In mathematics , a multiplication table (sometimes, less formally, a times table ) is a mathematical table used to define a multiplication operation for an algebraic system.
100-481: In computer science , a lookup table ( LUT ) is an array that replaces runtime computation with a simpler array indexing operation, in a process termed as direct addressing . The savings in processing time can be significant, because retrieving a value from memory is often faster than carrying out an "expensive" computation or input/output operation. The tables may be precalculated and stored in static program storage, calculated (or "pre-fetched" ) as part of
200-512: A LOOKUP function among its original 20 functions. This has been followed by subsequent spreadsheets, such as Microsoft Excel , and complemented by specialized VLOOKUP and HLOOKUP functions to simplify lookup in a vertical or horizontal table. In Microsoft Excel the XLOOKUP function has been rolled out starting 28 August 2019. Although the performance of a LUT is a guaranteed O ( 1 ) {\displaystyle O(1)} for
300-570: A discipline, computer science spans a range of topics from theoretical studies of algorithms and the limits of computation to the practical issues of implementing computing systems in hardware and software. CSAB , formerly called Computing Sciences Accreditation Board—which is made up of representatives of the Association for Computing Machinery (ACM), and the IEEE Computer Society (IEEE CS) —identifies four areas that it considers crucial to
400-724: A distinct academic discipline in the 1950s and early 1960s. The world's first computer science degree program, the Cambridge Diploma in Computer Science , began at the University of Cambridge Computer Laboratory in 1953. The first computer science department in the United States was formed at Purdue University in 1962. Since practical computers became available, many applications of computing have become distinct areas of study in their own rights. Although first proposed in 1956,
500-465: A grayscale picture of the planet Saturn could be transformed into a color image to emphasize the differences in its rings. In image processing, lookup tables are often called LUT s (or 3DLUT), and give an output value for each of a range of index values. One common LUT, called the colormap or palette , is used to determine the colors and intensity values with which a particular image will be displayed. In computed tomography , "windowing" refers to
600-472: A high degree of precision: However, this can be expensive to compute, especially on slow processors, and there are many applications, particularly in traditional computer graphics , that need to compute many thousands of sine values every second. A common solution is to initially compute the sine of many evenly distributed values, and then to find the sine of x we choose the sine of the value closest to x through array indexing operation. This will be close to
700-410: A lookup operation, no two entities or values can have the same key k {\displaystyle k} . When the size of universe ∪ {\displaystyle \cup } —where the keys are drawn—is large, it might be impractical or impossible to be stored in memory . Hence, in this case, a hash table would be a preferable alternative. For a trivial hash function lookup,
800-619: A lookup table was responsible for Intel's infamous floating-point divide bug . Functions of a single variable (such as sine and cosine) may be implemented by a simple array. Functions involving two or more variables require multidimensional array indexing techniques. The latter case may thus employ a two-dimensional array of power[x][y] to replace a function to calculate x for a limited range of x and y values. Functions that have more than one result may be implemented with lookup tables that are arrays of structures. As mentioned, there are intermediate solutions that use tables in combination with
900-423: A lookup table which in turn uses trivial hash function for better performance. The bits array, bits_set with 256 entries is constructed by giving the number of one bits set in each possible byte value (e.g. 0x00 = 0, 0x01 = 1, 0x02 = 1, and so on). Although a runtime algorithm can be used to generate the bits_set array, it's an inefficient usage of clock cycles when the size is taken into consideration, hence
1000-448: A lookup table. The table is built with very fast memory instead of being stored on slower external memory, and maintains two pieces of data for a sub-range of bits composing an external memory (or disk) address (notably the lowest bits of any possible external address): A single (fast) lookup is performed to read the tag in the lookup table at the index specified by the lowest bits of the desired external storage address, and to determine if
1100-464: A mathematical discipline argue that computer programs are physical realizations of mathematical entities and programs that can be deductively reasoned through mathematical formal methods . Computer scientists Edsger W. Dijkstra and Tony Hoare regard instructions for computer programs as mathematical sentences and interpret formal semantics for programming languages as mathematical axiomatic systems . A number of computer scientists have argued for
SECTION 10
#17329416063191200-443: A mathematics emphasis and with a numerical orientation consider alignment with computational science . Both types of departments tend to make efforts to bridge the field educationally if not across all research. Despite the word science in its name, there is debate over whether or not computer science is a discipline of science, mathematics, or engineering. Allen Newell and Herbert A. Simon argued in 1975, Computer science
1300-463: A network while using concurrency, this is known as a distributed system. Computers within that distributed system have their own private memory, and information can be exchanged to achieve common goals. This branch of computer science aims to manage networks between computers worldwide. Computer security is a branch of computer technology with the objective of protecting information from unauthorized access, disruption, or modification while maintaining
