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Why does 0.9999... = 1 ?This answer is adapted from an entry in the sci.math Frequently Asked Questions file, which is Copyright (c) 1994 Hans de Vreught (firstname.lastname@example.org).
The first thing to realize about the system of notation that we
use (decimal notation) is that things like the number 357.9
So in modern mathematics, the string of symbols
One can show that this limit is
Proof: 0.9999... = Sum 9/10^n (n=1 -> Infinity) = lim sum 9/10^n (m -> Infinity) (n=1 -> m) = lim .9(1-10^-(m+1))/(1-1/10) (m -> Infinity) = lim .9(1-10^-(m+1))/(9/10) (m -> Infinity) = .9/(9/10) = 1Not formal enough? In that case you need to go back to the construction of the number system. After you have constructed the reals (Cauchy sequences are well suited for this case, see [Shapiro75]), you can indeed verify that the preceding proof correctly shows
0.9999... = 1 Thus x = 0.9999... 10x = 9.9999... 10x - x = 9.9999... - 0.9999... 9x = 9 x = 1.Another informal argument is to notice that all periodic numbers such as
R.V. Churchill and J.W. Brown. Complex Variables and Applications.
E. Hewitt and K. Stromberg. Real and Abstract Analysis. Springer-Verlag, Berlin, 1965.
W. Rudin. Principles of Mathematical Analysis. McGraw-Hill, 1976.
L. Shapiro. Introduction to Abstract Algebra. McGraw-Hill, 1975.
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