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Digit Extraction Algorithm
A spigot algorithm is an algorithm for computing the value of a transcendental number (such as or ''e'') that generates the digits of the number sequentially from left to right providing increasing precision as the algorithm proceeds. Spigot algorithms also aim to minimize the amount of intermediate storage required. The name comes from the sense of the word "spigot" for a tap or valve controlling the flow of a liquid. Spigot algorithms can be contrasted with algorithms that store and process complete numbers to produce successively more accurate approximations to the desired transcendental. Interest in spigot algorithms was spurred in the early days of computational mathematics by extreme constraints on memory, and such an algorithm for calculating the digits of ''e'' appeared in a paper by Sale in 1968. In 1970, Abdali presented a more general algorithm to compute the sums of series in which the ratios of successive terms can be expressed as quotients of integer functions of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algorithm
In mathematics and computer science, an algorithm () is a finite sequence of Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can use Conditional (computer programming), conditionals to divert the code execution through various routes (referred to as automated decision-making) and deduce valid inferences (referred to as automated reasoning). In contrast, a Heuristic (computer science), heuristic is an approach to solving problems without well-defined correct or optimal results.David A. Grossman, Ophir Frieder, ''Information Retrieval: Algorithms and Heuristics'', 2nd edition, 2004, For example, although social media recommender systems are commonly called "algorithms", they actually rely on heuristics as there is no truly "correct" recommendation. As an e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transcendental Number
In mathematics, a transcendental number is a real or complex number that is not algebraic: that is, not the root of a non-zero polynomial with integer (or, equivalently, rational) coefficients. The best-known transcendental numbers are and . The quality of a number being transcendental is called transcendence. Though only a few classes of transcendental numbers are known, partly because it can be extremely difficult to show that a given number is transcendental. Transcendental numbers are not rare: indeed, almost all real and complex numbers are transcendental, since the algebraic numbers form a countable set, while the set of real numbers and the set of complex numbers are both uncountable sets, and therefore larger than any countable set. All transcendental real numbers (also known as real transcendental numbers or transcendental irrational numbers) are irrational numbers, since all rational numbers are algebraic. The converse is not true: Not all irrational numbers are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
E (mathematical Constant)
The number is a mathematical constant approximately equal to 2.71828 that is the base of a logarithm, base of the natural logarithm and exponential function. It is sometimes called Euler's number, after the Swiss mathematician Leonhard Euler, though this can invite confusion with Euler numbers, or with Euler's constant, a different constant typically denoted \gamma. Alternatively, can be called Napier's constant after John Napier. The Swiss mathematician Jacob Bernoulli discovered the constant while studying compound interest. The number is of great importance in mathematics, alongside 0, 1, Pi, , and . All five appear in one formulation of Euler's identity e^+1=0 and play important and recurring roles across mathematics. Like the constant , is Irrational number, irrational, meaning that it cannot be represented as a ratio of integers, and moreover it is Transcendental number, transcendental, meaning that it is not a root of any non-zero polynomial with rational coefficie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tap (valve)
A tap (also spigot or faucet: see usage variations) is a valve controlling the release of a fluid. Nomenclature United Kingdom * Tap is used in the United Kingdom and most of the Commonwealth for any everyday type of valve, particularly the fittings that control water supply to bathtubs and sinks. United States * Faucet is the most common term in the US, similar in use to "tap" in British English, e.g. "water faucet" (although the term "tap" is also used in the US). * Spigot is used by professionals in the trade (such as plumbers), and typically refers to an outdoor fixture. * Silcock (and sillcock), same as "spigot", referring to a "cock" (as in stopcock and petcock) that penetrates a foundation sill. * Bib, bibcock, or hose bib, usually a freeze-resistant version of a "spigot". * Wall hydrant, same as "hose bib". * Tap generally refers to a keg or barrel tap, though also commonly refers to a faucet that supplies either hot or cold water and not both. It also appears ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stanley Rabinowitz
Stanley may refer to: Arts and entertainment Film and television * ''Stanley'' (1972 film), an American horror film * ''Stanley'' (1984 film), an Australian comedy * ''Stanley'' (1999 film), an animated short * ''Stanley'' (1956 TV series), an American situation comedy * ''Stanley'' (2001 TV series), an American animated series Other uses in arts and entertainment * ''Stanley'' (play), by Pam Gems, 1996 * Stanley Award, an Australian Cartoonists' Association award * '' Stanley: The Search for Dr. Livingston'', a video game Businesses * Stanley, Inc., an American information technology company * Stanley Aviation, an American aerospace company * Stanley Black & Decker, formerly The Stanley Works, an American hardware manufacturer ** Stanley Hand Tools, a division of Stanley Black & Decker * Stanley bottle, a brand of food and beverage containers * Stanley Electric, a Japanese manufacturer of electric lights * Stanley Furniture, an American furniture manufacturer * The Sta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stan Wagon
