Recurring Digits in Irrational Decimals

Extract

Is it always possible to determine a real number to an assigned number of decimal places? Can we always tell how many terms of a convergent sequence must be taken to determine its limit to, say, four decimal places? How much information does the convergence of a sequence give us about the decimal expansion of the limit? These are the questions we shall consider; we shall find that the answer to the first two is “no”. To the third question the answer is that convergence itself gives very little information but that “most” convergent sequences have in fact a subtler property which gives precise information about the expansion of the limit, the rate at which the successive digits of the expansion can be determined and the number of times a digit can recur. The questions we have raised are of practical importance for it is with the approximations-to-so-many-places, not with real numbers themselves, that all practical mathematics is concerned.