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I'm a student of 10th std. Recently our teacher asked a Question that

"Is 4.999...equal to 5 or not?" Everyone said that is isn't equal or it is approximately equal. Teacher too agreed to that. But I did't agree. I opposed the teacher as I think it is precisely equal to 5. I also prove but then too she isn't satisfied. Can you please explain what is right and what is wrong in this case?Please Justify.

   Proof that I showed : let x=4.999...  (a)
                           therefore, 10x = 49.999...
                                      10x -x = 49.999... -4.999...
                                      9x     =  45
                                      x = 45%9
                                      X= 5     (b)
                         hence (a) =(b).
                                    And therefore, ***4.999... = 5***
Samved Joshi
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  • Well you cannot just substract $x$ on the one side and roughly about $\frac{1}{10}$ of the value on the right side.. – Tesla May 17 '16 at 05:34
  • This is very similar to the the non-terminating decimal $0.999...$ being equal to $1$. You are correct in this case that $4.999...=5$. – Ethan Hunt May 17 '16 at 05:37
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    Your teacher should be fired. 4.9999.... = 5 exactly and that is basic for all studiers of math. Your explanation is a good argument although although it begs the question as to what 4.99999.... actually means. I won't go into details but you should google "real numbers completeness" and "infinite decimal expansions and converging sequences". But ... you are right and your teach is absolutely unequivically and totally wrong. – fleablood May 17 '16 at 05:38
  • @fleablood This. Any statement involving "$\ldots$" requires the author to specify unambiguously what is meant. Do the dots only stand "for a couple of nines" or"infinitely many nines"? – Hagen von Eitzen May 17 '16 at 05:42
  • You need to define what is meant by $4.\dot{9}$ first, then you can discuss equality. With the usual limit definitions you are correct. And your teacher certainly needs a refresher... – copper.hat May 17 '16 at 05:47
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    It is understood and universally and unwaveringly accepted in the entire worldwide math community that "...." mean "infinitely many nines" and this is unambiguous and absolute. And as the reals are complete with least upper bound property 4.9999.... with an infinite number of 9s is = 5 is not a matter for discussion. It is correct and undisputed.... however, I will grant you that at the OP's level 4.999... does have to be defined and the technical definition and meaning of an infinite number of 9s is usually hand waved and elided over. – fleablood May 17 '16 at 06:00
  • I strongly agree with fleablood. Hardly anyone writes $\dots$ and means a finite amount of stuff, especially when dealing with decimals or series expansions. – Simply Beautiful Art May 26 '16 at 21:22

1 Answers1

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Yes, you are right in this... every terminating decimal fraction $n$ has two representations, namely $n$ and $n-1+0.999\dots$. This can be shown using a geometric series. In your case, we have that $4.999\dots = 4 + \frac{9}{10} + \frac{9}{100} + \frac{9}{1000} = 4+\frac{9}{10}(1+\frac{1}{10} +\frac{1}{100}\cdots)$
We now apply the formula $1+r+r^2+r^3\cdots = \frac{1}{1-r}$ to the RHS of the equation above to get
$$4 + \frac{9}{10(1-\frac{1}{10})} = 4+\frac{9}{9} = 5$$

Brevan Ellefsen
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