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I have studied that any real number $x$ can be approximated by rationals since the rationals are dense on the real line.

I am searching for an example . Can anyone show this with an example?

Thanks for any help.

Srijan
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    What definition of the real numbers are you using where this isn't true by definition? – Qiaochu Yuan May 22 '12 at 19:58
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    Not really close, the other question asks for cheap-but-good rational approximations, this one asks about the fact of approximability. – André Nicolas May 22 '12 at 20:02
  • @Andre I have reopened it so that the community can decide on duplicity. – Bill Dubuque May 22 '12 at 22:56
  • @BillDubuque: Thank you. No big issue involved, of course, just a matter of accuracy. The other question is connected with very interesting musical theory. This one is not. – André Nicolas May 23 '12 at 01:06
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    @QiaochuYuan: The term "real number" came a couple of centuries before it had a clear definition. This is a perfectly reasonable question to ask without knowing any rigorous definition of the reals. Anyway, one thing to point out is that there are several separate properties involved: (1) Given a real number, one can approximate it to any given, finite precision as a finite decimal expansion. (2) Given any infinite decimal expansion, there is always at least one real number that it approximates arbitrarily well... (3) ...and no more than one. The hyperreals satisfy 1 and 2 but not 3. –  May 23 '12 at 01:22
  • @Qiaochu: I don't know why srijan didn't answer, but I thought it worth mentioning that a number of introductory real analysis texts introduce the reals axiomatically, e.g. as an ordered field with the least upper bound property, or an ordered field in which bounded monotone sequences converge. From there one derives the Archimedean property and consequent density of the rationals early. The actual construction(s) may be included as appendices or optional sections or guided exercises, and they do tend to make density of the rationals more immediately apparent, but are typically not emphasized. – Jonas Meyer May 25 '12 at 06:45
  • @JonasMeyer Sorry sir i didn't reply. I am cleared with this now.Ya The Archimedean property leads to the density of rationals in $\mathbb{R}$" and density of irrationals in $mathbb{R}$" i.e. between any two distinct real numbers there is a rational number and also between any two distinct real numbers there is an irrational number. – Srijan May 25 '12 at 06:55

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$3$ is rational number that approximates $\pi$ with an error less than $1$.

$3.1$ is rational number that approximates $\pi$ with an error less than $1/10$.

$3.14$ is rational number that approximates $\pi$ with an error less than $1/100$.

$3.141$ is rational number that approximates $\pi$ with an error less than $1/1000$.

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