5

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 $\pi$)?

emacs drives me nuts
  • 10,390
  • 2
  • 12
  • 31
hasleron
  • 161
  • 8
    "Last discovered" and "rounded" sounds like a contradiction to me. If it is "discovered" it should be the true value. – Dietrich Burde Jul 01 '22 at 14:11
  • @DietrichBurde I guess my question aims at how new decimal places are actually discovered, i.e. how does one know how many places of the result are correct. – hasleron Jul 01 '22 at 14:13
  • Also, logically, why would anyone do that? Rounding is only important when used in actual practical calculation, and at that level of precision, it can't possibly be useful for calculation. – Deepak Jul 01 '22 at 14:14
  • Say, if you know that $3<\pi<\frac{22}{7}$, then which decimal digits of $\pi$ are "proved"? Or better, if you know $$\frac{333}{106}<\pi<\frac{355}{133}$$ – Dietrich Burde Jul 01 '22 at 14:15
  • I am sure that they generate all the actual digits, unrounded. That is what it means to be a digit of $\pi$. – TonyK Jul 01 '22 at 14:18
  • @DietrichBurde none, and I guess 3.1 < π < 22/7 would prove the first digit – hasleron Jul 01 '22 at 14:20
  • @hasleron: I would say one, not none. $3$ is a decimal digit (but not a decimal place). – TonyK Jul 01 '22 at 14:22
  • 1
    In case you are interested, see this post for a few estimates like this. – Dietrich Burde Jul 01 '22 at 14:23
  • 4
    I'm gonna contradict everyone here and claim it's a good question. I agree with most here that it's likely the exact digit. The methods to establish digits tell you the error between some approximation and the real $\pi$ is less than $10^{-n}$ and while that doesn't mean that you can give $n$ exact digit, you can just remove some digits until your sure. (You'll have to remove a lot of digits only if you're unlucky and have a lot of 9s in the trail of your approximation) – Bananach Jul 01 '22 at 14:27
  • Thanks to DietrichBurde and @Bananach! I think I got a basic understanding of the discovery process now. – hasleron Jul 01 '22 at 14:33
  • 1
    Upvoted ! I think, the right way to see this is that the "last Digit" is not the only information gathered, but the "Bounds" are included in the information gathered. Eg when 50 trillion Digits are known, ending in "...284" and then this sequence comes in the newer calculations "...284 356" but we only know that it must be between "...284 356" & "...284 358" , then the new Digits are either 356 or 357 or 358. In this case, only 35 will be known Digits; with next Digit known with Bounds "6,7 or 8", which is not "Exactly" known. It will not be rounded to 36, which is losing information gathered. – Prem Jul 01 '22 at 14:51

1 Answers1

0

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.

Thusle Gadelankz
  • 3,416
  • 1
    "and thus I do not have anything clever to say about the corresponding error analysis" I'm sorry, but I'd imagine that would be a strong reason not to say anything at all. Have you heard of quadratically convergent algorithms for $\pi$, or "spigot" algorithms (both discovered last century)? The latter really compute (binary) digits, one by one, without any rounding. – wasn't me Jul 01 '22 at 14:59
  • Yes, user619894, thanks for pointing that out. That was a typo, and it is now corrected.

    As to your comment, wasn’t me, I think our philosophies for these kinds of questions are fundamentally different. Of course I could make this answer more mathematically interesting by including more sophisticated material. However, I believe it is evident that the OP seeks an answer at more basic level. I truly believe that we should answer the question the way we believe it was intended rather than to please those who already knows the answer to the question.

     – Thusle Gadelankz Jul 01 '22 at 15:27