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Cross Multiplication
In mathematics, specifically in elementary arithmetic and elementary algebra, given an equation between two fractions or rational expressions, one can cross-multiply to simplify the equation or determine the value of a variable. The method is also occasionally known as the "cross your heart" method because lines resembling a heart outline can be drawn to remember which things to multiply together. Given an equation like : \frac a b = \frac c d, where and are not zero, one can cross-multiply to get : ad = bc \quad \text \quad a = \fracd. In Euclidean geometry the same calculation can be achieved by considering the ratios as those of similar triangles. Procedure In practice, the method of cross-multiplying means that we multiply the numerator of each (or one) side by the denominator of the other side, effectively crossing the terms over: : \frac a b \nwarrow \frac c d, \quad \frac a b \nearrow \frac c d. The mathematical justification for the method is from the following ... [...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] |
Turn (angle)
A turn is a unit of plane angle measurement equal to radians, 360 degrees or 400 gradians. Subdivisions of a turn include half-turns, quarter-turns, centiturns, milliturns, etc. The closely related terms ''cycle'' and ''revolution'' are not equivalent to a turn. Subdivisions A turn can be divided in 100 centiturns or milliturns, with each milliturn corresponding to an angle of 0.36°, which can also be written as 21′ 36″. A protractor divided in centiturns is normally called a "percentage protractor". Binary fractions of a turn are also used. Sailors have traditionally divided a turn into 32 compass points, which implicitly have an angular separation of 1/32 turn. The ''binary degree'', also known as the ''binary radian'' (or ''brad''), is turn. The binary degree is used in computing so that an angle can be represented to the maximum possible precision in a single byte. Other measures of angle used in computing may be based on dividin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Odds Ratio
An odds ratio (OR) is a statistic that quantifies the strength of the association between two events, A and B. The odds ratio is defined as the ratio of the odds of A in the presence of B and the odds of A in the absence of B, or equivalently (due to symmetry), the ratio of the odds of B in the presence of A and the odds of B in the absence of A. Two events are independent if and only if the OR equals 1, i.e., the odds of one event are the same in either the presence or absence of the other event. If the OR is greater than 1, then A and B are associated (correlated) in the sense that, compared to the absence of B, the presence of B raises the odds of A, and symmetrically the presence of A raises the odds of B. Conversely, if the OR is less than 1, then A and B are negatively correlated, and the presence of one event reduces the odds of the other event. Note that the odds ratio is symmetric in the two events, and there is no causal direction implied (correlation does not imply causat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cross-ratio
In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points ''A'', ''B'', ''C'' and ''D'' on a line, their cross ratio is defined as : (A,B;C,D) = \frac where an orientation of the line determines the sign of each distance and the distance is measured as projected into Euclidean space. (If one of the four points is the line's point at infinity, then the two distances involving that point are dropped from the formula.) The point ''D'' is the harmonic conjugate of ''C'' with respect to ''A'' and ''B'' precisely if the cross-ratio of the quadruple is −1, called the ''harmonic ratio''. The cross-ratio can therefore be regarded as measuring the quadruple's deviation from this ratio; hence the name ''anharmonic ratio''. The cross-ratio is preserved by linear fractional transformations. It is essentially the only projective invarian ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sylvie And Bruno
''Sylvie and Bruno'', first published in 1889, and its second volume ''Sylvie and Bruno Concluded'' published in 1893, form the last novel by Lewis Carroll published during his lifetime. Both volumes were illustrated by Harry Furniss. The novel has two main plots: one set in the real world at the time the book was published (the Victorian era), the other in the fantasy world of Fairyland. While the latter plot is a fairy tale with many nonsense elements and poems, similar to Carroll's ''Alice'' books, the story set in Victorian Britain is a social novel, with its characters discussing various concepts and aspects of religion, society, philosophy and morality. Origin Two short pieces, "Fairy Sylvie" and "Bruno's Revenge", originally appeared in ''Aunt Judy's Magazine'' in 1867. Some years later, in 1873 or 1874, Carroll had the idea to use these as the core for a longer story. Much of the rest of the novel he compiled from notes of ideas and dialogue which he had collected over ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Mad Gardener's Song
"The Mad Gardener's Song" is a poem by Lewis Carroll that appears in his book '' Sylvie and Bruno'' (1889, 1893). Structure The poem consists of nine stanzas, each of six lines. Each stanza contains alternating lines of iambic tetrameter and iambic trimeter, with the trimetric lines rhyming with each other. Each verse is scattered around the novel ''Sylvie and Bruno'', with eight verses in the first volume and one in the second, ''Sylvie and Bruno Concluded''. Text Reception In his ''Bright Dreams Journal'', Gary R. Hess called the poem "the only bright part of the book." In ''The Aesthetics of Children's Poetry: A Study of Children's Verse in English'', Katherine Wakely-Mulroney described the poem as "an incantatory, cyclical poem which reflects and even prefigures aspects of the prose narrative." Adaptations "The Mad Gardener's Song" featured on the BBC show '' Play School'' in 1981. Composer Stuart Findlay set "The Mad Gardener's Song" to viola, clarinet and piano in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lewis Carroll