1400-540: A number of terms for the practitioners of the field of computing were suggested in the Communications of the ACM — turingineer , turologist , flow-charts-man , applied meta-mathematician , and applied epistemologist . Three months later in the same journal, comptologist was suggested, followed next year by hypologist . The term computics has also been suggested. In Europe, terms derived from contracted translations of
1500-495: A particular kind of mathematically based technique for the specification , development and verification of software and hardware systems. The use of formal methods for software and hardware design is motivated by the expectation that, as in other engineering disciplines, performing appropriate mathematical analysis can contribute to the reliability and robustness of a design. They form an important theoretical underpinning for software engineering, especially where safety or security
1600-410: A precomputed table is used—although a compile time script could be used to dynamically generate and append the table to the source file . Sum of ones in each byte of the integer can be calculated through trivial hash function lookup on each byte; thus, effectively avoiding branches resulting in considerable improvement in performance. "Lookup tables (LUTs) are an excellent technique for optimizing
1700-534: A program's initialization phase ( memoization ), or even stored in hardware in application-specific platforms. Lookup tables are also used extensively to validate input values by matching against a list of valid (or invalid) items in an array and, in some programming languages, may include pointer functions (or offsets to labels) to process the matching input. FPGAs also make extensive use of reconfigurable, hardware-implemented, lookup tables to provide programmable hardware functionality. LUTs differ from hash tables in
1800-466: A prohibitively long time, it may make the use of a lookup table an inappropriate solution. As previously stated however, tables can be statically defined in many cases. Most computers only perform basic arithmetic operations and cannot directly calculate the sine of a given value. Instead, they use the CORDIC algorithm or a complex formula such as the following Taylor series to compute the value of sine to
1900-403: A related concept for determining how to display the intensity of measured radiation. A classic example of reducing run-time computations using lookup tables is to obtain the result of a trigonometry calculation, such as the sine of a value. Calculating trigonometric functions can substantially slow a computing application. The same application can finish much sooner when it first precalculates
2000-512: A significant amount of computer science does not involve the study of computers themselves. Because of this, several alternative names have been proposed. Certain departments of major universities prefer the term computing science , to emphasize precisely that difference. Danish scientist Peter Naur suggested the term datalogy , to reflect the fact that the scientific discipline revolves around data and data treatment, while not necessarily involving computers. The first scientific institution to use
2100-415: A small amount of computation, often using interpolation . Pre-calculation combined with interpolation can produce higher accuracy for values that fall between two precomputed values. This technique requires slightly more time to be performed but can greatly enhance accuracy in applications that require it. Depending on the values being precomputed, precomputation with interpolation can also be used to shrink
SECTION 20
#17329416063192200-410: A specific application. Codes are used for data compression , cryptography , error detection and correction , and more recently also for network coding . Codes are studied for the purpose of designing efficient and reliable data transmission methods. Data structures and algorithms are the studies of commonly used computational methods and their computational efficiency. Programming language theory
2300-406: A way that, to retrieve a value v {\displaystyle v} with key k {\displaystyle k} , a hash table would store the value v {\displaystyle v} in the slot h ( k ) {\displaystyle h(k)} where h {\displaystyle h} is a hash function i.e. k {\displaystyle k}
2400-6989: Is commutative , many schools use a smaller table as below. Some schools even remove the first column since 1 is the multiplicative identity . The traditional rote learning of multiplication was based on memorization of columns in the table, arranged as follows. 0 × 0 = 0 1 × 0 = 0 2 × 0 = 0 3 × 0 = 0 4 × 0 = 0 5 × 0 = 0 6 × 0 = 0 7 × 0 = 0 8 × 0 = 0 9 × 0 = 0 10 × 0 = 0 11 × 0 = 0 12 × 0 = 0 0 × 1 = 0 1 × 1 = 1 2 × 1 = 2 3 × 1 = 3 4 × 1 = 4 5 × 1 = 5 6 × 1 = 6 7 × 1 = 7 8 × 1 = 8 9 × 1 = 9 10 × 1 = 10 11 × 1 = 11 12 × 1 = 12 0 × 2 = 0 1 × 2 = 2 2 × 2 = 4 3 × 2 = 6 4 × 2 = 8 5 × 2 = 10 6 × 2 = 12 7 × 2 = 14 8 × 2 = 16 9 × 2 = 18 10 × 2 = 20 11 × 2 = 22 12 × 2 = 24 0 × 3 = 0 1 × 3 = 3 2 × 3 = 6 3 × 3 = 9 4 × 3 = 12 5 × 3 = 15 6 × 3 = 18 7 × 3 = 21 8 × 3 = 24 9 × 3 = 27 10 × 3 = 30 11 × 3 = 33 12 × 3 = 36 0 × 4 = 0 1 × 4 = 4 2 × 4 = 8 3 × 4 = 12 4 × 4 = 16 5 × 4 = 20 6 × 4 = 24 7 × 4 = 28 8 × 4 = 32 9 × 4 = 36 10 × 4 = 40 11 × 4 = 44 12 × 4 = 48 0 × 5 = 0 1 × 5 = 5 2 × 5 = 10 3 × 5 = 15 4 × 5 = 20 5 × 5 = 25 6 × 5 = 30 7 × 5 = 35 8 × 5 = 40 9 × 5 = 45 10 × 5 = 50 11 × 5 = 55 12 × 5 = 60 0 × 6 = 0 1 × 6 = 6 2 × 6 = 12 3 × 6 = 18 4 × 6 = 24 5 × 6 = 30 6 × 6 = 36 7 × 6 = 42 8 × 6 = 48 9 × 6 = 54 10 × 6 = 60 11 × 6 = 66 12 × 6 = 72 0 × 7 = 0 1 × 7 = 7 2 × 7 = 14 3 × 7 = 21 4 × 7 = 28 5 × 7 = 35 6 × 7 = 42 7 × 7 = 49 8 × 7 = 56 9 × 7 = 63 10 × 7 = 70 11 × 7 = 77 12 × 7 = 84 0 × 8 = 0 1 × 8 = 8 2 × 8 = 16 3 × 8 = 24 4 × 8 = 32 5 × 8 = 40 6 × 8 = 48 7 × 8 = 56 8 × 8 = 64 9 × 8 = 72 10 × 8 = 80 11 × 8 = 88 12 × 8 = 96 0 × 9 = 0 1 × 9 = 9 2 × 9 = 18 3 × 9 = 27 4 × 9 = 36 5 × 9 = 45 6 × 9 = 54 7 × 9 = 63 8 × 9 = 72 9 × 9 = 81 10 × 9 = 90 11 × 9 = 99 12 × 9 = 108 0 × 10 = 0 1 × 10 = 10 2 × 10 = 20 3 × 10 = 30 4 × 10 = 40 5 × 10 = 50 6 × 10 = 60 7 × 10 = 70 8 × 10 = 80 9 × 10 = 90 10 × 10 = 100 11 × 10 = 110 12 × 10 = 120 0 × 11 = 0 1 × 11 = 11 2 × 11 = 22 3 × 11 = 33 4 × 11 = 44 5 × 11 = 55 6 × 11 = 66 7 × 11 = 77 8 × 11 = 88 9 × 11 = 99 10 × 11 = 110 11 × 11 = 121 12 × 11 = 132 0 × 12 = 0 1 × 12 = 12 2 × 12 = 24 3 × 12 = 36 4 × 12 = 48 5 × 12 = 60 6 × 12 = 72 7 × 12 = 84 8 × 12 = 96 9 × 12 = 108 10 × 12 = 120 11 × 12 = 132 12 × 12 = 144 This form of writing
