[math-fun] Math history questions: infinite decimals
It's well known that Simon Stevin came up with the concept of decimal representation. But did he consider infinite decimals? Who first asserted that 1/3 could be written as .333... (and come to think of it, who came up with "point three bar")? Who first considered decimal expansions of irrational numbers? And who first thought about .999...? Thanks, Jim Propp
"It's well known that Simon Stevin came up with the concept of decimal representation. But did he consider infinite decimals?" http://www.archive.org/stream/larithmetiqvedes00stev#page/353/mode/1up The last three lines imply recognition of infinite decimals. Newton appreciated "the correspondence between decimal numbers and algebraic terms continued to infinity" (1671) and Wallis expressed the idea as "continued approximation" (1685). "who came up with 'point three bar'" Edward Hatton's "An intire system of arithmetic" (1721) uses "r" after a finitely expressed repeating decimal to indicate periodicity. Thus 1/3 is .3r, "or if a digit repeat after others are in the quotient" a number after the r expresses the number of terms repeated.
Thanks, Hans! Can someone whose French is better than mine comment on what Stevin says, and what he does not say? Thanks, Jim On Mon, Aug 17, 2015 at 12:23 PM, Hans Havermann <gladhobo@teksavvy.com> wrote:
"It's well known that Simon Stevin came up with the concept of decimal representation. But did he consider infinite decimals?"
http://www.archive.org/stream/larithmetiqvedes00stev#page/353/mode/1up
The last three lines imply recognition of infinite decimals. Newton appreciated "the correspondence between decimal numbers and algebraic terms continued to infinity" (1671) and Wallis expressed the idea as "continued approximation" (1685).
"who came up with 'point three bar'"
Edward Hatton's "An intire system of arithmetic" (1721) uses "r" after a finitely expressed repeating decimal to indicate periodicity. Thus 1/3 is .3r, "or if a digit repeat after others are in the quotient" a number after the r expresses the number of terms repeated.
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"Can someone whose French is better than mine comment on what Stevin says..." 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".
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".
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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"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?" Alexander Malcom's "A new system of arithmetick" (1730) states matter-of-factly: "If the repetend of any circulate is 9, the value or sum of that series is a unit of the place next to that repetend on the left hand, so, .9 [a dot over the 9] = 1". Richard Gadesby (A treatise of decimal arithmetic, 1757): "'Tis evident that .9 = 9/10 wants only 1/10 of unity; and .99 wants 1/100; .999 wants 1/1000; so that if the series were continued to infinity, the difference between that series of nines and an unit, would be equal to unity divided by infinity, that is, nothing at all." S.F. Lacroix (An elementary treatise on arithmetic, 1825 translation of 1818) concedes the result but feels it necessary to explain that unity is "but a limit".
participants (3)
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Hans Havermann -
James Propp -
Veit Elser