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floating point | A Wisdom Archive on floating point | | floating point A selection of articles related to floating point | |
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Floating point
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| ARTICLES RELATED TO floating point | | | |  floating point: Encyclopedia II - Division by zero - Algebraic interpretationIt is generally regarded among mathematicians that a natural way to interpret division by zero is to first define division in terms of other arithmetic operations. Under the standard rules for arithmetic on integers, rational numbers, real numbers and complex numbers, division by zero is undefined. Division by zero must be left undefined in any mathematical system that obeys the axioms of a field. The reason is that division is defined to be the inverse operation of multiplication. This means that the value of is the solution x of th ...
See also:Division by zero, Division by zero - Early attempts, Division by zero - Algebraic interpretation, Division by zero - Fallacies based on division by zero, Division by zero - Abstract algebra, Division by zero - Limits and division by zero, Division by zero - Formal interpretation, Division by zero - Other number systems, Division by zero - Real projective line, Division by zero - Riemann sphere, Division by zero - Non-standard analysis, Division by zero - Abstract algebra, Division by zero - In mathematical analysis, Division by zero - Division by zero in computer arithmetic Read more here: » Division by zero: Encyclopedia II - Division by zero - Algebraic interpretation |
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| | | | | | | | floating point: Encyclopedia II - Trigonometric function - History The earliest systematic study of trigonometric functions and tabulation of their values was performed by Hipparchus of Nicaea (180-125 BC), who tabulated the lengths of circle arcs (angle A times radius r) with the lengths of the subtending chords (2r sin(A/2)). Later, Ptolemy (2nd century) expanded upon this work in his Almagest, deriving addition/subtraction formulas for the equivalent of sin(A + B) and cos(A + B). Ptolemy also derived the equivalent of the half-angle formula s ...
See also:Trigonometric function, Trigonometric function - History, Trigonometric function - Right triangle definitions, Trigonometric function - Mnemonics, Trigonometric function - Slope definitions, Trigonometric function - Unit-circle definitions, Trigonometric function - Series definitions, Trigonometric function - Relationship to exponential function, Trigonometric function - Definitions via differential equations, Trigonometric function - The significance of radians, Trigonometric function - Other definitions, Trigonometric function - Computation, Trigonometric function - Inverse functions, Trigonometric function - Identities, Trigonometric function - Properties and applications, Trigonometric function - Law of sines, Trigonometric function - Law of cosines, Trigonometric function - Law of tangents Read more here: » Trigonometric function: Encyclopedia II - Trigonometric function - History |
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| | | | | | | | | | floating point: Encyclopedia II - History of mathematics - Mathematics in prehistory Long before the earliest written records, there are drawings that indicate a knowledge of mathematics and of measurement of time based on the stars. For example, paleontologists have discovered ochre rocks in a cave in South Africa adorned with scratched geometric patterns dating back more than 70,000 years [1]. Also prehistoric artifacts discovered in Africa and France, dated between 35000 BC and 20000 BC, indicate early attempts to quantify time Evidence exists that early counting involved women who kept records of their monthly biological ...
See also:History of mathematics, History of mathematics - Mathematics in prehistory, History of mathematics - Egyptian and Babylonian mathematics 2000 BC - 600 BC, History of mathematics - Ancient Indian mathematics 800 BC - 200 BC, History of mathematics - Greek and Hellenistic mathematics 550 BC - 200 BC, History of mathematics - Chinese mathematics 200 BC - AD 1200, History of mathematics - Classical Indian mathematics 200 BC - AD 1600, History of mathematics - Arabic and Persian mathematics 650 - 1500, History of mathematics - European Renaissance mathematics 1200 - 1600, History of mathematics - 17th century, History of mathematics - 18th century, History of mathematics - Complex numbers, History of mathematics - Miscellaneous historical notes, History of mathematics - Notes Read more here: » History of mathematics: Encyclopedia II - History of mathematics - Mathematics in prehistory |
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| | | | | floating point: Encyclopedia II - Genetic algorithm - Operation of a GA An individual, or solution to the problem to be solved, is represented by a list of parameters, called chromosome or genome. Chromosomes are typically represented as simple strings of data and instructions, although a wide variety of other data structures for storing chromosomes may also be used.
Initially several such individuals are randomly generated to form the first initial population. The user of the algorithm may seed the gene pool with "hint ...
See also:Genetic algorithm, Genetic algorithm - Operation of a GA, Genetic algorithm - Pseudo-code algorithm, Genetic algorithm - Observations, Genetic algorithm - Variants, Genetic algorithm - Problem domains, Genetic algorithm - History, Genetic algorithm - Applications, Genetic algorithm - Related techniques, Genetic algorithm - Building block hypothesis Read more here: » Genetic algorithm: Encyclopedia II - Genetic algorithm - Operation of a GA |
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| | | | | floating point: Encyclopedia II - Texas Instruments - History Texas Instruments was founded by Cecil H. Green, J. Erik Jonsson, Eugene McDermott and Patrick E. Haggerty. On December 6, 1941, the four men purchased Geophysical Service Incorporated (GSI), a pioneering provider of seismic exploration services to the petroleum industry. During World War II, GSI built electronics for the U.S. Army Signal Corps and the U.S. Navy. After the war, GSI continued to produce electronics, and in 1951 the company changed its name to Texas Instruments; GSI became a wholly-owned subsidiary of the new company. An early ...
