how to get the integer part of very large integer multiples of irrational numbers such as $\pi$?

Question

Do I need exact value of $\pi $ upto say $100$ digits if my multiplier is order $10^{10}$.

Answer 1

The answer is: it depends. Say you have a multiplier of $N \approx 10^{10}$ (it sounds like it's not exactly $10^{10}$ because that would be fairly easy). You're asking whether $\lfloor N\pi \rfloor$ can be different from $\lfloor N\pi^*\rfloor$ when $|\pi-\pi^*|$ is as small as $10^{-100}$. For simplicity let's assume $\pi^*$ is a truncated (not rounded) approximation to $\pi$ so that $\pi^*<\pi$.

Well, the contents of those two floors differ by only $10^{-100}N \approx 10^{-90}$. That may seem like an insignificant difference, but it still matters when $N\pi$ is very close to an integer. Most of the time you can compute $N\pi^*$ and you can see that it is at least $10^{-90}$ away from being an integer, and safely conclude that $N\pi$ has the same integer part. But what if $N\pi^*$ is just, just below an integer and $N\pi$ is just, just above that same integer (resulting in different floors)? What we'd really like to know is,

How close can $N\pi$ be to an integer? In other words, how small could $\|N\pi\| := \min \{|N\pi-M| : M \in \mathbb Z\}$ be relative to $N$?

This is the subject of an active body of research and is related to the notion of irrationality measure. It is known from continued fraction theory that for any irrational constant $\alpha$ there are values of $N$ where $\|N\alpha\|$ is smaller than $1/N$, but in the case of $\pi$ it occasionally gets much smaller (well-known example: $113\pi \approx 354.99997$).

In practice this is something we can control. We already know enough digits of $\pi$ that we can compute (efficiently by using continued fractions) that for $N$ of order $10^{10}$ the above is essentially the worst case, and that $N\pi$ will never be less than $1/(300N)$ away from any integer. That means you never need more than about 13 decimal places of $\pi$ to get the integer part for such smallish $N$, even if you are very unlucky and choose an $N$ where $N\pi$ is close than integer.

If we extend your question to much larger values of $N$ then this is where irrationality measure comes in. We believe the irrationality measure for $\pi$ is $2$, as is known to be true for "most" irrational numbers. A low irrationality measure for $\pi$ means that $\|N\pi\|$ doesn't ever get really, really small, so lower is better for our purposes. But we only how to prove that it is at most $7.6063$, and that is a recent result of Salikhov (2008).

Salikhov's bound means that $\|N\pi\|$ cannot be smaller than about $N^{-6.61}$, at least for very large $N$. Then for $N$ of order $10^m$ you shouldn't need more than about $7m$ digits of $\pi$ (as long as $m$ is large enough – I honestly don't know for sure how large it needs to be but in practice it hardly matters).

Of course, if you carefully choose the "wrong" irrational number $\alpha$ then you can easily rig it so that you may need a very large number of decimal places to correctly get the integer part of $N\alpha$: some numbers do have infinite irrationality measure (Liouville numbers).