2500-415: Is a branch of computer science that deals with the design, implementation, analysis, characterization, and classification of programming languages and their individual features . It falls within the discipline of computer science, both depending on and affecting mathematics, software engineering, and linguistics . It is an active research area, with numerous dedicated academic journals. Formal methods are
2600-500: Is an efficient way of encoding Boolean logic functions, and LUTs with 4-6 bits of input are in fact the key component of modern field-programmable gate arrays (FPGAs) which provide reconfigurable hardware logic capabilities. In data acquisition and control systems , lookup tables are commonly used to undertake the following operations in: In some systems, polynomials may also be defined in place of lookup tables for these calculations. Computer science Computer science
2700-422: Is an empirical discipline. We would have called it an experimental science, but like astronomy, economics, and geology, some of its unique forms of observation and experience do not fit a narrow stereotype of the experimental method. Nonetheless, they are experiments. Each new machine that is built is an experiment. Actually constructing the machine poses a question to nature; and we listen for the answer by observing
2800-484: Is an open problem in the theory of computation. Information theory, closely related to probability and statistics , is related to the quantification of information. This was developed by Claude Shannon to find fundamental limits on signal processing operations such as compressing data and on reliably storing and communicating data. Coding theory is the study of the properties of codes (systems for converting information from one form to another) and their fitness for
2900-475: Is associated in the popular mind with robotic development , but the main field of practical application has been as an embedded component in areas of software development , which require computational understanding. The starting point in the late 1940s was Alan Turing's question " Can computers think? ", and the question remains effectively unanswered, although the Turing test is still used to assess computer output on
3000-543: Is connected to many other fields in computer science, including computer vision , image processing , and computational geometry , and is heavily applied in the fields of special effects and video games . Information can take the form of images, sound, video or other multimedia. Bits of information can be streamed via signals . Its processing is the central notion of informatics, the European view on computing, which studies information processing algorithms independently of
3100-409: Is considered by some to have a much closer relationship with mathematics than many scientific disciplines, with some observers saying that computing is a mathematical science. Early computer science was strongly influenced by the work of mathematicians such as Kurt Gödel , Alan Turing , John von Neumann , Rózsa Péter and Alonzo Church and there continues to be a useful interchange of ideas between
Lookup table - Misplaced Pages Continue
3200-555: Is continuous and has continuous first derivative , one should use the cubic Hermite spline . When using interpolation, the size of the lookup table can be reduced by using nonuniform sampling , which means that where the function is close to straight, we use few sample points, while where it changes value quickly we use more sample points to keep the approximation close to the real curve. For more information, see interpolation . Storage caches (including disk caches for files, or processor caches for either code or data) work also like
3300-508: Is determining what can and cannot be automated. The Turing Award is generally recognized as the highest distinction in computer science. The earliest foundations of what would become computer science predate the invention of the modern digital computer . Machines for calculating fixed numerical tasks such as the abacus have existed since antiquity, aiding in computations such as multiplication and division. Algorithms for performing computations have existed since antiquity, even before
3400-630: Is generally considered the province of disciplines other than computer science. For example, the study of computer hardware is usually considered part of computer engineering , while the study of commercial computer systems and their deployment is often called information technology or information systems . However, there has been exchange of ideas between the various computer-related disciplines. Computer science research also often intersects other disciplines, such as cognitive science , linguistics , mathematics , physics , biology , Earth science , statistics , philosophy , and logic . Computer science
3500-584: Is intended to organize, store, and retrieve large amounts of data easily. Digital databases are managed using database management systems to store, create, maintain, and search data, through database models and query languages . Data mining is a process of discovering patterns in large data sets. The philosopher of computing Bill Rapaport noted three Great Insights of Computer Science : Programming languages can be used to accomplish different tasks in different ways. Common programming paradigms include: Many languages offer support for multiple paradigms, making
