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Is it possible to express $0.999...$, a repeating number, as a fraction? Or as a ratio of two numbers?

Basically all (my) attempts at the problem cancels all the terms and returns $1$. Is it even possible? Or has it been proven to be an exercise of futility?

José Carlos Santos
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John Glenn
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    If by $0.999\cdots$ you mean $\sum_{n=1}^\infty 9/10^n$ then that's $1$. – Angina Seng Feb 24 '18 at 11:59
  • $0.999\ldots=1$ as said many times before. – Mathematician 42 Feb 24 '18 at 12:00
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    Go here $\longrightarrow$ https://www.livescience.com/57849-greatest-mathematical-equations.html and go through all the slides with the $11$ most beautiful equations in mathematics. You will see one equation in particular, namely $$1 = 0.999999999999999999\ldots$$ – Mr Pie Feb 24 '18 at 12:08

4 Answers4

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Yes, it can: $0.999999999999\ldots=\dfrac11$.

José Carlos Santos
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Let $$x=0.99999\ldots ,$$ now $$10x=9.99999\ldots ,$$ then $$10x-x=9x=9\Rightarrow x=1.$$

Botond
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thesmallprint
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$$0.9999\dots= 3*0.3333\dots= 3 * \frac 1 3=1$$

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0.9 lacks 1 by 0.1

0.99 lacks 1 by 0.01

0.999 lacks 1 by 0.001

0.9999 lacks 1 by 0.0001

0.9999.... lacks 1 by 0.0000...

Hence it is equal to 1

Atul Mishra
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