Charles Lutwidge Dodgson (; 27 January 1832 – 14 January 1898), better known by his pen name Lewis Carroll, was an English author, poet and mathematician. His most notable works are ''Alice's Adventures in Wonderland'' (1865) and its sequel ''Through the Looking-Glass'' (1871). He was noted for his facility with word play, logic, and fantasy. His poems ''Jabberwocky'' (1871) and ''The Hunting of the Snark'' (1876) are classified in the genre of literary nonsense. Carroll came from a family of high-church Anglicanism, Anglicans, and developed a long relationship with Christ Church, Oxford, where he lived for most of his life as a scholar and teacher. Alice Liddell, the daughter of Christ Church's dean Henry Liddell, is widely identified as the original inspiration for ''Alice in Wonderland'', though Carroll always denied this. An avid puzzler, Carroll created the word ladder puzzle (which he then called "Doublets"), which he published in his weekly column for ''Vanity Fair ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cocker's Arithmetick
''Cocker's Arithmetick'', also known by its full title "Cocker's Arithmetick: Being a Plain and Familiar Method Suitable to the Meanest Capacity for the Full Understanding of That Incomparable Art, As It Is Now Taught by the Ablest School-Masters in City and Country", is a grammar school mathematics textbook written by Edward Cocker (1631–1676) and published posthumously by John Hawkins in 1678. ''Arithmetick'' along with companion volume, '' Decimal Arithmetick'' published in 1684, were used to teach mathematics in schools in the United Kingdom for more than 150 years. Some controversy exists over the authorship of the book. Augustus De Morgan claimed the work was written by Hawkins, who merely used Cocker's name to lend the authority of his reputation to the book. Ruth Wallis, in 1997, wrote an article in '' Annals of Science'', claiming De Morgan's analysis was flawed and Cocker was the real author. The popularity of ''Arithmetick'' is unquestioned by its more than 130 edi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thirteen Colonies
The Thirteen Colonies, also known as the Thirteen British Colonies, the Thirteen American Colonies, or later as the United Colonies, were a group of Kingdom of Great Britain, British Colony, colonies on the Atlantic coast of North America. Founded in the 17th and 18th centuries, they began fighting the American Revolutionary War in April 1775 and formed the United States of America by United States Declaration of Independence, declaring full independence in July 1776. Just prior to declaring independence, the Thirteen Colonies in their traditional groupings were: New England (Province of New Hampshire, New Hampshire; Province of Massachusetts Bay, Massachusetts; Colony of Rhode Island and Providence Plantations, Rhode Island; Connecticut Colony, Connecticut); Middle (Province of New York, New York; Province of New Jersey, New Jersey; Province of Pennsylvania, Pennsylvania; Delaware Colony, Delaware); Southern (Province of Maryland, Maryland; Colony of Virginia, Virginia; Provin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clearing Fractions
In mathematics, the method of clearing denominators, also called clearing fractions, is a technique for simplifying an equation equating two expressions that each are a sum of rational expressions – which includes simple fractions. Example Consider the equation : \frac x 6 + \frac y = 1. The smallest common multiple of the two denominators 6 and 15''z'' is 30''z'', so one multiplies both sides by 30''z'': : 5xz + 2y = 30z. \, The result is an equation with no fractions. The simplified equation is not entirely equivalent to the original. For when we substitute and in the last equation, both sides simplify to 0, so we get , a mathematical truth. But the same substitution applied to the original equation results in , which is mathematically meaningless. Description Without loss of generality, we may assume that the right-hand side of the equation is 0, since an equation may equivalently be rewritten in the form . So let the equation have the form :\sum_^n \frac = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elementary Arithmetic
The operators in elementary arithmetic are addition, subtraction, multiplication, and division. The operators can be applied on both real numbers and imaginary numbers. Each kind of number is represented on a number line designated to the type. Digits Digits are the set of symbols used to represent numbers. In a numeral system, each digit represents a value. The Arabic numerals (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) are the most common set of symbols, and the most frequently used form of these digits is the Western style. A numeral system defines the value of all numbers that contain more than one digit, most often by adding the value of adjacent digits. The Hindu–Arabic numeral system includes positional notation to determine the value of any numeral. In this type of system, the increase in value of an additional digit includes one or more multiplications with the radix value and the result is added to the value of an adjacent digit. For example, with Arabic numerals, the radix ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