3600-426: Is involved. Formal methods are a useful adjunct to software testing since they help avoid errors and can also give a framework for testing. For industrial use, tool support is required. However, the high cost of using formal methods means that they are usually only used in the development of high-integrity and life-critical systems , where safety or security is of utmost importance. Formal methods are best described as
3700-545: Is mathematical and abstract in spirit, but it derives its motivation from practical and everyday computation. It aims to understand the nature of computation and, as a consequence of this understanding, provide more efficient methodologies. According to Peter Denning, the fundamental question underlying computer science is, "What can be automated?" Theory of computation is focused on answering fundamental questions about what can be computed and what amount of resources are required to perform those computations. In an effort to answer
3800-512: Is necessary to memorize the table up to 9 × 9. The oldest known multiplication tables were used by the Babylonians about 4000 years ago. However, they used a base of 60. The oldest known tables using a base of 10 are the Chinese decimal multiplication table on bamboo strips dating to about 305 BC, during China's Warring States period. The multiplication table is sometimes attributed to
3900-519: Is of high quality, affordable, maintainable, and fast to build. It is a systematic approach to software design, involving the application of engineering practices to software. Software engineering deals with the organizing and analyzing of software—it does not just deal with the creation or manufacture of new software, but its internal arrangement and maintenance. For example software testing , systems engineering , technical debt and software development processes . Artificial intelligence (AI) aims to or
4000-524: Is on a wax tablet dated to the 1st century AD and currently housed in the British Museum . In 493 AD, Victorius of Aquitaine wrote a 98-column multiplication table which gave (in Roman numerals ) the product of every number from 2 to 50 times and the rows were "a list of numbers starting with one thousand, descending by hundreds to one hundred, then descending by tens to ten, then by ones to one, and then
4100-432: Is possible to construct a lookup table for a required operation. One is the amount of memory that is available: one cannot construct a lookup table larger than the space available for the table, although it is possible to construct disk-based lookup tables at the expense of lookup time. The other is the time required to compute the table values in the first instance; although this usually needs to be done only once, if it takes
Lookup table - Misplaced Pages Continue
4200-575: Is required to synthesize goal-orientated processes such as problem-solving, decision-making, environmental adaptation, learning, and communication found in humans and animals. From its origins in cybernetics and in the Dartmouth Conference (1956), artificial intelligence research has been necessarily cross-disciplinary, drawing on areas of expertise such as applied mathematics , symbolic logic, semiotics , electrical engineering , philosophy of mind , neurophysiology , and social intelligence . AI
4300-432: Is the field of study and research concerned with the design and use of computer systems , mainly based on the analysis of the interaction between humans and computer interfaces . HCI has several subfields that focus on the relationship between emotions , social behavior and brain activity with computers . Software engineering is the study of designing, implementing, and modifying the software in order to ensure it
4400-783: Is the field of study concerned with constructing mathematical models and quantitative analysis techniques and using computers to analyze and solve scientific problems. A major usage of scientific computing is simulation of various processes, including computational fluid dynamics , physical, electrical, and electronic systems and circuits, as well as societies and social situations (notably war games) along with their habitats, among many others. Modern computers enable optimization of such designs as complete aircraft. Notable in electrical and electronic circuit design are SPICE, as well as software for physical realization of new (or modified) designs. The latter includes essential design software for integrated circuits . Human–computer interaction (HCI)
4500-417: Is the quick development of this relatively new field requires rapid review and distribution of results, a task better handled by conferences than by journals. Multiplication table The decimal multiplication table was traditionally taught as an essential part of elementary arithmetic around the world, as it lays the foundation for arithmetic operations with base-ten numbers. Many educators believe it
4600-562: Is the study of computation , information , and automation . Computer science spans theoretical disciplines (such as algorithms , theory of computation , and information theory ) to applied disciplines (including the design and implementation of hardware and software ). Algorithms and data structures are central to computer science. The theory of computation concerns abstract models of computation and general classes of problems that can be solved using them. The fields of cryptography and computer security involve studying
4700-454: Is used to compute the slot, while in the case of LUT, the value v {\displaystyle v} is stored in slot k {\displaystyle k} , thus directly addressable. Before the advent of computers, lookup tables of values were used to speed up hand calculations of complex functions, such as in trigonometry , logarithms , and statistical density functions. In ancient (499 AD) India, Aryabhata created one of
