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| | Hi, everybody! <br>I'm Turkish female ;=). <br>I really love Seaglass collecting!<br><br>Also visit my web site :: [http://www.imariogames.net/profile/roi59.html wordpress backup] |
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| In [[algebra]], '''componendo and dividendo''' (or '''componendo et dividendo''') is a method of [[simplification]] based on fractions provided that they are in proportion. It states that<ref>Bhamra, ''Partial Differential Equations''. PHI Learning Pvt. Ltd. ISBN 978-81-203-3917-0
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| </ref>
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| <ref>
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| http://www.qc.edu.hk/math/Junior%20Secondary/Componendo%20et%20Dividendo.htm
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| </ref>
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| :<math> \text{If } \frac{a}{b} = \frac{c}{d} \text{ and } a \neq b \text{, then } \frac{a+b}{a-b} = \frac{\frac{a}{b} + 1}{\frac{a}{b} - 1} = \frac{\frac{c}{d} + 1}{\frac{c}{d} - 1} = \frac{c+d}{c-d}. </math>
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| ==Comment on the proof==
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| We can similarly deduce the much more general fact that the value of any fraction
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| :<math>\frac{x_0 + \cdots + x_n}{y_0 + \cdots +y_n}</math>
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| in which <math>x_0</math> and <math>y_0</math> are nonzero and can be expressed in terms of the values of
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| :<math>\frac{x_1}{x_0}, \ldots, \frac{x_n}{x_0}, \frac{y_1}{y_0}, \ldots, \frac{y_n}{y_0} </math> | |
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| and the value of <math>\frac{x_0}{y_0}</math>, and so depends only on the values of those 2''n'' + 1 fractions:
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| :<math> \frac{x_0 + \cdots + x_n}{y_0 + \cdots +y_n} | |
| = \frac{x_0}{y_0} \left(\frac{1 + \frac{x_1}{x_0} + \cdots + \frac{x_n}{x_0}}{1 + \frac{y_1}{y_0} + \cdots + \frac{y_n}{y_0}}\right)</math>
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| The original result is essentially a special case of this fact, because
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| :<math>\frac{x+y}{x-y} = \frac{x+y}{x+(-y)}</math>
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| can be regarded as a fraction of the above form.
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| ==Example==
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| This method can be used in various situations.
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| For instance :
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| :<math>\frac{\sqrt{3} + x}{\sqrt{3} - x} = 2</math>
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| Find the value of ''x''.
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| Solution :
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| Applying C and D
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| : <math>\frac{(\sqrt{3} + x) + (\sqrt{3} - x)}{(\sqrt{3} + x) - (\sqrt{3} - x)} = \frac{2 + 1}{2 - 1}</math>
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| : <math>=> \frac{2 \sqrt{3}}{2 x} = \frac{3}{1}</math>
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| : <math>=> \frac{\sqrt{3}}{x} = 3</math>
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| : <math>=> x = \frac{1}{\sqrt{3}}</math>
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| ==References==
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| <references/>
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| ==See also==
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| * [[Reduction (mathematics)]]
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| * [[Fraction (mathematics)]]
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| {{DEFAULTSORT:Componendo And Dividendo}}
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| [[Category:Fractions]]
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| [[Category:Algebra]]
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| {{algebra-stub}}
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Hi, everybody!
I'm Turkish female ;=).
I really love Seaglass collecting!
Also visit my web site :: wordpress backup