Slashdot Mirror
Pi Calculated To Record 2.5 Trillion Digits
Joshua writes "Researchers from Japan have calculated Pi to over 2.5 trillion decimals using the T2K Open Supercomputer (which is currently ranked 47th in the world according to a June, 2009 report from Top500.org). This new number more than doubles the previous record of about 1.2 trillion decimals set in 2002 by another Japanese research team. Unfortunately, there still seems to be no pattern."
These researchers are now in possession of the most useless piece of information in science.
3.14 was very useful. 3.1415? Even more so. But after that it's diminishing returns, baby. 2.5 trillion digits? Good heavens. Of course it never repeats - we kind of knew that already.
Pointless mathematical dick-sizing. Problem is, this dick is so huge no vagina will ever make use of it.
...have they found the circle yet?
http://michaelsmith.id.au
Otherwise how would you calculate it? The "pattern" is it matches the stream of digits produced by a simple algorithm!
Researchers will find that Pi begins to repeat after 2,500,000,000,001 digits.
Otherwise it would mean other non-predictable numbers could actually be predictable, potentially make breaking cryptography easier (much like finding out that a prime really isnt), would generally disrupt a bunch of mathematical theorems probably pissing off a whole sect of mathematicians, and turn a lot of things we think we know upside down.
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A nice little article on why it's useless to know pi to more than 50 digits in this universe.
http://everything2.com/title/Too%2520small%2520a%2520Universe%2520to%2520memorize%2520Pi
if they found a repeat at say, 3 trillion digits?*
just so that certain science/ math completists/ perfectionists, who would consider it their duty to know pi exactly, their brains would explode in an attempt to remember the digits
(*i don't think it is possible for pi to repeat at all, i think pi's irrationality is essential to what pi represents)
intellectual property law is philosophically incoherent. it is your moral duty to ignore it or sabotage it
Of course there's a pattern. In fact, an infinite number of them. My favourite is the one in the generalised continued fraction expansion of pi.
entropy happens
Since Pi is irrational, does that mean that a "perfect" circle cannot actually exist? If you don't understand my question, think about it like this. Let's say I want to construct a circle of radius R. To create a "perfect" circle, it seems like I would need a length of material to build the circle out of that was exactly 2*Pi*R, but since Pi is irrational, it seems that you could never actually get any length which is an exact multiple of Pi? If Pi really expands out infinitely, even a circle with a radius the size of a galaxy, or a cluster of galaxies, could never be *exactly* the right length?
Perhaps value derives from the lack of pattern in this particular instance. Some math junkie might look at the problem from that point of view and see what pops up.
Everything in the Universe sucks: It's the law!
Of course there's a pattern. I mean, otherwise, I wouldn't be able to match it with 3.[0-9]{1,}
I fail to see how not understanding the word "seems" is insightful.
How about a pattern appearing when Pi is expressed in another base?
Of course there's a pattern, even a simple and elegant one. It's equal to:
4 * (1 -1/3 + 1/5 -1/7 +1/9 -1/11 +1/13 -1/15 etc., etc., etc.)
Just because the pattern doesn't come out pretty in a decimal representation doesn't mean it's not elegant or not a pattern.
Haven't you seen Contact? Get out your different-sized graph papers!
I'm glad you took the time to post that, instead of say, curing cancer.
Control is an illusion, order our comforting lie. From chaos, through chaos, into chaos we fly
They can't find a pattern because they're doing their calculations using base 10. They should expand their minds and try using another base, perhaps 2. I bet there's a pattern using binary!!
pi=sigma{n:0-infinity}[(1/16)^n][(4/(8n+1))-(2/(8n +4))-(1/ (8n+5))-(1/(8n+6))]
2.5 Trillion digits?
That's nothing. Chuck Norris knows the last digit.
I heard somewhere it's equal to the circumference of a circle divided by it's diameter...
Free the Quark 3 from asymptotic confinement! Bring your charm! Don't get down! All colours and flavours welcome!
I'm glad there is more than one computer.
Ever stopped to think that throwing more computing power at a problem is about as productive as throwing more money at a problem or more man power? You can only do so much before an effort becomes either redundant or the return on investment is as dismal as the stock market has been this past year.
I don't honestly know what the practical value of knowing Pi to the 2.5 trillionth digit is but I'd like to think that there are enough resources in play that the fight for cancer isn't going to miss this one.