4800-424: The population function . For example, the decimal number "37" is "00100101" in binary, so it contains three bits that are set to binary "1". A simple example of C code, designed to count the 1 bits in a int , might look like this: The above implementation requires 32 operations for an evaluation of a 32-bit value, which can potentially take several clock cycles due to branching . It can be " unrolled " into
4900-475: The "technocratic paradigm" (which might be found in engineering approaches, most prominently in software engineering), and the "scientific paradigm" (which approaches computer-related artifacts from the empirical perspective of natural sciences , identifiable in some branches of artificial intelligence ). Computer science focuses on methods involved in design, specification, programming, verification, implementation and testing of human-made computing systems. As
5000-460: The "tens" unit as you would normally do as usual to the "ones" unit. For example, to recall all the multiples of 7: Tables can also define binary operations on groups , fields , rings , and other algebraic systems . In such contexts they are called Cayley tables . For every natural number n , addition and multiplication in Z n , the ring of integers modulo n , is described by an n by n table. (See Modular arithmetic .) For example,
5100-554: The 100th anniversary of the invention of the arithmometer, Torres presented in Paris the Electromechanical Arithmometer , a prototype that demonstrated the feasibility of an electromechanical analytical engine, on which commands could be typed and the results printed automatically. In 1937, one hundred years after Babbage's impossible dream, Howard Aiken convinced IBM, which was making all kinds of punched card equipment and
SECTION 50
#17329416063195200-451: The 2nd of the only two designs for mechanical analytical engines in history. In 1914, the Spanish engineer Leonardo Torres Quevedo published his Essays on Automatics , and designed, inspired by Babbage, a theoretical electromechanical calculating machine which was to be controlled by a read-only program. The paper also introduced the idea of floating-point arithmetic . In 1920, to celebrate
5300-618: The Analytical Engine, Ada Lovelace wrote, in one of the many notes she included, an algorithm to compute the Bernoulli numbers , which is considered to be the first published algorithm ever specifically tailored for implementation on a computer. Around 1885, Herman Hollerith invented the tabulator , which used punched cards to process statistical information; eventually his company became part of IBM . Following Babbage, although unaware of his earlier work, Percy Ludgate in 1909 published
5400-703: The Japanese multiplication table uses non-standard pronunciations for numbers in some specific instances (such as the replacement of san roku with saburoku ). A bundle of 21 bamboo slips dated 305 BC in the Warring States period in the Tsinghua Bamboo Slips (清華簡) collection is the world's earliest known example of a decimal multiplication table. In 1989, the National Council of Teachers of Mathematics (NCTM) developed new standards which were based on
5500-547: The Machine Organization department in IBM's main research center in 1959. Concurrency is a property of systems in which several computations are executing simultaneously, and potentially interacting with each other. A number of mathematical models have been developed for general concurrent computation including Petri nets , process calculi and the parallel random access machine model. When multiple computers are connected in
5600-546: The UK (as in the School of Informatics, University of Edinburgh ). "In the U.S., however, informatics is linked with applied computing, or computing in the context of another domain." A folkloric quotation, often attributed to—but almost certainly not first formulated by— Edsger Dijkstra , states that "computer science is no more about computers than astronomy is about telescopes." The design and deployment of computers and computer systems
5700-516: The accessibility and usability of the system for its intended users. Historical cryptography is the art of writing and deciphering secret messages. Modern cryptography is the scientific study of problems relating to distributed computations that can be attacked. Technologies studied in modern cryptography include symmetric and asymmetric encryption , digital signatures , cryptographic hash functions , key-agreement protocols , blockchain , zero-knowledge proofs , and garbled circuits . A database
5800-457: The address signal and whose inputs are the values of the elements contained in the array. These values can either be hard-wired, as in an ASIC whose purpose is specific to a function, or provided by D latches which allow for configurable values. ( ROM , EPROM , EEPROM , or RAM .) An n -bit LUT can encode any n -input Boolean function by storing the truth table of the function in the LUT. This
5900-515: The ancient Greek mathematician Pythagoras (570–495 BC). It is also called the Table of Pythagoras in many languages (for example French, Italian and Russian), sometimes in English. The Greco-Roman mathematician Nichomachus (60–120 AD), a follower of Neopythagoreanism , included a multiplication table in his Introduction to Arithmetic , whereas the oldest surviving Greek multiplication table
6000-433: The application of a fairly broad variety of theoretical computer science fundamentals, in particular logic calculi, formal languages , automata theory , and program semantics , but also type systems and algebraic data types to problems in software and hardware specification and verification. Computer graphics is the study of digital visual contents and involves the synthesis and manipulation of image data. The study
