Anomalous cancellation

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in calculus

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 cancelling 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 cases of cancelling trailing zeros or where all of the digits are equal are 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):

  • [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 and those with different numbers of digits (98/392 = 8/32).

Elementary properties[edit]

When the base is prime no two-digit solutions exist. This can be proved by contradiction: suppose a solution exists, and without loss of generality we can say that this solution is

where the line indicates digit concatenation. Thus we have

But as they are digits in base yet which means that so therefore the right hand side is zero which means the left hand side must also be zero, i.e. , a contradiction.

Another property is that the numbers of solutions in a base is odd if and only if is an even square. This can be proved similarly to the above: suppose that we have a solution

Then doing the same manipulation we get

Suppose that . Then note that is also a solution to the equation. This almost sets up an involution from the set of solutions to itself, but a problem arises when . In this case, we can substitute in to get so this only has solutions when is a square. Let . Square rooting and rearranging yields . Since the greatest common divisor of is one, we know that . Noting that , this has precisely the solutions i.e. it has an odd number of solutions when is an even square. The converse of the statement may be proved by noting that these solutions all satisfy the initial requirements.

See also[edit]


  1. ^ Weisstein, Eric W. "Anomalous Cancellation". 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.