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Math and Arithmetic
Calculus
Gems and Precious Stones

Why is point nine repeating equal to one?

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August 09, 2008 8:41PM

To answer that question we should first talk about why any non-termination decimal number is equal to whatever it is. And to talk about that, we should first talk about the value of ordinary terminating decimals. Consider a terminating decimal, say 0.314. This decimal represents the sum of the fractions 3/10 + 1/100 + 4/1000; and longer (but still terminating) decimals can be computed in a similar way. But how do we decide what value a non-terminating decimal represents, say 0.314159265458979... and so on with a never-ending sequence of digits? By analogy, it should be equal to 3/10 + 1/100 + 4/1000 + 1/10,000 + ... and so on; but how can we figure out what such a never-ending sum adds up to? Well, one way of looking at it is as follows: Whatever value the decimal has, we know that (say)0.314 is off by no more than 0.001, since 0.314159... - 0.314 = 0.000159..., and 0.000159... is clearly < 0.001. Likewise, 0.3141 is off by no more than 0.0001, and 0.31415 is off by no more than 0.00001, and so on. In other words, the sequence of (terminating) decimals, 0.3, 0.31, 0.314, 0.3141, 0.31415, etc. gives us a list of better and better approximations to the ultimate value of the non-terminating decimal; and in fact by taking enough decimal places, the error in the approximation can be made as small as you like. If you've studied calculus, you may recognize this sort of discussion--it means that the value of the non-terminating decimal acts like the limit of the sequence of terminating decimals. In fact, it just *is* the limit of the sequence. So mathematicians have chosen to define the value of a non-terminating decimal as the limit of the sequence of approximations. Now we can talk about the specific case of 0.9 repeating: It turns out that the limit of the sequence 0.9, 0.99, 0.999, ... is just equal to 1, exactly (which should not be too hard to convince yourself of) and therefore the value of the non-terminating decimal 0.9 repeating is, by definition, equal to 1.