I wonder how Stifel felt about 1/3 and 0.333... Clearly 1/3 is a true number, yet 0.333... conceals 1/3 under a fog of infinity... But what I'm really curious about is 0.999... Who first noticed that Stevin's system gave certain numbers two different decimal expansions? How was the anomaly dealt with? (I can well imagine that infinite sequences of trailing 9's might have been explicitly banned.) Jim Propp Jim On Mon, Aug 17, 2015 at 1:41 PM, Hans Havermann <gladhobo@teksavvy.com> wrote:
Here's a little more of "Numbers" preceding that translation:
"M. Stifel still wrote, in his 'Arithmetica integra' of 1544... 'Just as an infinite number is no number, so an irrational number is not a true number, because it is so to speak concealed under a fog of infinity'. This 'fog of infinity' is already defined rather more precisely by Stevin... as an infinite sequence of decimal fractions, representing a sequence of nested intervals, which he develops, for example, in finding successive approximations to the solution of the equation x^3 = 300x + 33900000."
A Springer book called "Numbers" (1991, Ebbinghaus, et al.) translates "Et procedant ainsi infiniment, l’on approche infiniment plus pres au requis" as "and proceeding in this way unendingly, one approaches infinitely closer to the required value".
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