6100-585: The belief that all students should learn higher-order thinking skills, which recommended reduced emphasis on the teaching of traditional methods that relied on rote memorization, such as multiplication tables. Widely adopted texts such as Investigations in Numbers, Data, and Space (widely known as TERC after its producer, Technical Education Research Centers) omitted aids such as multiplication tables in early editions. NCTM made it clear in their 2006 Focal Points that basic mathematics facts must be learned, though there
SECTION 60
#17329416063196200-410: The binary number system. In 1820, Thomas de Colmar launched the mechanical calculator industry when he invented his simplified arithmometer , the first calculating machine strong enough and reliable enough to be used daily in an office environment. Charles Babbage started the design of the first automatic mechanical calculator , his Difference Engine , in 1822, which eventually gave him the idea of
6300-565: The cache may also become a problem. Table accesses for large tables will almost certainly cause a cache miss . This phenomenon is increasingly becoming an issue as processors outpace memory. A similar issue appears in rematerialization , a compiler optimization . In some environments, such as the Java programming language , table lookups can be even more expensive due to mandatory bounds-checking involving an additional comparison and branch for each lookup. There are two fundamental limitations on when it
6400-402: The correct value because sine is a continuous function with a bounded rate of change. For example: Unfortunately, the table requires quite a bit of space: if IEEE double-precision floating-point numbers are used, over 16,000 bytes would be required. We can use fewer samples, but then our precision will significantly worsen. One good solution is linear interpolation , which draws a line between
6500-750: The design and principles behind developing software. Areas such as operating systems , networks and embedded systems investigate the principles and design behind complex systems . Computer architecture describes the construction of computer components and computer-operated equipment. Artificial intelligence and machine learning aim to synthesize goal-orientated processes such as problem-solving, decision-making, environmental adaptation, planning and learning found in humans and animals. Within artificial intelligence, computer vision aims to understand and process image and video data, while natural language processing aims to understand and process textual and linguistic data. The fundamental concern of computer science
6600-467: The development of sophisticated computing equipment. Wilhelm Schickard designed and constructed the first working mechanical calculator in 1623. In 1673, Gottfried Leibniz demonstrated a digital mechanical calculator, called the Stepped Reckoner . Leibniz may be considered the first computer scientist and information theorist, because of various reasons, including the fact that he documented
6700-583: The discipline of computer science: theory of computation , algorithms and data structures , programming methodology and languages , and computer elements and architecture . In addition to these four areas, CSAB also identifies fields such as software engineering, artificial intelligence, computer networking and communication, database systems, parallel computation, distributed computation, human–computer interaction, computer graphics, operating systems, and numerical and symbolic computation as being important areas of computer science. Theoretical computer science
6800-424: The distinction more a matter of style than of technical capabilities. Conferences are important events for computer science research. During these conferences, researchers from the public and private sectors present their recent work and meet. Unlike in most other academic fields, in computer science, the prestige of conference papers is greater than that of journal publications. One proposed explanation for this
6900-459: The distinction of three separate paradigms in computer science. Peter Wegner argued that those paradigms are science, technology, and mathematics. Peter Denning 's working group argued that they are theory, abstraction (modeling), and design. Amnon H. Eden described them as the "rationalist paradigm" (which treats computer science as a branch of mathematics, which is prevalent in theoretical computer science, and mainly employs deductive reasoning),
7000-412: The evaluation of functions that are expensive to compute and inexpensive to cache. ... For data requests that fall between the table's samples, an interpolation algorithm can generate reasonable approximations by averaging nearby samples." In data analysis applications, such as image processing , a lookup table (LUT) can be used to transform the input data into a more desirable output format. For example,
7100-511: The expression "automatic information" (e.g. "informazione automatica" in Italian) or "information and mathematics" are often used, e.g. informatique (French), Informatik (German), informatica (Italian, Dutch), informática (Spanish, Portuguese), informatika ( Slavic languages and Hungarian ) or pliroforiki ( πληροφορική , which means informatics) in Greek . Similar words have also been adopted in
7200-457: The first programmable mechanical calculator , his Analytical Engine . He started developing this machine in 1834, and "in less than two years, he had sketched out many of the salient features of the modern computer". "A crucial step was the adoption of a punched card system derived from the Jacquard loom " making it infinitely programmable. In 1843, during the translation of a French article on
7300-404: The first sine tables , which he encoded in a Sanskrit-letter-based number system. In 493 AD, Victorius of Aquitaine wrote a 98-column multiplication table which gave (in Roman numerals ) the product of every number from 2 to 50 times and the rows were "a list of numbers starting with one thousand, descending by hundreds to one hundred, then descending by tens to ten, then by ones to one, and then
