Turing was aware that this definition is equivalent to the formula_25-approximation definition given above. The argument proceeds as follows: if a number is computable in the Turing sense, then it is also computable in the formula_25 sense: if formula_37, then the first "n" digits of the decimal expansion for "a" provide an formula_25 approximation of "a". For the converse, we pick an formula_25 computable real number "a" and generate increasingly precise approximations until the "n"th digit after the decimal point is certain. This always generates a decimal expansion equal to "a" but it may improperly end in an infinite sequence of 9's in which case it must have a finite (and thus computable) proper decimal expansion.