Dedicated Cthulhu Cultist since 4523 BC.
While I think that the computing horsepower was misdirected (covered elsewhere), and the last trillion digits could have waited, this post is mostly here for me to be arrogantly dismissive and make dick / vagina jokes.
Pi Calculated = Pi in your face!
I always found the Basel problem to be the most elegant converging series involving pi (being the square root of six times the sum of the reciprocals of the squares), probably because there are so many (elegant) proofs of this (pdf), because it's so simple to understand yet not so simple to prove on a cursory inspection, and because it's the specific case that generalized to one of the most important unsolved problems in mathematics.
Sorry to burst your bubbles but pi is an irrational number so it's impossible for it to have a pattern.
First of all, Pi appears to be normal (that is the digits actually meet certain statistical tests for randomness). That is a pattern in some sense. In any event, digits to any base (even base 2 or base 3) are in many ways a very artificial way of thinking about numbers. A far more natural way is to represent numbers as continued fractions http://en.wikipedia.org/wiki/Continued_fraction, When considering generalized continued fractions, Pi has a variety of different very elegant patterns.
I wonder, who needs that?
FYI
The reason the Babylonians, and the Egyptians, and we use 360 degrees is this:
355/113 = 3.14159292035
pi `= 3.14159265359
A difference of 8.5x10-6%
Which makes 355/113 close enough to pi. 360 is close to 355 which is why we use 360 degrees for angles and time.
The pattern just isn't in base 10. It's in base e. Why does anyone expect to see a numerical pattern in an arbitrary number base like 10? Just because we have 10 fingers doesn't make it the "correct" base for anything.
having effectively zero size, your girlfriend must wish you were throwing a hotdog through the halway :P
Just because nobody has detected a pattern doesn't mean there isn't one.
Don't you think that's an irrational conclusion?
Want to improve your Karma? Instead of "Post Anonymously", try the "Post Humously" option.
"Knowing (thinking) that something doesn't repeat and PROVING that it doesn't repeat are two ENTIRELY different things. I am guessing your maths/science education either stopped very early or you didn't do too well in either."
I wish I'd noticed this earlier, so I could belittle you proper. I'll leave it to you and your superior science/math brain to figure out why I find this amusing.
One day, someone's gonna see it out of nowhere in the first 100 digits, and shit their pants laughing at everyone else.
However, nobody else will know about that person's great discovery, because right after they shit their pants laughing, their head will fucking explode.
If you aren't suspicious of your government's actions, you aren't doing your job as a responsible citizen.
Better ways to represent that.... \[4\cdot\sum_{n=0}^{\infty}\left(\frac{\left(-1\right)^{n}}{2\cdot n+1}\right)=\pi\] was trying for a more elegant representation, but I'm going to first have to figure out how to make slashdot accept mathml...
In God we trust,
everyone else we firewall!!
00000001 110000000
00001110 001110000
00110000 000001100
01000000 000000010
01000000 000000010
01000000 000000010
00110000 000001100
00001110 001110000
00000001 110000000
About two trillin digits down the line, in base 2, scientists discovered a curious pattern... is it purely random, or perhaps a message from the Creators?
Pi = 3.141592653589793helpimtrappedinauniversefactory7108914...
There's a good argument that the choice of pi = (circumference / diameter) was unfortunate; it should have been (circumference / radius). In the light of modern mathematics it seems clear that the radius is more "fundamental" than the diameter; choosing pi = (circumference / radius) = 6.28... gives a number of nice things like:
A = (1/2)pi r^2, just as E = (1/2)m v^2 or d = (1/2)a t^2, and for the same reason.
In general, in the current convention, 2pi seems to show up a lot more than pi, e.g. there are 2pi radians in a circle, sin(x) has period 2pi, etc. All these would become simply pi with the (circumference / radius) convention
Because if there's a pattern in one base, there's a pattern in all bases. It's just maybe less obvious and easy to describe in some.
Wait, we can record a ridiculous amount of data (2.5 trillion digits!) just by calculating pi?
Best.
Compression Algorithm.
Evar!
A pattern is a pattern no matter which number base you use. Changing the base you work in only makes some patterns stand out more than others. (i.e. a multiply-by-10 pattern stands out much better in decimal, and a multiply-by-8 pattern stands out better in octal.)
Pi calculated to 2.5 Trillion Digits: Still thought to be between 3.1 and 3.2...