See also:Texas Instruments, Texas Instruments - History, Texas Instruments - Consumer electronics and computers, Texas Instruments - Defense electronics, Texas Instruments - TI today, Texas Instruments - Semiconductors, Texas Instruments - DLP products, Texas Instruments - Sensors and controls, Texas Instruments - Educational and productivity solutions Read more here: » Texas Instruments: Encyclopedia II - Texas Instruments - History |
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| | | | | floating point: Encyclopedia II - RISC - Meanwhile... While the RISC philosophy was coming into its own, new ideas about how to dramatically increase performance of the CPUs were starting to develop.
In the early 1980s it was thought that existing design was reaching theoretical limits. Future improvements in speed would be primarily through improved semiconductor "process", that is, smaller features (transistors and wires) on the chip. The complexity of the chip would remain largely the same, but the smaller size would allow it to run at higher clock rates. A considerable amount of effo ...
See also:RISC, RISC - RISC design philosophy, RISC - Pre-RISC design philosophy, RISC - Meanwhile..., RISC - Early RISC, RISC - Later RISC, RISC - Alternative term Read more here: » RISC: Encyclopedia II - RISC - Meanwhile... |
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| | | | | | floating point: Encyclopedia II - Real number - Properties
Real number - Completeness.
The main reason for introducing the reals is that the reals contain all limits. More technically, the reals are complete (in the sense of metric spaces or uniform spaces, which is a different sense than the Dedekind completeness of the order in the previous section). This means the following:
A sequence (xn) of real numbers is called a Cauchy sequence if for any ε > 0 there exists an integer N (possibly depending on ε) such t ...
See also:Real number, Real number - History, Real number - Definition, Real number - Construction from the rational numbers, Real number - Axiomatic approach, Real number - Properties, Real number - Completeness, Real number - The complete ordered field, Real number - Advanced properties, Real number - Generalizations and extensions Read more here: » Real number: Encyclopedia II - Real number - Properties |
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| | | | | floating point: Encyclopedia II - Googol - The shrinking googol Back when it was named in 1938, the googol was undeniably large. However, with the invention of fast computers and fast algorithms, computation with numbers the size of a googol has become routine. For example, even the difficult problem of prime factorization is now fairly accessible for 100 digit numbers.
The largest number that can be represented by a typical pocket calculator for high school or scientific use is slightly less than a googol (e.g. 9.9999999 E+99, i.e. 9.99999991099, or 0.99999999 googol). However, some mo ...
See also:Googol, Googol - Writing out a googol, Googol - Relation to -illion number names, Googol - The shrinking googol, Googol - Trivia, Googol - Googolplex Read more here: » Googol: Encyclopedia II - Googol - The shrinking googol |
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| | | | | floating point: Encyclopedia II - Motorola 68881 - Overview The 68020 and 68030 CPUs were designed with the separate 68881 chip in mind. Their instruction sets reserved the "F-line" instructions — that is, all opcodes beginning with the hexadecimal digit "F" were "traps" which would throw an interrupt, handing control to the computer's operating system. If a 68881 were present in the system, the CPU would allow it to execute the instruction. If not, the OS would either call an FPU emulator to execute the instruction using 68020 integer-based software code, or ...
See also:Motorola 68881, Motorola 68881 - Overview, Motorola 68881 - Selected statistics, Motorola 68881 - 68881, Motorola 68881 - 68882, Motorola 68881 - 68040 Read more here: » Motorola 68881: Encyclopedia II - Motorola 68881 - Overview |
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| | | | | | floating point: Encyclopedia II - Genetic algorithm - Operation of a GA Two elements are required for any problem before a genetic algorithm can be used to search for a solution: First, there must be a method of representing a solution in a manner that can be manipulated by the algorithm. Traditionally, a solution can be represented by a string of bits, numbers or characters. Second, there must be some method of measuring the quality of any proposed solution, using a fitness function.
For instance, if the problem involves fitting as many different weights as possible into a knapsack without ...
See also:Genetic algorithm, Genetic algorithm - Operation of a GA, Genetic algorithm - Initialization, Genetic algorithm - Selection, Genetic algorithm - Reproduction, Genetic algorithm - Termination, Genetic algorithm - Pseudo-code algorithm, Genetic algorithm - Observations, Genetic algorithm - Variants, Genetic algorithm - Problem domains, Genetic algorithm - History, Genetic algorithm - Applications, Genetic algorithm - Related techniques, Genetic algorithm - Building block hypothesis Read more here: » Genetic algorithm: Encyclopedia II - Genetic algorithm - Operation of a GA |
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| | | | | floating point: Encyclopedia II - Radix sort - Recursion A recursively subdividing radix sort algorithm works as follows:
take the most significant digit of each key.
sort the list of elements based on that digit, grouping elements with the same digit into one bucket.
recursively sort each bucket, starting with the next most significant digit.
concatenate the buckets together in order.
This recursive method can be interpreted as a generalization of quicksort from strings with two possible symbols to strings with any number of possib ...
See also:Radix sort, Radix sort - An example, Radix sort - Iterative version using queues, Radix sort - Recursion, Radix sort - A Recursive Forward Radix Sort Example, Radix sort - Efficiency, Radix sort - Sample implementations, Radix sort - C#, Radix sort - C++ Read more here: » Radix sort: Encyclopedia II - Radix sort - Recursion |
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