Monthly Archives: November 2013

Why .9999… (Repetend) Is Not Equal to 1

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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Filed under Mathematics, Philosophy, Philosophy of Mathematics, Pure Philosophy