Even people that believe in pre-destiny look both ways before crossing the street.
We could pay nearly 1/4 of the US deficit.
Mine is Good
\frac{1}{2^6}\sum_{n=0}^\infty \frac{(-1)^n}{2^{10n}} \left( -frac{2^5}{4n+1} - \frac{1}{4n+3} + \frac{2^8}{10n+1} - \frac{2^6}{10n+3} - \frac{2^2}{10n+5} - \frac{2^2}{10n+7} + \frac{1}{10n+9} \right)
This gives pi in binary, and there is a definite pattern there.
640k decimal places should be enough for anybody.
No, Pi is not a formula. Pi is the name given to a certain irrational number, it's a quantity not a formula.
There are various formulas that lead to Pi.
And even as numerical techniques are used to find Pi to increasing levels of accuracy there is a very strong expectation in the community that no pattern of the individual digits exists.
FWIW, it isn't required that you need infinitely many line segments to define something to make the parameterization ratio an irrational number...
For example, take a right triangle. If both legs are the same length (say "1"), the length of the hypotenuse is sqrt(2) which is an irrational number w/o a repetitive pattern in the numerical representation. However, if one of the legs is length "3" and the other is length "4", the hypotenuse is of course "5" which is not an irrational number. So we have a case of the length of one line segment giving us both rational and irrational numbers.
However, as it turns out, the converse is true that the straight line is the only curve that has a rational parameterization for arc-length, but that takes a bit more math to prove it...
http://portal.acm.org/citation.cfm?id=1523523.1523896
Base e, really? You should look at it in base pi. That fraction terminates real fast.
Rankmaniac 2010
Ok so pi = 1.000000000... in base pi. That's a nice pattern. What does that pattern look like in base 10, even if it stands out a little less?
Before you herald this progress, spare a thought for the poor memory-trick savants who will now have to spend half of their lives memorizing another trillion digits of Pi.
Yes, and just because there is static on the TV it does not mean the TV doesn't show an image, right?
While being technically correct, these sort of statements don't help understanding anything. After all, what concept does the word pattern embody if not something that is clearly discernible in some well known format/notation?
Note that I do not argue that there is not pattern, but just that we haven't found it yet, and that what you describe most certainly isn't it (if there is one at all).
They better keep on going, 'cos what if the pattern is that the SECOND three trillion digits are the same as the FIRST three trillion digits, except like BACKWARDS! :O
Man, that'd be SO AWESOME.
Sam! If you will let me be,
I will try them.
You will see.
The f1r5t p0st is right. Just b/c we haven't found one yet doesn't mean there isn't one. However, the fact that Johann Lambert proved it in 1768...does.
but have you considered the following argument: shut up.
22/7 ?
"Unfortunately, there still seems to be no pattern."
Bullshit. The pattern is (most simply) the number that is the circumference of a diameter-1 circle. Or various and sundry Taylor series, etc. Perhaps you mean that it does not repeat, which is a fact known since 1768.
I was actually predicting some numbnut would say that very thing, just not in the article summary itself.
We know where leadership by an anti-intellectual "strongman" who scapegoats minorities and likes boisterous rallies goes
There is no pattern in pi. Simple: 0 degrees to 90 degrees, 'the rules are ever changing'. Can you look for a pattern if something is not constant?
Yes axioms bla bla bla. 0,9 to 1 is not infinite.
Here be signatures
It's a great way to test the performance of these supercomputers, to ensure that their calculations are correct. The calculation of pi to additional decimal places beyond what was previously known is never done with just a single method--otherwise, it is impossible to verify the additional digits. It is always done with two different algorithms to ensure that the result is valid. There are many rapidly converging algorithms (e.g., variations on AM-GM methods can be quadratically convergent or better; BBP-type digit extraction methods; and of course, classic Ramanujan series-type methods). However, computing pi to so many decimal places has much less to do with the chosen algorithm than it has to do with the memory- and computing time-efficient implementations of such algorithms in massively parallel architectures. Thus these calculations serve as very good tests for the robustness of supercomputers. The result is also verifiable to previously known digits, and even beyond the previous record, it is possible to perform statistical analyses to determine whether there are any significant deviations in the distribution of digit frequencies.
So, in summary, it is hardly a useless computation. Not that you're going to get an explanation like this from your usual news sources, which generally do not write for technical audiences.