7400-477: The first question, computability theory examines which computational problems are solvable on various theoretical models of computation . The second question is addressed by computational complexity theory , which studies the time and space costs associated with different approaches to solving a multitude of computational problems. The famous P = NP? problem, one of the Millennium Prize Problems ,
7500-452: The fractions down to 1/144" Modern school children are often taught to memorize " times tables " to avoid calculations of the most commonly used numbers (up to 9 x 9 or 12 x 12). Early in the history of computers, input/output operations were particularly slow – even in comparison to processor speeds of the time. It made sense to reduce expensive read operations by a form of manual caching by creating either static lookup tables (embedded in
7600-538: The fractions down to 1/144." In his 1820 book The Philosophy of Arithmetic , mathematician John Leslie published a multiplication table up to 1000 × 1000, which allows numbers to be multiplied in triplets of digits at a time. Leslie also recommended that young pupils memorize the multiplication table up to 50 × 50. The illustration below shows a table up to 12 × 12, which is a size commonly used nowadays in English-world schools. Because multiplication of integers
7700-430: The lookup table size while maintaining accuracy. While often effective, employing a lookup table may nevertheless result in a severe penalty if the computation that the LUT replaces is relatively simple. Memory retrieval time and the complexity of memory requirements can increase application operation time and system complexity relative to what would be required by straight formula computation. The possibility of polluting
7800-502: The machine in operation and analyzing it by all analytical and measurement means available. It has since been argued that computer science can be classified as an empirical science since it makes use of empirical testing to evaluate the correctness of programs , but a problem remains in defining the laws and theorems of computer science (if any exist) and defining the nature of experiments in computer science. Proponents of classifying computer science as an engineering discipline argue that
7900-461: The means for secure communication and preventing security vulnerabilities . Computer graphics and computational geometry address the generation of images. Programming language theory considers different ways to describe computational processes, and database theory concerns the management of repositories of data. Human–computer interaction investigates the interfaces through which humans and computers interact, and software engineering focuses on
8000-414: The memory address is hit by the cache. When a hit is found, no access to external memory is needed (except for write operations, where the cached value may need to be updated asynchronously to the slower memory after some time, or if the position in the cache must be replaced to cache another address). In digital logic , a lookup table can be implemented with a multiplexer whose select lines are driven by
8100-402: The multiples of any number from 0 to 10, except 5. As you would start on the number you are multiplying, when you multiply by 0, you stay on 0 (0 is external and so the arrows have no effect on 0, otherwise 0 is used as a link to create a perpetual cycle). The pattern also works with multiples of 10, by starting at 1 and simply adding 0, giving you 10, then just apply every number in the pattern to
8200-444: The multiplication table in columns with complete number sentences is still used in some countries, such as Bosnia and Herzegovina, instead of the modern grids above. There is a pattern in the multiplication table that can help people to memorize the table more easily. It uses the figures below: Figure 1 is used for multiples of 1, 3, 7, and 9. Figure 2 is used for the multiples of 2, 4, 6, and 8. These patterns can be used to memorize
8300-475: The multiplication table may have been introduced to Japan through Chinese mathematical treatises such as the Sunzi Suanjing , because their expression of the multiplication table share the character 如 in products less than ten. Chinese and Japanese share a similar system of eighty-one short, easily memorable sentences taught to students to help them learn the multiplication table up to 9 × 9. In current usage,
8400-478: The principal focus of computer science is studying the properties of computation in general, while the principal focus of software engineering is the design of specific computations to achieve practical goals, making the two separate but complementary disciplines. The academic, political, and funding aspects of computer science tend to depend on whether a department is formed with a mathematical emphasis or with an engineering emphasis. Computer science departments with
8500-437: The program) or dynamic prefetched arrays to contain only the most commonly occurring data items. Despite the introduction of systemwide caching that now automates this process, application level lookup tables can still improve performance for data items that rarely, if ever, change. Lookup tables were one of the earliest functionalities implemented in computer spreadsheets , with the initial version of VisiCalc (1979) including
8600-484: The reliability of computational systems is investigated in the same way as bridges in civil engineering and airplanes in aerospace engineering . They also argue that while empirical sciences observe what presently exists, computer science observes what is possible to exist and while scientists discover laws from observation, no proper laws have been found in computer science and it is instead concerned with creating phenomena. Proponents of classifying computer science as
8700-409: The scale of human intelligence. But the automation of evaluative and predictive tasks has been increasingly successful as a substitute for human monitoring and intervention in domains of computer application involving complex real-world data. Computer architecture, or digital computer organization, is the conceptual design and fundamental operational structure of a computer system. It focuses largely on
