I recently bumped into a graduate student in the economics department at the University of Pittsburgh, Shawn McCoy, and he brought to my attention that there are some folks who wish to claim that .9999…=1. That is, the decimal value, .9-repetend, which has infinitely many places of ‘9’ after the decimal, is equivalent to the whole number, 1. Any individual of sufficient commonsense and no real inclination toward contrarianism-for-the-sake-of-contrarianism will maintain that the claim is silly and move on. However, there is a bit of mathematical prestidigitation —and that’s precisely what it is, as I will show— presents an “argument” to the contrary of commonsense. The argument requires that we do the following:
Let x be .9999… Then, let the left-hand side (LHS) of our equation be 10x-x. Also, let the right-hand side of the equation (RHS) be the same, not in algebraic terms, but in numerical terms: 9.9999…-.9999…=9. Solving the LHS, we get 10x-x=9x. The conundrum is that it should be the case, of course, that LHS=RHS. However, if one divides LHS and RHS by 9, the consequent value is x=1, though, ab initio, we said that x=.9999… Therefore, some try to conclude, .9999…=1. Continue reading
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