Also note that distributed computing resources such as Folding@home, or even the Great Internet Mersenne Prime Search don't bother with calculating pi, as the purpose of these projects is to make new discovers in their respective fields of interest.
You mean the Creators are sending us a goatse?
Now I have to re-memorize it.
Nope, a recurring decimal is not irrational. You fail at maths, unless you think 0.0909090909... is irrational too.
If we can put a man on the moon, why can't we shoot people for Apollo-related non-sequiturs?
The algorithm they used is the pattern!
Just because nobody has detected a pattern doesn't mean there is one.
What's the pattern in base e? Cos I'm not seeing it.
10.101002020002111....
The most dangerous drug
An even better proof of the lack of pattern is that human brains are set up to find patterns, even in random data. We haven't found a pattern in pi, so its pretty good proof that it is REALLY random. If someone hasn't found Jesus in it yet, it is more random than tortilla chips, and that is pretty damn random.
A patriot must always be ready to defend his country against his government. -edward abbey
It can't be accurately defined no matter what number base you use either as an interger + mantissa or as a fractional value.
There are, however, irrational--indeed, transcendental--numbers that follow a discernible decimal pattern, like the Liouville constant.
Tomato wedge sperm darts that are Republican.
0.123456789012345678901234567890... = 1234567890 / 9999999999
Any recurring decimal can trivially be written as a fraction.
So kudos for the post, Antique Geekmeister. It's posts like yours that make /. a fun site to read.
Slashdot, fix your code or at least hire someone who is competent at it to do it for you.
Except that in the book, it was in base 11 and the figure was starting and ending with a line of 0 as well.
22/7 = 3.1428571428571 428571428571428571 42857142857142857142857 1428571428571428571428 57142857142857142857142 857142857142857142 85714285714285714
Looks like a pattern to me.
But I thought:
__
3.14 = 3.14141414141414141414141414141414141414141414
Why couldn't we all just be happy with 22/7?
I don't honestly know what the practical value of knowing Pi to the 2.5 trillionth digit is
The benefit is not knowing more about pi but rather knowing better the capabilities of the computer you are using. A test program, if you will. And once we ensure that the computer is accurate and functional, we can set it loose on problems like cancer. Pi is used because we already have extensive data on it and it's a simple matter to compare two computers' output; they should match perfectly.
-b
No offense, but I've stopped responding to AC's.
I'm surprised that nobody posted this yet.
"Unfortunanely, there seems to be no pattern yet", but what about secret messages?
and the amazing secret zero sent from the aliens just happens to display correctly on any random earth made display.
I've decided to Diversify my Holdings. I've divided my cash between my left and right pockets, instead of all in one.
I don't think the finite amount of information makes much sense. If you take a 2 meter long string and make a circle of it, the non-straight circle has a circumference of 2 meters. Well, I was wondering if ellipses can have integer major and minor axes while still having good circumference, but according to Wikipedia, they don't have a way of finding the circumference of an ellipse without going into calculus. Scary! They seemed so well defined.
In the example you give, perhaps you're thinking of Champernowne's number, 0.123456789101112....
This is an irrational number, and was the first number proven to be normal.
Slashdot: news for Apple. Stuff that Apple.
Computing all digits of 42 took a lot longer than that. Just imagine how long the reverse question would take: "What was the question whose answer is 3.141562...?"
cpghost at Cordula's Web.
...but I would be interested to know how often substrings of digits repeat.
Here's an interesting problem: Do arbitrarily long substrings of digits occur an infinite number of times in the decimal expansion of PI? If not, what is the length of the longest substring S that repeats infinitely often?
Slashdot: news for Apple. Stuff that Apple.
This gives pi in binary, and there is a definite pattern there.
Oh my gods, you're right! It's all ONES and ZEROES!!!!!
Hang on a second...
My cousin's sister's uncle's...
Your cousin's sister is also your cousin.
You are Dark Helmet, and I claim my five pounds.
KIKOMAN
Can someone please post the resulting value for PI this computer generated? For some reason it's not in the fucking article...
Thanks,
aeeee
Mmmmmmmmm! Pie!
My digital TV doesn't get static any more......it's either a picture (sometimes with artifacts) or nothing (black). I miss snow.....
decimal expansion is infinite in all bases
Besides the fact that "decimal expansion" is a number's representation in base-10 only, pi need not have an infinite representation in all bases -- it's perfectly valid to have a number system with base pi, for example.
