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So,I was wondering this: In the decimal representation, if we start writing the 10 numerals in such a way that the decimal portion never ends and never repeats; then am I getting closer and closer to some irrational number?

- Thread starter reek
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- Thread starter
- #1

So,I was wondering this: In the decimal representation, if we start writing the 10 numerals in such a way that the decimal portion never ends and never repeats; then am I getting closer and closer to some irrational number?

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- Jan 26, 2012

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On the other hand, an irrational number can be approximated to an arbitrary number of digits by a rational number. For instance, given any irrational number $\pi$, we can trivially approximate it to $n$ decimals as:If a decimal never repeats and never terminates, then we cannot express it as the ratio of two integers, and so it is called irrational.

$$\overset{\approx}{\pi_n} = \frac{\lfloor 10^n \pi \rceil}{10^n}$$

However, the numerator and denominator will endlessly grow as $n \to \infty$.

In fact, assuming the digits of $\pi$ are randomly distributed, then the numerator is uniform in $0 \leq \lfloor 10^n \pi \rceil \leq 10^n \pi$. Note this is generally not true (I don't even think it is

Then, the probability that $\lfloor 10^n \pi \rceil$ is divisible by $10$ is basically $\frac{1}{10}$, which is subcritical, therefore it is clear that the numerator and denominator will grow exponentially with $n$, largely unsimplified.