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Least Common Denominator
In mathematics, the lowest common denominator or least common denominator (abbreviated LCD) is the lowest common multiple of the denominators of a set of fractions. It simplifies adding, subtracting, and comparing fractions. Description The lowest common denominator of a set of fractions is the lowest number that is a multiple of all the denominators: their lowest common multiple. The product of the denominators is always a common denominator, as in: : \frac+\frac\;=\;\frac+\frac\;=\;\frac but it is not always the lowest common denominator, as in: : \frac+\frac\;=\;\frac+\frac\;=\;\frac Here, 36 is the least common multiple of 12 and 18. Their product, 216, is also a common denominator, but calculating with that denominator involves larger numbers: : \frac+\frac=\frac+\frac=\frac. With variables rather than numbers, the same principles apply: : \frac+\frac\;=\;\frac+\frac\;=\;\frac Some methods of calculating the LCD are at . Role in arithmetic and algebra The same ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polymeter
In music, metre (British spelling) or meter (American spelling) refers to regularly recurring patterns and accents such as bars and beats. Unlike rhythm, metric onsets are not necessarily sounded, but are nevertheless implied by the performer (or performers) and expected by the listener. A variety of systems exist throughout the world for organising and playing metrical music, such as the Indian system of '' tala'' and similar systems in Arabic and African music. Western music inherited the concept of metre from poetry, where it denotes the number of lines in a verse, the number of syllables in each line, and the arrangement of those syllables as long or short, accented or unaccented. The first coherent system of rhythmic notation in modern Western music was based on rhythmic modes derived from the basic types of metrical unit in the quantitative metre of classical ancient Greek and Latin poetry. Later music for dances such as the pavane and galliard consisted of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Fraction Decomposition
In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator. The importance of the partial fraction decomposition lies in the fact that it provides algorithms for various computations with rational functions, including the explicit computation of antiderivatives, Taylor series expansions, inverse Z-transforms, and inverse Laplace transforms. The concept was discovered independently in 1702 by both Johann Bernoulli and Gottfried Leibniz. In symbols, the ''partial fraction decomposition'' of a rational fraction of the form \frac, where and are polynomials, is the expression of the rational fraction as \frac=p(x) + \sum_j \frac where is a polynomial, and, for each , the denominator is a power of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Greatest Common Divisor
In mathematics, the greatest common divisor (GCD), also known as greatest common factor (GCF), of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers , , the greatest common divisor of and is denoted \gcd (x,y). For example, the GCD of 8 and 12 is 4, that is, . In the name "greatest common divisor", the adjective "greatest" may be replaced by "highest", and the word "divisor" may be replaced by "factor", so that other names include highest common factor, etc. Historically, other names for the same concept have included greatest common measure. This notion can be extended to polynomials (see ''Polynomial greatest common divisor'') and other commutative rings (see ' below). Overview Definition The ''greatest common divisor'' (GCD) of integers and , at least one of which is nonzero, is the greatest positive integer such that is a divisor of both and ; that is, there are integers and such that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anomalous Cancellation
An anomalous cancellation or accidental cancellation is a particular kind of arithmetic procedural error that gives a numerically correct answer. An attempt is made to reduce a fraction A fraction (from , "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, thre ... by cancelling individual digits in the numerator and denominator. This is not a legitimate operation, and does not in general give a correct answer, but in some rare cases the result is numerically the same as if a correct procedure had been applied. The trivial cases of cancelling trailing zeros or where all of the digits are equal are ignored. Examples of anomalous cancellations which still produce the correct result include (these and their inverses are all the cases in base 10 with the fraction different from 1 and with two digits): The a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Collins English Dictionary
The ''Collins English Dictionary'' is a printed and online dictionary of English. It is published by HarperCollins in Glasgow. It was first published in 1979. Corpus The dictionary uses language research based on the Collins Corpus, which is continually updated and has over 20 billion words. Editions * The current edition is the 14th; it was published on 31 August 2023, with more than 732,000 words, meanings, and phrases (not 730,000 headwords) and 9,500 place names and 7,300 biographies. A newer edition of the 14th edition was published 7 May 2024. * The previous edition was the 13th edition, which was published in November 2018. * A special "30th Anniversary" 10th edition was published in 2010. * Earlier editions were published once every 3 or 4 years. History The 1979 edition of the dictionary, with Patrick Hanks as editor and Laurence Urdang as editorial director, was the first British English dictionary to be typeset from the output from a computer database in a specif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metre (music)