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0.123456789101112131415161718192021....
http://www.geoffreylandis.com
Also note that pi is irrational so its decimal expansion is infinite in all bases.
Well, that's true for all integer bases, but not for all bases. Pi has a finite decimal expansion in some non-integer bases. For example:
pi is exactly 10 in base pi
pi is exactly 11 in base pi-1
pi is exactly 100 in base sqrt(pi)
Many years ago, before electronic calculators, I spent hours in high school math classes converting numbers to base pi (or base e or base phi, etc.) by hand. I was one of the first to finish in-class assignments, which left me with lots of time to kill. For example, e is approximately 2.20212010021 in base pi, and pi is approximately 10.101002020002 in base e.
Those who can make you believe absurdities can make you commit atrocities. - Voltaire
Just because nobody has detected a pattern doesn't mean there isn't one.
:*(
I read the original as "Just because nobody has detected a patent doesn't mean there isn't one." The sad part is that I didn't even blink.
WHY?!? That was my initial gut reaction to this, and, when it comes to gut reactions, mine is fairly substantial... But all glibness aside, why calculate it out that far? Granted my degree is not in mathematics -- it's in engineering. I tend to look for the application of research. I'm not criticizing the people that did this, I just wondered why/how people got funding to do this.
Assuming no coding errors...
You're assuming that the circumference of a circle will always have an irrational length. Not so. There's no reason you couldn't have a circle with a circumference of exactly one meter. Of course, to do so it would have to have a radius of irrational length, but you can't have everything
But according to the bible, both the diameter and the circumference of a circle can even be integers! If the diameter is 10 then the circumference is 30 (not 31 or 31 and a bit as godless mathematicians would have you believe). Clearly, pi must be 3.0, so those guys with their trillions of digits got it wrong in the second digit.
1 Kings 7:23 blurts out: "He made the sea of cast metal, circular in shape, measuring ten cubits from rim to rim and five cubits high. It took a line of thirty cubits to measure around it."
Those who can make you believe absurdities can make you commit atrocities. - Voltaire
If PI is in-fact infinite then eventually any set of numbers that might construe intelligence could be found eventually.
"Be sure to drink your Ovaltine"
"Freedom in the USA is not the ability to do what you want. It is the ability to stop others from doing what THEY want"
My reaction was to the article which said they were applying for a world record. Also according to the article: "Whether a pattern has been found or if the number will become scientifically useful has yet to be announced." They didn't mention comparing the output of two computers, but that might be a viable point. However, if no one has ever calculated it out that far, how can you be 100% positive that the result is correct? Couldn't you theoretically get the same wrong answer using the same program on two computers? I suppose that would at least prove consistency, but not necessarily accuracy.
No if about it. It does has an infinite number of decimal places. It is irrational. Interesting idea that, say, a Turing machine for "intelligence" could be found in the decimal expansion of any irrational number (which requires Strong AI to be true).
Interesting.
Pi is an irrational number. This means it goes on forever. Thus it's safe to assume the set of numbers after 3.14.. are infinite. Is it right to say that it contains ALL possible combination of ALL numbers imaginable? For example. I could search for 1234567890 and it would be found in the numbers trailing Pi because they are infinite and at SOME point they will be present. Is my line of thinking correct?
Also extend this line of thought one step farther. The universe is infinite. There are an infinite amount of galaxies and within these infinite amount of galaxies are infinite number of planets. Of these infinite amounts of planets there are bound to be an infinite amount similar to earth. Since it's infinite and contains ALL possible combination of molecules/atoms is it safe to assume that SOMEWHERE in the universe there is a parallel earth with people exactly like me doing the exact same thing right now except one thing is different. Like a molecule or something .
Think of the TV show sliders, lots of similar parallel dimensions.
Can someone smarter than me chime in on this? (Sorry if this post is one big clump of text. The preview box isn't reflecting my HTML formatting)
Never know, perhaps pi is the code for an intelligence built into the fabric of the universe. Just need to compile it after we find the pattern :P
It should be nice and neat in base pi.
I read the internet for the articles.
All your pattern are belong to us?
Before you herald this progress, spare a thought for the poor memory-trick freaks who will now have to spend half of their lives memorizing another trillion digits of Pi.
fixed that for you
There's a pattern, it's just encrypted and we haven't found the key.
