Is the last discovered digit of $\pi$ rounded or not?

Question

Since 2019 about 31.4 trillion decimal places of $\pi$ are known, as someone not having any idea how new digits are discovered, my question is about the last known digit. E.g. if there is a sequence in $\pi$ such as 351989, but only 3 more digits were discovered at the time, would those be 351 (the actual digits) or 352 (resulting in the closest approximation of the value of $\pi$)?

Answer 1

This answer is written under the assumption that the OP has little familiarity with higher level mathematics and numerical analysis and will address the question rather broadly.

The problem of approximating $\pi$ is addressed within the branch of mathematics known as numerical analysis. First of all it is worth noting why that is a problem at all. The constant $\pi$ is a so-called irrational number, meaning that it is not an integral fraction. That also means that it has no expansion in a any base which is finite or ends in infinite repetition of some finite string of digits.

There are several ways of approximating irrational numbers. One way is to use series. These are limits of sums. For example $\pi$ cam be expressed as the series $$\pi = 4 \sum_{n=0}^{\infty} \frac{(-1)^{n}}{2n+1}.$$ Knowing this one can approximate $\pi$ by truncating the series, i. e. $$\pi \approx 4 \sum_{n=0}^{N} \frac{(-1)^n}{2n+1}$$ for some integer $N$. Another way of approximating $\pi$ would be to express it as an integral and then try to approximate that integral. There are several general methods for doing this. There are also other ways of doing this, for example iteratively, but there is only so much I can say here. A procedure for approximating something like this is called a numerical algorithm or scheme and the result, which is meant as an approximating of the relevant quantity, is called an estimate.

Now your question arises. What exactly does $\approx$ mean here? How close is the sum the right to $\pi$, how many digits will be correct? Figuring this out is called error analysis and it is an important and necessary part of developing and applying any numerical algorithm. It often involves some «hard core» mathematical analysis (derivatives, integrals, series, inequalities, etc. ) to do this, but in the end you end up with another estimate (…) hopefully telling you how far off your inital estimate is from what you want to calculate. This can again be translated into number of exact digits as explained by Bananach in the comments.

Now, to answer your question more concretely, I am not familiar with the particular scheme used to calculate the currently best known estimate of $\pi$, and thus I do not have anything clever to say about the corresponding error analysis. However, I do agree with the comments that semantically you should interpret «the last known digit» to that digit being exact, not rounded.

I hope this was informative and gives you a better understanding.