Recurring Nines

We learn to compare ordinary finite decimals more or less by eye: thus .09 and .1 not only look different, they really are different—.09 being less than .1, since 9/100 ( = .09) is less than 10/100 ( = .1). But we have gone out of our way to stress the fact that, unlike ordinary finite decimals, infinite decimals do not correspond to decimal fractions; instead they have to be interpreted in a completely new way as endless sums. We therefore have to resist any temptation to assume that procedures which work with finite decimals will automatically carry over to infinite decimals.