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] The trivial case of canceling trailing zeros is ignored.

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. Boas, R. P. "Anomalous Cancellation." Ch. 6 in Mathematical Plums (Ed. R. Honsberger). Washington, DC: Math. Assoc. Amer., pp. 113–129, 1979.