Sunday, December 1, 2013

The perils of dividing by zero

In a middle school math competition a few years ago, the following question was asked: given an n by m matrix of numbers whose average row sum is R and average column sum is C, what is the ratio R/C.  Since R = T/n and C = T/m, where T is the sum of all the entries in the matrix, it would appear that R/C = m/n. This is the expected answer for this question and is correct in almost all cases.  But one must be careful when dividing by a number to ensure that the divisor is not zero.  We assume the matrix is nondegenerate, i.e. m > 0 and n > 0, so dividing by n and m is okay and R, C are well-defined.  However, it is possible that C = 0, in which case R/C is not well-defined.  This cases occurs when T = 0, in which case R is also 0.  A simple example is when the matrix consists of all zeros.  The use of undefined operations such as dividing by 0 can lead to an alleged "proof" of 1 = 2 which is not good as we do not want an inconsistent theory.