DailyDirt: Pi Math
from the urls-we-dig-up dept
National Pie Day is not actually March 14th (although it really should be, if only to make it more memorable). But here’s to the number, not the delicious dessert.
- Does pi contain every set of finite number sequences? The answer to that question may not be known, but the first trillion or so digits of pi appear to be statistically random — with 0-9 appearing with even distributed frequency. [url]
- It’s possible to calculate the nth digit of pi without calculating every previous digit. So the gazillionth digit of pi can be verified, if you really need to know it. [url]
- If you’re thinking about coming up with a new way to calculate pi, you can check your work for the first several trillion digits. Beyond about 10 trillion digits, you’re into record breaking territory, and you’ll need to adopt some other strategies. [url]
If you’d like to read more awesome and interesting stuff, check out this unrelated (but not entirely random!) Techdirt post.
Filed Under: calculations, irrational, math, numbers, pi
Comments on “DailyDirt: Pi Math”
Pi
Pi is an irrational number, which means that it cannot be expressed exactly as a ratio of two integers (such as 22/7 or other fractions that are commonly used to approximate π); consequently, its decimal representation never ends and never settles into a permanent repeating pattern.
Does pi contain every set of finite number sequences? The answer to that question may not be known
WRONG, IT IS KNOW… YES
…
Re: Pi
Just because it never ends and never settles into a permanent repeating pattern, does not mean it contains every finite number sequence.
For example, if you converted pi to base 9 but then treated that pattern of numbers as base 10, you would still have a sequence of numbers that never ends and never settles into a permanent repeating pattern – but there would be no 9’s in it, and that number would therefore not contain every set of finite number sequences.
Unless you have some other proof? I’m sure the mathematical world would be glad to hear it.
Re: Re: Pi
@AnonCwrd: you’re NOT converting it to base9 and then treating it as base10, that’s just stupid. Either convert it to base9 and treat it as base9 or leave it as base10. In base9, every possible sequence of the digits 0-8 will be there.
It’s the same as if the universe was infinite. Every possible combination of particles would be duplicated an infinite number of times so there’d be an infinite number of me’s out there. Wow, I might finally be able to get some work done around the house 🙂
Re: Re: Re: Pi
No, the original article is correct. While the digits of pi do pass statistical tests for randomness, they are not truly random. If they were truly random, we would not be able to calculate them, by definition.
It is possible (even probable, I’d say) that it contains all possible digit sequences, but this is conjecture.
Re: Re: Re:2 Pi
who said it was, or HAD to be random ??
Re: Re: Re:2 Pi
that’s why it’s a random distribution, not in itself a random number.
it is far from random as you correctly stated because you can calculate it, but once you do that calculation you find the distribution of any number position to have a random distribution.
and as it meets the definition of a random distribution, given a string long enough all possible sequence combinations of any length will occur, 100% of the time.
and infinitely long random distribution of numbers IS LONG ENOUGH.
even a small number like 100 million random numbers give a 100% chance of getting 1 to 5 length string matches.
and 100 million is TINY, compared to 10000000 trillion which is tiny (equally tiny) compared to infinity.
you could put a digit on every cell in your body (approx. 50 million million) or 50 trillion cells and you would not even come close to finding that length sequence of numbers in an infinite sequence of random (distributed) numbers.
if fact the set of sequences of random number that would occur in an infinite set of random numbers is itself infinite.
so the probability of any sequence of any length is 100% that it will occur and infinite number of time within that infinite sequence. (of randomly distributed decimal numbers).
The length of the string of randomly distributed decimal numbers to ensure 100% chance of a sequence of ANY size can be calculated.
