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This just feels wrong. My professor is claiming that .999... = 1 with the following proof:

$x = .999\dots$

$10x = 9.999\dots$

$9x = 10x - x = 9.999\dots = 9$

$x = 1$

Is this actually correct and I'm just going crazy?

Asinomás
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Ken W
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    The first thing is to sit down and find out/recall, what the notation $0.999\dotsc$ means. How is it to be interpreted? The standard interpretation is that it is the sum of the infinite series $$\sum_{k=1}^\infty \frac{9}{10^k},$$ and that sum is $1$. – Daniel Fischer May 02 '14 at 17:56
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    Does $1 = (3)(\frac{1}{3}) = (3)(0.333\ldots) = 0.999\ldots$ also feel wrong? – Dave L. Renfro May 02 '14 at 18:01
  • Nope, those are great explanations, it just felt wrong putting in x on one side and .999... on the other. This is definitely a duplicate question. Go ahead and close it someone. – Ken W May 02 '14 at 18:02
  • See http://math.stackexchange.com/questions/11/does-99999-1 – user35603 May 02 '14 at 18:05
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    Its sickening that a horde of "oberlehrers" mark questions dealt with the last time four years ago as duplicates, and so can throw them off the board. For the second time today an answer of mine bringing up new aspects to a question has been killed. – Christian Blatter May 02 '14 at 18:25

1 Answers1

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Two real numbers are equal, if their difference is 0. The difference between a and b is 0 if $\forall \epsilon>0 \, |a-b|<\epsilon $. This has to do with how real numbers are constructed. There are several ways. One is e.g. that a real-number is a cauchy sequence of rational numbers. Then, two real-numbers $a_n, b_n$ so defined, are equal if the sequence $c_n = a_n - b_n$ has the limit $0$. Decimal numbers are just one way to represent a real number, and as your example suggests, for some numbers there is more than one decimal representation.

Edit: The proof looks good. The second-last line should read $9x = 10x - x = 9.99999... - 0.99999... = 9$

Michael
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  • The expression does not mean anything in itself. It is a series of symbols and must be defined. There are many possible definitions, of which I offer 3:

    1. 0.99999… is a string of symbols only.

    2. 0.99999… is a limit of the series 0.9 ; 0.99;0.999;0.9999 etc …The limit is obviously 1.

    3. It is a so called “actual infinity” of decimal nines following a leading 0. This needs to be defined in turn. It is not by definition the same as the answer 2 (but most persons believe so). There is a problem is related to the implications for the general concept of actual infinity.

     – Mikael Jensen May 02 '14 at 18:30