". Proprietary software is rarely distributed as source code. Although the term open-source software literally refers to public access to the source code , open-source software has additional requirements: free redistribution, permission to modify the source code and release derivative works under

7038-443: The first actual element. The indexing expression for a 1-based index would then be Hence, the efficiency benefit at run time of zero-based indexing is not inherent, but is an artifact of the decision to represent an array with the address of its first element rather than the address of the fictitious zeroth element. However, the address of that fictitious element could very well be the address of some other item in memory not related to

7140-495: The first week (of eight) is considered to begin with the following Sunday. The four days before that Sunday are often referred to as "zero week". 24-hour clocks and the international standard ISO 8601 use 0 to denote the first (zeroth) hour of the day, consistent with using the 0 to denote the first (zeroth) minute of the hour and the first (zeroth) second of the minute. Also, the 12-hour clocks used in Japan use 0 to denote

7242-408: The government charged that bundled software was anti-competitive . While some software might always be free, there would henceforth be a growing amount of software produced primarily for sale. In the 1970s and early 1980s, the software industry began using technical measures (such as only distributing binary copies of computer programs ) to prevent computer users from being able to study or adapt

7344-502: The hour immediately after midnight and noon in contrast to 12 used elsewhere, in order to avoid confusion whether 12 a.m. and 12 p.m. represent noon or midnight . Robert Crumb 's drawings for the first issue of Zap Comix were stolen, so he drew a whole new issue, which was published as issue 1. Later he re-inked his photocopies of the stolen artwork and published it as issue 0. The Brussels ring road in Belgium

7446-418: The index r in the linear array to access a specific element with L[ r ] = A[ z ][ y ][ x ] in zero-based indexing, i.e. [0 ≤ x < P ] , [0 ≤ y < N ] , [0 ≤ z < M ] , and [0 ≤ r < M ⋅ N ⋅ P ] , is calculated by Organizing all arrays with 1-based indices ( [1 ≤ x′ ≤ P ] , [1 ≤ y′ ≤ N ] , [1 ≤ z′ ≤ M ] , [1 ≤ r′ ≤ M ⋅ N ⋅ P ] ), and assuming an analogous arrangement of

7548-423: The index numbers count from 0, the expression becomes This simpler expression is more efficient to compute at run time . However, a language wishing to index arrays from 1 could adopt the convention that every array address is represented by a ′ = a – s ; that is, rather than using the address of the first array element, such a language would use the address of a fictitious element located immediately before

7650-526: The index origin to 0 or 1 during runtime programmatically. Some recent languages, such as Lua and Visual Basic , have adopted the same convention for the same reason. Zero is the lowest unsigned integer value, one of the most fundamental types in programming and hardware design. In computer science, zero is thus often used as the base case for many kinds of numerical recursion . Proofs and other sorts of mathematical reasoning in computer science often begin with zero. For these reasons, in computer science it

7752-657: The indirection optimization provided by these arrays was done at compile time. The optimization was nevertheless important. In 1982 Edsger W. Dijkstra in his pertinent note Why numbering should start at zero argued that arrays subscripts should start at zero as the latter being the most natural number . Discussing possible designs of array ranges by enclosing them in a chained inequality, combining sharp and standard inequalities to four possibilities, demonstrating that to his conviction zero-based arrays are best represented by non-overlapping index ranges, which start at zero, alluding to open, half-open and closed intervals as with

7854-456: The initial element is sometimes termed the zeroth element, rather than the first element; zeroth is a coined ordinal number corresponding to the number zero . In some cases, an object or value that does not (originally) belong to a given sequence, but which could be naturally placed before its initial element, may be termed the zeroth element. There is no wide agreement regarding the correctness of using zero as an ordinal (nor regarding

7956-475: The instructions can be carried out. After being compiled, the program can be saved as an object file and the loader (part of the operating system) can take this saved file and execute it as a process on the computer hardware. Some programming languages use an interpreter instead of a compiler. An interpreter converts the program into machine code at run time , which makes them 10 to 100 times slower than compiled programming languages. Software quality