> Unfortunately, there still seems to be no pattern.
No repeating pattern. There already is a pattern -- the formula for pi.
There won't be a repeating pattern because, if there were, it could be represented as a proper fraction, which has been disproven long ago by the Greeks. That pi could not be represented with a fraction, i.e. a ratio, gave rise to the term "irrational", ir-rational.
Now if they're looking for other, non-repeating patterns ala the ending of the original Contact novel, then that's a different story. :)
(-1: Post disagrees with my already-settled worldview) is not a valid mod option.
No, 0.1101001000100001000001... is irrational and does not encode useful information in any known computer language. Just because the number is not an integer fraction does not mean it contain all digit combinations.
Its odd that the poster says, "unfortunately". if we found a true pattern in pi after a certain number of digits (and not just a statistical anomaly), then the entire basis for our mathematical understanding of this universe would seriously be threatened. I for one dont welcome our new masters who didnt create a universe with internal consistency. that would be like living in star wars or start trek. eep.
You hear about the person who didn't rely on anecdotal evidence to support his belief system?
The computer worked out the 2.5 trillionth digit of pi on a Euclidean surface. For those of us in the real world this level of accuracy is not only overkill, but its wrong. You will never live in a place where pi actually equals this number.
1368127 is prime!
Of course it is, if you're computing in base pi.
Help stamp out iliturcy.
Imagine the poor guy who finds that in the math class. I wouldn't want to be the guy who gets the chair after he does.
Is 1563649 a prime number?
Then you zoom out a bit and realize you've been goatse'd by the Creators.
Now if only God provided source code! Instead we get these damn executables...
0.123456789012345678901234567890... = 1234567890 / 9999999999
Or 411522630 / 3333333333.
Or 137174210 / 1111111111.
Alexander Peter Kristopeit bought his basement from his mommy for one dollar.
I believe that formula is actually the hexadecimal digit extraction formula for Pi. As I understand it, it can be used to determine the the value of the Nth hexadecimal digit of Pi without calculating all the previous digits.
Stylish sheet to fix many problems in Slashdot's D3: https://gist.github.com/801524
WTF?
...the future crusty old bastards are already drinking the Kool-Aid.
No, Steve... not pie... Pi!
That which does not kill us makes us... st
I prefer the use of base 13 myself to find the answer sent by the creators to the ultimate question of life, the universe and everything.
"All tyranny needs to gain a foothold is for people of good conscience to remain silent." [Thomas Jefferson]
That's why they use multiple methods whenever they do a calculation.
When you say "base 10", what base is the "10" using?
I guess every base is really base 10...but not necessarily base ten.
But, I wanted socialized health insurance!
OMG WTF try some education?
Here be signatures
I suppose it would have to be normal, and irrational?
Interesting.
I remember seeing this or a related set of patterns in an encyclopedia from the late 50's. The Americana that my parent had featured it.
The Americana from the 80's dropped the equations from what I could tell. Perhaps calculators were a cause of the change.
The main usefulness of PI to that many digits is to provide speed trials for computers.
A second reason is to provide mathematicians with more numbers to base papers on.
What about 0.9999...? It's just 1!
There's no such recurring decimal as 0.9999[...].
Alexander Peter Kristopeit bought his basement from his mommy for one dollar.
Decades ago, while playing around with a scientific calculator, I determined that you could calculate the volume of the Milky Way galaxy in angstroms on it and not overload the limits of the calculator, at least as far as powers of ten are concerned.
*****
Milky Way galaxy diameter = 100,000 light years
1 light year = 9.5 E 15 meters
Milky Way galaxy diameter = 9.5 E 20 meters
1 Angstrom = 1E-10m
Milky Way galaxy diameter = 9.5 E 30 angstroms
Milky Way galaxy radius = 4.75 E 30 angstroms
Milky Way galaxy area = PI * radius squared = 7.1 E 61 angstroms squared
Milky Way galaxy thickness = 1,000 light years = 9.5 E 28 angstroms
Milky Way galaxy volume = Area * Thickness = 6.9 E 90 angstroms cubed
*****
Given the above, you might need to know PI to more than 90 digits if you want to be able to do accurate positional calculations of each atom in our little galaxy, using polar coordinates. You would need a lot more digits to cover the known universe. (Ignoring, of course, the Heisenberg uncertainty principle.)