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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 |
| | </ref> |
| | <ref> |
| | http://www.qc.edu.hk/math/Junior%20Secondary/Componendo%20et%20Dividendo.htm |
| | </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> |
| | |
| | ==Comment on the proof== |
| | We can similarly deduce the much more general fact that the value of any fraction |
| | |
| | :<math>\frac{x_0 + \cdots + x_n}{y_0 + \cdots +y_n}</math> |
| | |
| | in which <math>x_0</math> and <math>y_0</math> are nonzero and can be expressed in terms of the values of |
| | |
| | :<math>\frac{x_1}{x_0}, \ldots, \frac{x_n}{x_0}, \frac{y_1}{y_0}, \ldots, \frac{y_n}{y_0} </math> |
| | |
| | and the value of <math>\frac{x_0}{y_0}</math>, and so depends only on the values of those 2''n'' + 1 fractions: |
| | |
| | :<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 |
| | |
| | :<math>\frac{x+y}{x-y} = \frac{x+y}{x+(-y)}</math> |
| | |
| | can be regarded as a fraction of the above form. |
| | |
| | ==Example== |
| | This method can be used in various situations. |
| | |
| | For instance : |
| | |
| | :<math>\frac{\sqrt{3} + x}{\sqrt{3} - x} = 2</math> |
| | |
| | Find the value of ''x''. |
| | |
| | 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> |
| | : <math>=> \frac{2 \sqrt{3}}{2 x} = \frac{3}{1}</math> |
| | : <math>=> \frac{\sqrt{3}}{x} = 3</math> |
| | : <math>=> x = \frac{1}{\sqrt{3}}</math> |
| | |
| | ==References== |
| | <references/> |
| | |
| | ==See also== |
| | * [[Reduction (mathematics)]] |
| | * [[Fraction (mathematics)]] |
| | |
| | {{DEFAULTSORT:Componendo And Dividendo}} |
| | [[Category:Fractions]] |
| | [[Category:Algebra]] |
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| | |
| | {{algebra-stub}} |
Template:Multiple issues
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[1]
[2]
We can similarly deduce the much more general fact that the value of any fraction
in which and are nonzero and can be expressed in terms of the values of
and the value of , and so depends only on the values of those 2n + 1 fractions:
The original result is essentially a special case of this fact, because
can be regarded as a fraction of the above form.
Example
This method can be used in various situations.
For instance :
Find the value of x.
Solution :
Applying C and D
References
See also
Template:Algebra-stub