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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 fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polymeter
In music, metre ( Commonwealth 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 m ... [...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 its expression as \frac=p(x) + \sum_j \frac where is a polynomial, and, for each , the denominator is a power of an irreducible polynomial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Greatest Common Divisor
In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is denoted \gcd (x,y). For example, the GCD of 8 and 12 is 4, that is, \gcd (8, 12) = 4. 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 (hcf), 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 two nonzero integers and is the greatest positive integer such that is a divisor of both and ; that is, there are integers and such that and , and is the largest s ... [...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 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 article by Boas analyzes two-digit cases in bases other than base 10, e.g., 32/13 = 2/1 and its inverse are the only solutions in base 4 with two digits. The anomalous cancellation happens also with more digits, e.g. 165/462 = 15/42 and those with differ ... [...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. The edition of the dictionary in 1979 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 specified format. This meant that every aspect of an entry was handled by a different editor using different forms or templates. Once all the entries for an entry had been assembled, they were passed on to be keyed into the slowly assembled dictionary database which was completed for the typesetting of the first edition. In a later edition, they increasingly used the Bank of English established by John Sinclair at COBUILD to provide typical citations rather than examples composed by the lexicographer. Editions The current edition is the 13th edition, which was published in November 2018. The previous edition was the 12th edition, which was pu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metre (music)
In music, metre ( Commonwealth 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 m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Count Time
Count (feminine: countess) is a historical title of nobility in certain European countries, varying in relative status, generally of middling rank in the hierarchy of nobility. Pine, L. G. ''Titles: How the King Became His Majesty''. New York: Barnes & Noble, 1992. p. 73. . The etymologically related English term "county" denoted the territories associated with the countship. Definition The word ''count'' came into English from the French ''comte'', itself from Latin ''comes''—in its accusative ''comitem''—meaning “companion”, and later “companion of the emperor, delegate of the emperor”. The adjective form of the word is "comital". The British and Irish equivalent is an earl (whose wife is a "countess", for lack of an English term). In the late Roman Empire, the Latin title ''comes'' denoted the high rank of various courtiers and provincial officials, either military or administrative: before Anthemius became emperor in the West in 467, he was a military ''comes ... [...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 musicologist Arthur Morris Jones (1889–1980). It refers to when 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 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 master drummer and scholar C.K. Ladzekpo, and in the writings of David Locke. Jones observes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Least Common Multiple
In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers ''a'' and ''b'', usually denoted by lcm(''a'', ''b''), 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 lcm is the "lowest common denominator" (lcd) that can be used before fractions can be added, subtracted or compared. The least common multiple of more than two integers ''a'', ''b'', ''c'', . . . , usually denoted by lcm(''a'', ''b'', ''c'', . . .), is also well defined: It is 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 ... [...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 receive a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tessellation
A tessellation or tiling is the covering of a surface, often a plane (mathematics), plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to high-dimensional spaces, 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. 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 a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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