Difference between revisions of "Reduced fraction"
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| − | A '''reduced fraction''', also known as a [[fraction]] written in ''lowest terms'', is a [[rational number]] written in the form <math>\frac a b</math> where <math>a</math> and <math>b</math> are [[integer]]s with no [[common | + | A '''reduced fraction''', also known as a [[fraction]] written in ''lowest terms'', is a [[rational number]] written in the form <math>\frac a b</math> where <math>a</math> and <math>b</math> are [[integer]]s with no common [[divisor]]s (that is, which are [[relatively prime]]). |
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| + | Given an unreduced fraction, one may reduce it by cancelling common factors of the [[numerator]] and [[denominator]] in accordance with the rules of [[arithmetic]]. For example, <math>\frac {15}{27}</math> is not a reduced fraction because both 15 and 27 are divisible by 3. So in order to reduce, we write <math>\frac{15}{27} = \frac{3 \cdot 5}{3\cdot 9} = \frac5 9</math>, and 5 and 9 are relatively prime, so this fraction is reduced. | ||
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| + | Sometimes, it might take several steps to reduce a fraction (because we don't notice all the common factors of the numerator and denominator, for example). For instance, in reducing the fraction <math>\frac{357}{273}</math>, we might first notice that the numerator and denominator are divisble by 3 and so reduce to <math>\frac{357}{273} = \frac{3\cdot 119}{3\cdot 91} = \frac{119}{91}</math>. We have now reduced our original fraction, but it can be reduced further: both 119 and 91 are divisble by 7, and the fraction reduces again to <math>\frac{17}{13}</math>. In this case, we call the intermediate steps ''partially reduced'' and the final, reduced fraction ''fully reduced''. | ||
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Revision as of 23:41, 14 September 2006
A reduced fraction, also known as a fraction written in lowest terms, is a rational number written in the form 
where 
and 
are integers with no common divisors (that is, which are relatively prime).
Given an unreduced fraction, one may reduce it by cancelling common factors of the numerator and denominator in accordance with the rules of arithmetic. For example, 
is not a reduced fraction because both 15 and 27 are divisible by 3. So in order to reduce, we write 
, and 5 and 9 are relatively prime, so this fraction is reduced.
Sometimes, it might take several steps to reduce a fraction (because we don't notice all the common factors of the numerator and denominator, for example). For instance, in reducing the fraction 
, we might first notice that the numerator and denominator are divisble by 3 and so reduce to 
. We have now reduced our original fraction, but it can be reduced further: both 119 and 91 are divisble by 7, and the fraction reduces again to 
. In this case, we call the intermediate steps partially reduced and the final, reduced fraction fully reduced.
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