In music, metre (British spelling) or meter (American spelling) refers to regularly recurring patterns and accents such as bars and beats. Unlike rhythm, metric onsets are not necessarily sounded, but are nevertheless implied by the performer (or performers) and expected by the listener. A variety of systems exist throughout the world for organising and playing metrical music, such as the Indian system of '' tala'' and similar systems in Arabic and African music. Western music inherited the concept of metre from poetry, where it denotes the number of lines in a verse, the number of syllables in each line, and the arrangement of those syllables as long or short, accented or unaccented. The first coherent system of rhythmic notation in modern Western music was based on rhythmic modes derived from the basic types of metrical unit in the quantitative metre of classical ancient Greek and Latin poetry. Later music for dances such as the pavane and galliard consisted of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Count Time
In music, counting is a system of regularly occurring sounds that serve to assist with the performance or audition of music by allowing the easy identification of the beat. Commonly, this involves verbally counting the beats in each measure as they occur, whether there be 2 beats, 3 beats, 4 beats, or even 5 beats. In addition to helping to normalize the time taken up by each beat, counting allows easier identification of the beats that are stressed. Counting is most commonly used with rhythm (often to decipher a difficult rhythm) and form and often involves subdivision. Introduction to systems: numbers and syllables The method involving numbers may be termed ''count chant'', "to identify it as a unique instructional process." In lieu of simply counting the beats of a measure, other systems can be used which may be more appropriate to the particular piece of music. Depending on the tempo, the divisions of a beat may be vocalized as well (for slower times), or skipping numbers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cross-rhythm
In music, a cross-beat or cross-rhythm is a specific form of polyrhythm. The term ''cross rhythm '' was introduced in 1934 by the Musicology, musicologist Arthur Morris Jones (1889–1980). It refers to a situation where the rhythmic conflict found in polyrhythms is the basis of an entire musical piece. Etymology The term "cross rhythm" was introduced in 1934 by the Musicology, musicologist Arthur Morris Jones (1889–1980), who, with Klaus Wachsmann, took-up extended residence in Zambia and Uganda, respectively, as missionaries, educators, musicologists, and museologists. African music One main system African cross-rhythm is most prevalent within the greater Niger-Congo linguistic group, which dominates the continent south of the Sahara Desert. (Kubik, p. 58) Cross-rhythm was first identified as the basis of sub-Saharan rhythm by A.M. Jones. Later, the concept was more fully explained in the lectures of Ewe drumming, Ewe master drummer and scholar C.K. Ladzekpo, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Least Common Multiple
In arithmetic and number theory, the least common multiple (LCM), lowest common multiple, or smallest common multiple (SCM) of two integers ''a'' and ''b'', usually denoted by , is the smallest positive integer that is divisible by both ''a'' and ''b''. Since division of integers by zero is undefined, this definition has meaning only if ''a'' and ''b'' are both different from zero. However, some authors define lcm(''a'', 0) as 0 for all ''a'', since 0 is the only common multiple of ''a'' and 0. The least common multiple of the denominators of two fractions is the "lowest common denominator" (lcd), and can be used for adding, subtracting or comparing the fractions. The least common multiple of more than two integers ''a'', ''b'', ''c'', . . . , usually denoted by , is defined as the smallest positive integer that is divisible by each of ''a'', ''b'', ''c'', . . . Overview A multiple of a number is the product of that number and an integer. For example, 10 is a multiple of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Work Schedule
A schedule (, ) or a timetable, as a basic time-management tool, consists of a list of times at which possible tasks, events, or actions are intended to take place, or of a sequence of events in the chronological order in which such things are intended to take place. The process of creating a schedule — deciding how to order these tasks and how to commit resources between the variety of possible tasks — is called scheduling,Ofer Zwikael, John Smyrk, ''Project Management for the Creation of Organisational Value'' (2011), p. 196: "The process is called scheduling, the output from which is a timetable of some form". and a person responsible for making a particular schedule may be called a scheduler. Making and following schedules is an ancient human activity. Some scenarios associate this kind of planning with learning life skills. Schedules are necessary, or at least useful, in situations where individuals need to know what time they must be at a specific location to rec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tessellation
A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of geometries. A periodic tiling has a repeating pattern. Some special kinds include '' regular tilings'' with regular polygonal tiles all of the same shape, and '' semiregular tilings'' with regular tiles of more than one shape and with every corner identically arranged. The patterns formed by periodic tilings can be categorized into 17 wallpaper groups. A tiling that lacks a repeating pattern is called "non-periodic". An '' aperiodic tiling'' uses a small set of tile shapes that cannot form a repeating pattern (an aperiodic set of prototiles). A '' tessellation of space'', also known as a space filling or honeycomb, can be defined in the geometry of higher dimensions. A real physical tessellation is a tiling made of materials such as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