Stanley Wagon is a Canadian-American mathematician, a professor emeritus of mathematics at Macalester College in Minnesota. He is the author of multiple books on number theory, geometry, and computational mathematics, and is also known for his snow sculpture. Biography Wagon was born in Montreal, to Sam and Diana (Idlovitch) Wagon. His sister Lila (Wagon) Hope-Simpson died in 2021. Wagon did his undergraduate studies at McGill University in Montreal, graduating in 1971. He earned his Ph.D. in 1975 from Dartmouth College, under the supervision of James Earl Baumgartner. He married mathematician Joan Hutchinson, and the two of them shared a single faculty position at Smith College and again at Macalester, where they moved in 1990. Books *'' The Banach–Tarski Paradox'' (Cambridge University Press, 1985) *''Old and New Unsolved Problems in Plane Geometry and Number Theory'' (with Victor Klee, Mathematical Association of America, 1991)''Mathematica® in Action: Problem Solving Thro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bailey–Borwein–Plouffe Formula
The Bailey–Borwein–Plouffe formula (BBP formula) is a formula for . It was discovered in 1995 by Simon Plouffe and is named after the authors of the article in which it was published, David H. Bailey, Peter Borwein, and Plouffe. The formula is: : \pi = \sum_^\left frac \left(\frac-\frac-\frac-\frac\right)\right/math> The BBP formula gives rise to a spigot algorithm for computing the ''n''th base-16 (hexadecimal) digit of (and therefore also the ''4n''th binary digit of ) without computing the preceding digits. This does ''not'' compute the ''n''th decimal digit of (i.e., in base 10). But another formula discovered by Plouffe in 2022 allows extracting the ''n''th digit of in decimal. BBP and BBP-inspired algorithms have been used in projects such as PiHex for calculating many digits of using distributed computing. The existence of this formula came as a surprise. It had been widely believed that computing the ''n''th digit of is just as hard as computing the first ''n'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Logarithm
The natural logarithm of a number is its logarithm to the base of a logarithm, base of the e (mathematical constant), mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if the base is implicit, simply . Parentheses are sometimes added for clarity, giving , , or . This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity. The natural logarithm of is the exponentiation, power to which would have to be raised to equal . For example, is , because . The natural logarithm of itself, , is , because , while the natural logarithm of is , since . The natural logarithm can be defined for any positive real number as the Integral, area under the curve from to (with the area being negative when ). The simplicity of this definition, which is matched in many other formulas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Arithmetic
In mathematics, modular arithmetic is a system of arithmetic operations for integers, other than the usual ones from elementary arithmetic, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book '' Disquisitiones Arithmeticae'', published in 1801. A familiar example of modular arithmetic is the hour hand on a 12-hour clock. If the hour hand points to 7 now, then 8 hours later it will point to 3. Ordinary addition would result in , but 15 reads as 3 on the clock face. This is because the hour hand makes one rotation every 12 hours and the hour number starts over when the hour hand passes 12. We say that 15 is ''congruent'' to 3 modulo 12, written 15 ≡ 3 (mod 12), so that 7 + 8 ≡ 3 (mod 12). Similarly, if one starts at 12 and waits 8 hours, the hour hand will be at 8. If one instead waited twice as long, 16 hours, the hour hand would be on 4. This ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Exponentiation
Modular exponentiation is exponentiation performed over a modulus. It is useful in computer science, especially in the field of public-key cryptography, where it is used in both Diffie–Hellman key exchange and RSA public/private keys. Modular exponentiation is the remainder when an integer (the base) is raised to the power (the exponent), and divided by a positive integer (the modulus); that is, . From the definition of division, it follows that . For example, given , and , dividing by leaves a remainder of . Modular exponentiation can be performed with a ''negative'' exponent by finding the modular multiplicative inverse of modulo using the extended Euclidean algorithm. That is: :, where and . Modular exponentiation is efficient to compute, even for very large integers. On the other hand, computing the modular discrete logarithm – that is, finding the exponent when given , , and – is believed to be difficult. This one-way function behavior makes modular e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Precision (arithmetic)
Significant figures, also referred to as significant digits, are specific digits within a number that is written in positional notation that carry both reliability and necessity in conveying a particular quantity. When presenting the outcome of a measurement (such as length, pressure, volume, or mass), if the number of digits exceeds what the measurement instrument can resolve, only the digits that are determined by the resolution are dependable and therefore considered significant. For instance, if a length measurement yields 114.8 mm, using a ruler with the smallest interval between marks at 1 mm, the first three digits (1, 1, and 4, representing 114 mm) are certain and constitute significant figures. Further, digits that are uncertain yet meaningful are also included in the significant figures. In this example, the last digit (8, contributing 0.8 mm) is likewise considered significant despite its uncertainty. Therefore, this measurement contains f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Single Precision
Single-precision floating-point format (sometimes called FP32 or float32) is a computer number format, usually occupying 32 bits in computer memory; it represents a wide dynamic range of numeric values by using a floating radix point. A floating-point variable can represent a wider range of numbers than a fixed-point variable of the same bit width at the cost of precision. A signed 32-bit integer variable has a maximum value of 231 − 1 = 2,147,483,647, whereas an IEEE 754 32-bit base-2 floating-point variable has a maximum value of (2 − 2−23) × 2127 ≈ 3.4028235 × 1038. All integers with seven or fewer decimal digits, and any 2''n'' for a whole number −149 ≤ ''n'' ≤ 127, can be converted exactly into an IEEE 754 single-precision floating-point value. In the IEEE 754 standard, the 32-bit base-2 format is officially referred to as binary32; it was called single in IEEE 754-1985. IEEE 754 specifies additional floating-point types, such as 64-bit base-2 ''doubl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
