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The earliest mention of the converse, that all repeating decimals are rationals, comes late, surprisingly late, by Lambert in 1758, he justifies it by using the now familiar trick with the geometric series. In Euler's Elements of Algebra (1765), "one of the earliest books to set out algebra in the modern form... and one of Euler's few writings that are accessible to the general public" we find the same trick $9.999...=9+9/10+9/10^2+...=\frac{9}{1-1/10}=10$. However, according to the formal attitude of the 18th century, e.g. Euler's, we also have $1+2+2^2+\dots=\frac{1}{1-2}=-1$, by definition. It took some time before Cauchy tied summing series to convergence.

The earliest mention of the converse, that all repeating decimals are rationals, comes late, surprisingly late, by Lambert in 1758, he justifies it by using the now familiar trick with the geometric series. In Euler's Elements of Algebra (1765), "one of the earliest books to set out algebra in the modern form... and one of Euler's few writings that are accessible to the general public" we find the same trick $9.999...=9+9/10+9/10^2+...=\frac{9}{1-1/10}=10$. However, according to the formal attitude of the 18th century, e.g. Euler's, we also have $1+2+2^2+\dots=\frac{1}{1-2}=-1$, by definition. It took some time before Cauchy tied summing series to convergence.

In another book of Katz's the quoted passage is reproduced almost verbatim, except for one word:"one can potentially calculate an infinite decimal expansion of a number". This goes back to the view of Aristotle and Euclid that all infinity is only potential, not actual, not completed, which continued to exert heavy influence on mathematicians until late 19th century. Infinite strings of digits were mostly contemplated as processes, not finished objects. A notable exception was Stevin's De Thiende (1585), largely responsible for acceptance of the decimal system in Europe (it helped that Stevin was more engineer than mathematician). But rejection of actual infinities infected even the identification of repeating decimals with rationals, let alone of "lawless" strings with irrationals. This attitude is deep seated, even today many students have Aristotelian intuitions of infinity in repeating decimals. According to Tall "interviews revealed that students continued to conceive of $0.999...$ as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven’t specified how many places there are'".

In another book of Katz's the quoted passage is reproduced almost verbatim, except for one word:"one can potentially calculate an infinite decimal expansion of a number". This goes back to the view of Aristotle and Euclid that all infinity is only potential, not actual, not completed, which continued to exert heavy influence on mathematicians until late 19th century. Infinite strings of digits were mostly contemplated as processes, not finished objects. A notable exception was Stevin's Arithmetic (1585), largely responsible for acceptance of the decimal system in Europe (it helped that Stevin was more engineer than mathematician). But rejection of actual infinities infected even the identification of repeating decimals with rationals, let alone of "lawless" strings with irrationals. This attitude is deep seated, even today many students have Aristotelian intuitions of infinity in repeating decimals. According to Tall "interviews revealed that students continued to conceive of $0.999...$ as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven’t specified how many places there are'".

It is interesting to compare this to a similar issue for continued fractions. Fowler argues that already Theaetetus (c. 417 – 369) had a heuristic understanding that simple terminating continued fractions correspond to commensurable ratios and periodic ones to ratios "commensurable in square" (rationals and quadratic irrationals in modern terms). But it was stated and proved explicitly only around the same time as for repeating decimals, by Euler in De Fractionlous Continious (1737) and Lagrange in Sur la Resolution des Equations Numeriques (1767), respectively.

It is interesting to compare this to a similar issue for continued fractions. Fowler argues that already Theaetetus (c. 417–369 BC) had a heuristic understanding that simple terminating continued fractions correspond to commensurable ratios and periodic ones to ratios "commensurable in square" (rationals and quadratic irrationals in modern terms). But it was stated and proved explicitly only around the same time as for repeating decimals, by Euler in De Fractionlous Continious (1737) and Lagrange in Sur la Resolution des Equations Numeriques (1767), respectively.