8058-421: The list, but later added first in the list as it was considered very important. Freedoms 1 and 3 require source code to be available because studying and modifying software without its source code can range from highly impractical to nearly impossible. Thus, free software means that computer users have the freedom to cooperate with whom they choose, and to control the software they use. To summarize this into

8160-451: The lower initial capital investment and ability to freely customize the application packages. Most companies in the software business include free software in their commercial products if the licenses allow that. Free software is generally available at no cost and can result in permanently lower TCO ( total cost of ownership ) compared to proprietary software . With free software, businesses can fit software to their specific needs by changing

8262-420: The naming convention usually found in the United States of America. This makes a consistent set with underground floors marked with negative numbers. While the ordinal of 0 mostly finds use in communities directly connected to mathematics, physics, and computer science, there are also instances in classical music. The composer Anton Bruckner regarded his early Symphony in D minor to be unworthy of including in

8364-465: The productivity of computer programmers, the economic value of a code base, effort estimation for projects in development, and the ongoing cost of software maintenance after release. Source code is also used to communicate algorithms between people – e.g., code snippets online or in books. Computer programmers may find it helpful to review existing source code to learn about programming techniques. The sharing of source code between developers

8466-416: The programs they use; free software is a pure public good rather than a private good . Companies that contribute to free software increase commercial innovation . "We migrated key functions from Windows to Linux because we needed an operating system that was stable and reliable – one that would give us in-house control. So if we needed to patch, adjust, or adapt, we could." Official statement of

8568-416: The real numbers. Dijkstra's criteria for preferring this convention are in detail that it represents empty sequences in a more natural way ( a ≤ i < a  ?) than closed "intervals" ( a ≤ i ≤ ( a − 1) ? ), and that with half-open "intervals" of naturals, the length of a sub-sequence equals the upper minus the lower bound ( a ≤ i < b gives ( b − a ) possible values for i , with

8670-566: The same developer. The claim of incompatibility between commercial companies and free software is also a misunderstanding. There are several large companies, e.g. Red Hat and IBM (IBM acquired RedHat in 2019), which do substantial commercial business in the development of free software. Free software played a significant part in the development of the Internet, the World Wide Web and the infrastructure of dot-com companies . Free software allows users to cooperate in enhancing and refining

8772-457: The same license, and nondiscrimination between different uses—including commercial use. The free reusability of open-source software can speed up development. Zero-based numbering Zero-based numbering is a way of numbering in which the initial element of a sequence is assigned the index  0, rather than the index 1 as is typical in everyday non-mathematical or non-programming circumstances. Under zero-based numbering,

8874-511: The same set of software to be acceptable, but sometimes see copyleft as restrictive. They generally advocate permissive free software licenses , which allow others to use the software as they wish, without being legally forced to provide the source code. Their view is that this permissive approach is more free. The Kerberos , X11 , and Apache software licenses are substantially similar in intent and implementation. There are thousands of free applications and many operating systems available on

8976-431: The series moved to drivers carrying career-long personalised numbers, instead of team-allocated numbers, other than the defending champion still having the option to carry number 1. Therefore 0 is no longer used in this scenario. It is not clear if it is available as a driver's chosen number, or whether they must be between 2 and 99, but it has not been used to date under this system. Some team sports allow 0 to be chosen as

9078-435: The shift in climate surrounding the computer world and its users. In his initial declaration of the project and its purpose, he specifically cited as a motivation his opposition to being asked to agree to non-disclosure agreements and restrictive licenses which prohibited the free sharing of potentially profitable in-development software, a prohibition directly contrary to the traditional hacker ethic . Software development for

9180-461: The software applications as they saw fit. In 1980, copyright law was extended to computer programs. In 1983, Richard Stallman , one of the original authors of the popular Emacs program and a longtime member of the hacker community at the MIT Artificial Intelligence Laboratory , announced the GNU Project , the purpose of which was to produce a completely non-proprietary Unix-compatible operating system, saying that he had become frustrated with