That calculation does not equal infinity, with a finite sequence length it’s a finite number of randomly distributed numbers.
you could probably write a random number generator and have it generate a 100 digit number of random distribution.
very good odds you will have in it 10 1’s
make it 1000 and you will have close to 100 1’s
if it is a random distribution with 10 1’s on your 100 digit number one of those 10 1’s will be followed by the random number 2
so with a 100 length distribution you can expect 10 1’s and 1 followed by 2 occurring on one occasion.
therefore with a 100 length distribution you have an almost 100% chance of getting the sequence ’12’ or any other 2 length sequence.
for 3 digits match, you are going to need all the 2 digit sequences to have occurred at least 10 times (to cover 0 to 9) so you are going to need 1000 random numbers to ensure 100% chance for a 3 digit match (of non-repeating random numbers).
to ensure 4 digits match, 10,000
with pi only calculated to 100 million places you can have 100% probability that a 0 to 5 length sequence will occur.
so if you need 100 million to ensure 5 length sequence, you would need 10 times that value for your 5 length to occur assuming a random number between 0 and 9 fall as the next number, now if it’s only 10 times bigger there is a possibility of a number repeating, so if you make the size 100 times larger, your 5 sequence will occur 100 time within that sequence, out of that 100 time there is 100% chance the next digit will be a 0 to 9 digit.
so you can clearly see that you do not need a string of infinite length of distributed numbers to find any length finite size string match with 100% probability you just need an exponentially larger initial number.
No where even close to infinite
Re: Re: Re:3 Pi
Ahh, but has it been proven to be a truly random distribution, or does it just pass our current best statistical tests for being a random distribution? It’s not the same thing!
Re: Re: Re:4 Pi
yes it has..
there is no ‘current best statistical tests’ it’s either a random distribution or it follows some sequence.
but it is PURELY random distribution, that means if you have a large enough sample size you will get an equal number of all possible value.
so a number of 100 length will have an relative even distribution of each number and there will be approximately 10 of each.
That is what a random distribution means
you cannot ‘prove’ a truly random distribution, it’s either random or it is not.
Re: Re: Pi
Interesting theory – if you remove all nines from a set, that set will not contains any nines.
Re: Pi
Citation needed. Just because you have an infinitely long string of non-repeating numbers, it doesn’t necessarily mean that every sequence possible is contained in that string. Perhaps it is true that every irrational number has this property but I don’t think it’s been proven.
Re: Re: Pi
yes it does Michael. It means EXACTLY THAT.. infinite is not ‘just very big’ it means it goes on forever, NEVER ENDS, therefore as they are non-repeating numbers you WILL FIND somewhere along that infinite line (probably towards the end !!!) and sequence of numbers your social security number, your telephone number, date of birth.
easily your IQ.
if you could buy an infinite number of (non-repeating)lotto tickets the odds of you winning is 100%.
I guess you just cant get your head around what infinite actually means.
Re: Re: Re: Pi
Despite the insults, you have not proven your assertion. Non-repeating infinite strings of numbers do not necessarily contain every finite sequence of numbers. A simple example: there could be an infinitely long string of numbers that NEVER contained the sequence 1234567890. Perhaps pi does or does not contain that sequence, but there are an infinite number of finite sequences — it’s difficult to really know.
If you can prove it, by all means, post your solution here!
Re: Re: Re:2 Pi
despite your ignorance, you have failed to prove your assertion
your example does not make much sense, there cannot be an infinitely long string of (essentially random) that does not contain and equally infinite of infinite sequences of numbers.
if you have a string of infinite numbers non-repeating (essentially random). you will have 1234567890 occurring an infinite number of time. NOT zero times.
if you want to proof, you can work it out yourself. or just look up the web page that has it.
with only 100 million digits of pi, you have a 0.995% chance of finding your 1234567890.
almost 1% in only 100 million.
so multiply the number of digits by 1000 and you will have 100% chance of finding your 10 digit sequence.
1000 times 100 million falls very short of infinite, 100% is still 100%.
in fact you would expect to find that sequence 10 times in your pi to 100 billion decimal places.
And I don’t think 100 billion is very close to infinite, what do you think ???
Re: Re: Pi
I have in fact proved it several times, it’s your level of comprehension that is the problem.
If you cannot grasp this simple concept that is a problem for you to solve not me.
it’s a sad reflection on the quality (or lack of) the US education system.
