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:

\frac{64}{16} = \frac{(6)4}{1(6)} = \frac{4}{1} = 4
\frac{26}{65} = \frac{2(6)}{(6)5} = \frac{2}{5}
\frac{98}{49} = \frac{(9)8}{4(9)} = \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 is the only solution in base 4.[2]

References

  1. ^ Weisstein, Eric W., "Anomalous Cancellation" from MathWorld.
  2. ^ a b Boas, R. P. "Anomalous Cancellation." Ch. 6 in Mathematical Plums (Ed. R. Honsberger). Washington, DC: Math. Assoc. Amer., pp. 113–129, 1979.