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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 [[Reduction (mathematics)|reduce]] a [[Fraction (mathematics)|fraction]] by canceling individual [[Numerical digit|digit]]s 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.<ref>{{Mathworld|title=Anomalous Cancellation|urlname=AnomalousCancellation}}</ref> | |||
Examples of anomalous cancellations which still produce the correct result include: | |||
: <math>\frac{64}{16} = \frac{\!\!\!\not64}{1\!\!\!\not6} = \frac{4}{1} = 4</math> | |||
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: <math>\frac{26}{65} = \frac{2\!\!\!\not6}{\!\!\!\not65} = \frac{2}{5}</math> | |||
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: <math>\frac{98}{49} = \frac{\!\!\!\not98}{4\!\!\!\not9} = \frac{8}{4} = 2.</math><ref name=boas>[[Ralph P. Boas, Jr|Boas, R. P.]] "Anomalous Cancellation." Ch. 6 in ''Mathematical Plums'' (Ed. R. Honsberger). Washington, DC: [[Mathematical Association of America|Math. Assoc. Amer.]], pp. 113–129, 1979.</ref> | |||
The article by [[Ralph P. Boas, Jr|Boas]] analyzes two-digit cases in [[base (exponentiation)|base]]s other than [[base 10]], e.g., 32/13 = 2/1 is the only solution in base 4.<ref name=boas/> | |||
==References== | |||
{{reflist}} | |||
[[Category:Arithmetic]] | |||
<br /> | |||
{{numtheory-stub}} | |||
Latest revision as of 19:11, 3 December 2013
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 canceling 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.[1]
Examples of anomalous cancellations which still produce the correct result include:
The article by Boas analyzes two-digit cases in bases other than base 10, e.g., 32/13 = 2/1 is the only solution in base 4.[2]
References
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Here is my web site - cottagehillchurch.com - ↑ 2.0 2.1 Boas, R. P. "Anomalous Cancellation." Ch. 6 in Mathematical Plums (Ed. R. Honsberger). Washington, DC: Math. Assoc. Amer., pp. 113–129, 1979.