But Pi is wrong! http://tauday.com/
Re: Re:
Yes, it is a well known fact and legislators have attempted to correct this in the past – but those pesky mathematicians.
http://en.wikipedia.org/wiki/Indiana_Pi_Bill
For some reason an irrational number is just too much, rather than a circle it must be a square as this would be much more easier you see.
Round peg – insert – square hole == typical legislator.
Pi
In my hometown of Princeton, NJ, we celebrate Pi day, as well as Einstein’s birthday, each March 14. One of the highlights of the day is the Pi Recitation Contest. The winner is the person who can recite Pi to the most digits — from memory of course. Last year, the adult winner recited Pi to 2,222 digits.
See link – http://www.pidayprinceton.com/events
Two Pi Days
Could as easily be 22/7 (in the European-style notation) as 3/14
Getting rid of pi
There are some mathematicians attempting to replace pi with tau.
http://www.newscientist.com/article/mg20927944.300-pis-nemesis-mathematics-is-better-with-tau.html
42
The last two digits of Pi are 42
Go ahead and check for yourself.
pi is 3.1415
so 31415 would be a sequence of numbers.. RIGHT AT THE START..
it also occurs in position 88,008
73520514593330496265 31415 14138612443793588507
“1234567” occurs at position 9,470,344
LESS THAT 10 MILLION and you get 1234567
you get “12345678” at position 186,557,266
what about ‘88888888’ yes at position 46,663,520
and that is only in the first 200 million digits of Pi, infinity smaller than the size of pi. (which is infinite).
Re: Re:
http://www.angio.net/pi/piquery.html
Re: Re: Re:
and YES, it can be mathematically proven to be true, any number sequence will be found in pi, assuming your sample size is large enough.
infinite is large enough.
you have a 100% chance (statistically) of finding a 5 digit number sequence in pi within the tiny first 100 million digits of pi.
you have about a 1% chance of finding a 10 digit number in the first 100 million.
so make that 100 million 100 googles and recalculate !!!.
100 googles or 100 million googles is nothing compared to the actual size of pi. (again infinite).
so calculate pi to 100 times 100 million and you have a 100% chance of finding any 10 digit number combination. if not, then calculate pi 100 times again.. or a sooner or later you will get a number big enough to have a 100% statistical chance of finding any (finite) length number sequence in pi.
the answer to that question IS KNOW, just not by Ho.
Re: Re:
That is not proof. There are an infinite number of finite sequences…. Which infinity wins? Perhaps a thousand 5s in a row never appears in pi — can you tell me if it does with certainty?
Re: Re: Re:
There are an infinite number of finite sequences…. Which infinity wins? Perhaps a thousand 5s in a row never appears in pi — can you tell me if it does with certainty?
sorry, wrong, there is a FINITE number of FINITE Sequences
If you have a 10 digit (finite size) sequence, there is a finite number of combinations that you can make with those 10 digits.
So that means (it’s really not that hard) that there are a finite number of finite sequences, not an infinite number.
what infinity wins ??
what kind of stupid question is that ? Sorry but that is just plain dumb, there IS NOT WIN OR LOSE, no finish line. How on earth if you have even a basic understanding of ‘infinite’ would prompt you to ask such a stupid question??????
draw a circle on a bit of paper and tell me where the circle begins and where it ends, a circle has no beginning or ending. SO does making the circle bigger give it a beginning or an ending ?
I really don’t know how I can dumb it down any more for you, if you simply cannot comprehend this then I don’t know why you are bothering to engage in any argument.
you are clearly way out of your depth, all you guys are right into open and free and saying knowledge should be available to everyone, it’s just a shame you have not taken advantage of that..
Buy (or pirate) a book on maths and get an education please.
Because your making yourself look ridiculous with this display of ignorance.
And no I am not trying to be nasty, just pointing out what appears to be significant flaws in your level of knowledge and understanding. Does not fill me with hope, but I am pleased I have not been exposed to your clearly 3rd rate education system.
Re: Re: Re: Re:
Insults aside. Seriously?
sorry, wrong, there is a FINITE number of FINITE Sequences
IF that statement were true (it’s not).. what is the largest finite sequence? You can’t name it… because I’ll just add one more number to it. The set is INFINITE.
