Anomalous cancellation

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 (these and their inverses are all the cases in base 10 with the fraction different from 1 and with two digits):

\frac{64}{16} = \frac{\not64}{1\not6} = \frac{4}{1} = 4


\frac{26}{65} = \frac{2\not6}{\not65} = \frac{2}{5}


\frac{19}{95} = \frac{1\not9}{\not95} = \frac{1}{5}


\frac{98}{49} = \frac{\not98}{4\not9} = \frac{8}{4} = 2.[2]

The article by Boas analyzes two-digit cases in bases other than base 10, e.g., 32/13 = 2/1 and its inverse are the only solutions in base 4 with two digits.[2]

The anomalous cancellation happens also with more digits, e.g. 165/462 = 15/42.

References

  1. ↑ Weisstein, Eric W., "Anomalous Cancellation", MathWorld.
  2. ↑ 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.