# A basic question related with the approximation of real numbers by rationals

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.

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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
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 1:06
@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. – Ben Crowell May 23 '12 at 1:22

$3$ 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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Thank you very much – srijan May 22 '12 at 20:02
@srijan This is true for any real number, e.g. $1$ can be approximated by $1$ or $1$, also $1$ is close enough ;-) – dtldarek May 22 '12 at 20:03
+1 But let's not forget that $355/113$ is better than $314159/100000$. – Jyrki Lahtonen May 22 '12 at 21:07