9282-443: The software themselves or by hiring programmers to modify it for them. Free software often has no warranty, and more importantly, generally does not assign legal liability to anyone. However, warranties are permitted between any two parties upon the condition of the software and its usage. Such an agreement is made separately from the free software license. A report by Standish Group estimates that adoption of free software has caused

9384-414: The source code. Many IDEs support code analysis tools, which might provide metrics on the clarity and maintainability of the code. Debuggers are tools that often enable programmers to step through execution while keeping track of which source code corresponds to each change of state. Source code files in a high-level programming language must go through a stage of preprocessing into machine code before

9486-527: The source code." It states that while the term "Free Software" can lead to two different interpretations, at least one of them is consistent with the intended meaning unlike the term "Open Source". The loan adjective " libre " is often used to avoid the ambiguity of the word "free" in the English language , and the ambiguity with the older usage of "free software" as public-domain software. ( See Gratis versus libre . ) The first formal definition of free software

9588-416: The subscript refers to an offset from the starting position of an array, so the first element has an offset of zero. Referencing memory by an address and an offset is represented directly in computer hardware on virtually all computer architectures, so this design detail in C makes compilation easier, at the cost of some human factors. In this context using "zeroth" as an ordinal is not strictly correct, but

9690-696: The use of the term zeroth ), as it creates ambiguity for all subsequent elements of the sequence when lacking context. Numbering sequences starting at 0 is quite common in mathematics notation, in particular in combinatorics , though programming languages for mathematics usually index from 1. In computer science , array indices usually start at 0 in modern programming languages, so computer programmers might use zeroth in situations where others might use first , and so forth. In some mathematical contexts, zero-based numbering can be used without confusion, when ordinal forms have well established meaning with an obvious candidate to come before first ; for instance,

9792-495: The users' freedom effectively to modify the software and distribute modified versions. Also, since the blobs are undocumented and may have bugs , they pose a security risk to any operating system whose kernel includes them. The proclaimed aim of the campaign against blobs is to collect hardware documentation that allows developers to write free software drivers for that hardware, ultimately enabling all free operating systems to become or remain blob-free. The issue of binary blobs in

9894-675: Was a proprietary software licence. However, with version 0.12 in February 1992, he relicensed the project under the GNU General Public License . Much like Unix, Torvalds' kernel attracted the attention of volunteer programmers. FreeBSD and NetBSD (both derived from 386BSD ) were released as free software when the USL v. BSDi lawsuit was settled out of court in 1993. OpenBSD forked from NetBSD in 1995. Also in 1995, The Apache HTTP Server , commonly referred to as Apache,

9996-402: Was competing with the hardware manufacturer's bundled software products (free in that the cost was included in the hardware cost), leased machines required software support while providing no revenue for software, and some customers able to better meet their own needs did not want the costs of "free" software bundled with hardware product costs. In United States vs. IBM , filed January 17, 1969,

10098-416: Was published by FSF in February 1986. That definition, written by Richard Stallman , is still maintained today and states that software is free software if people who receive a copy of the software have the following four freedoms. The numbering begins with zero, not only as a spoof on the common usage of zero-based numbering in programming languages, but also because "Freedom 0" was not initially included in

10200-476: Was released under the Apache License 1.0 . All free-software licenses must grant users all the freedoms discussed above. However, unless the applications' licenses are compatible, combining programs by mixing source code or directly linking binaries is problematic, because of license technicalities . Programs indirectly connected together may avoid this problem. The majority of free software falls under

10302-480: Was simultaneous with the compilers needed to translate the source code automatically into machine code that can be directly executed on the computer hardware . Source code is the form of code that is modified directly by humans, typically in a high-level programming language. Object code can be directly executed by the machine and is generated automatically from the source code, often via an intermediate step, assembly language . While object code will only work on

10404-406: Was typically public-domain software . Software was commonly shared by individuals who used computers and by hardware manufacturers who welcomed the fact that people were making software that made their hardware useful. Organizations of users and suppliers, for example, SHARE , were formed to facilitate exchange of software. As software was often written in an interpreted language such as BASIC ,

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