8800-426: The sentences that express products less than ten include an additional particle in both languages. In the case of modern Chinese, this is 得 ( dé ); and in Japanese, this is が ( ga ). This is useful for those who practice calculation with a suanpan or a soroban , because the sentences remind them to shift one column to the right when inputting a product that does not begin with a tens digit . In particular,
8900-505: The sine of a number of values, for example for each whole number of degrees (The table can be defined as static variables at compile time, reducing repeated run time costs). When the program requires the sine of a value, it can use the lookup table to retrieve the closest sine value from a memory address, and may also interpolate to the sine of the desired value, instead of calculating by mathematical formula. Lookup tables can thus used by mathematics coprocessors in computer systems. An error in
9000-466: The tables for Z 5 are: For other examples, see group . Hypercomplex number multiplication tables show the non- commutative results of multiplying two hypercomplex imaginary units. The simplest example is that of the quaternion multiplication table. For further examples, see Octonion § Multiplication , Sedenion § Multiplication , and Trigintaduonion § Multiplication . Mokkan discovered at Heijō Palace suggest that
9100-559: The term computer came to refer to the machines rather than their human predecessors. As it became clear that computers could be used for more than just mathematical calculations, the field of computer science broadened to study computation in general. In 1945, IBM founded the Watson Scientific Computing Laboratory at Columbia University in New York City . The renovated fraternity house on Manhattan's West Side
9200-743: The term "computer science" appears in a 1959 article in Communications of the ACM , in which Louis Fein argues for the creation of a Graduate School in Computer Sciences analogous to the creation of Harvard Business School in 1921. Louis justifies the name by arguing that, like management science , the subject is applied and interdisciplinary in nature, while having the characteristics typical of an academic discipline. His efforts, and those of others such as numerical analyst George Forsythe , were rewarded: universities went on to create such departments, starting with Purdue in 1962. Despite its name,
9300-561: The term was the Department of Datalogy at the University of Copenhagen, founded in 1969, with Peter Naur being the first professor in datalogy. The term is used mainly in the Scandinavian countries. An alternative term, also proposed by Naur, is data science ; this is now used for a multi-disciplinary field of data analysis, including statistics and databases. In the early days of computing,
9400-443: The two fields in areas such as mathematical logic , category theory , domain theory , and algebra . The relationship between computer science and software engineering is a contentious issue, which is further muddied by disputes over what the term "software engineering" means, and how computer science is defined. David Parnas , taking a cue from the relationship between other engineering and science disciplines, has claimed that
9500-431: The two points in the table on either side of the value and locates the answer on that line. This is still quick to compute, and much more accurate for smooth functions such as the sine function. Here is an example using linear interpolation: Linear interpolation provides for an interpolated function that is continuous, but will not, in general, have continuous derivatives . For smoother interpolation of table lookup that
9600-481: The type of information carrier – whether it is electrical, mechanical or biological. This field plays important role in information theory , telecommunications , information engineering and has applications in medical image computing and speech synthesis , among others. What is the lower bound on the complexity of fast Fourier transform algorithms? is one of the unsolved problems in theoretical computer science . Scientific computing (or computational science)
9700-418: The unsigned raw data value is used directly as an index to a one-dimensional table to extract a result. For small ranges, this can be amongst the fastest lookup, even exceeding binary search speed with zero branches and executing in constant time . One discrete problem that is expensive to solve on many computers is that of counting the number of bits that are set to 1 in a (binary) number, sometimes called
9800-438: The way by which the central processing unit performs internally and accesses addresses in memory. Computer engineers study computational logic and design of computer hardware, from individual processor components, microcontrollers , personal computers to supercomputers and embedded systems . The term "architecture" in computer literature can be traced to the work of Lyle R. Johnson and Frederick P. Brooks Jr. , members of
9900-440: Was IBM's first laboratory devoted to pure science. The lab is the forerunner of IBM's Research Division, which today operates research facilities around the world. Ultimately, the close relationship between IBM and Columbia University was instrumental in the emergence of a new scientific discipline, with Columbia offering one of the first academic-credit courses in computer science in 1946. Computer science began to be established as
10000-536: Was also in the calculator business to develop his giant programmable calculator, the ASCC/Harvard Mark I , based on Babbage's Analytical Engine, which itself used cards and a central computing unit. When the machine was finished, some hailed it as "Babbage's dream come true". During the 1940s, with the development of new and more powerful computing machines such as the Atanasoff–Berry computer and ENIAC ,
#318681