And it’s actually possible that some infinite sets of numbers are larger than other infinite sets of numbers. For example, the set of irrational numbers (which is infinite) is larger than the set of rational numbers (but the set of rational numbers is ALSO infinite). There are also more real numbers than natural numbers.
I am admittedly not a mathematician, but you may wish to review the work of Georg Cantor.
There are an infinite number of finite sequences
REALLY !!!!!!!!!!
now I know your way out of your depth.
IF.. there is an infinite number of finite sequences it will be really easy to have all of those finite sequences contained within an infinite sequence of non-repeating numbers.
you’ve proven false your own (confused) argument, in fact that infinite set of ‘finite’ sequences will occur an infinite number of time within an infinite sequence of numbers (such as pi). If fact the set of infinite sequences with pi could be larger than the infinite size of pi.
Yes, set of infinites can be larger or smaller that other sets of infinities.
I hope that clears that up for you Mr Ho.
ok calculate pi to 1 decimal place you the value 3
you have your sequence of numbers to want to match, you make a guess, you have 0123456789 to choose from. you have a 10% chance of finding your ‘sequence’ in pi.
so if you calculate pi to 10 digits and you get
3.124567890 and your sequence size is still 1 digit (5 for example) you have 100% chance to find your sequence within your calculated value of pi.
(10% x 10)
so you increase your sequence size to 2 digits (eg 12), that would mean you have a 0.01% chance of finding your sequence. (10% of 10%).
but I think even you can see that if the original number is bit enough a point is reached when no matter how large your sequence is, it will exist somewhere in the original number 100% of the time..
I guess there is something to be said regarding the quality of mathematics education in the US..
it’s not even a valid question, clearly with an infinite number any number sequence even an infinite one will be found in the sequence.
what the question really is, is ‘how big does the number have to be to ensure that a specific size sequence will occur.
That can easily be worked out statistically.
for example.
with a sequence of 10 digits of random numbers you have a 100% chance of finding a 1 digit sequence in that. 1 in 10 for each digit.
so if your number sequence is 9876543210 you have a 10% chance for each digit to get a match to your sequence and a 100% chance your single digit sequence can be found in the number.
if your search sequence is 23 then you only have 10 possible numbers that can follow a 2 (0123456789), so if you have a sequence or 100 numbers there would be a 100% chance of 2 occurring 10 times, and only 10 different possibilities that can follow the 2 (3 being one of them).
therefore with 10 numbers you have 100% chance of finding a sequence of 1.
with 100 numbers you have a 100% chance of finding a sequence of 2. It’s very simple, the longer your required sequence the larger the initial number has to be.
of for infinite sequence of finite numbers joke.
lets try, we’ll use a finite number of digits (3) we’ll call them 1, 2, 3.
now according to you I can arrange that sequence an infinite number of ways !!!.. LETS TRY.. (this could take awile).
123
213
231
321
312
132
does not look infinite to me !!! yet every possible sequence is covered
0123456789
this is a number that contains every possible 1 digit sequence of numbers.
you cannot make up or think of a number that is not contained in this number.
for a 2 digit sequence you would need to have a 0123456789 after every occurrence of 0123456789 from the original 10 digit number.
ie.
00010203…… 09….10111213….19…2021222324…29…..
….
….
90919293949596979899
no matter what combination of 2 digit sequence you choose it will be contained in this one fixed (not random) FINITE NUMBER..
if your sequence is 3 digits long, for every 2 digit sequence (ie 12) you will need a 0 to 9 after it..
120121122123124125126127…..
that gives me a number that contains every possible combination of 3 digit sequence.
you notice that as long as you have a finite sequence (a specific length, it will within a finite number length of random digits.
you don’t need an infinity of numbers you only need an even statistical distribution and a large number set to contain ANY number sequence of any finite length. You may need a very large number but you do not need an infinite number.
in an infinite number a sequence of infinite length will occur an infinite number of times.
which I believe is a few more than none !!!!!.
you can have a sequence of infinite length (such as pi0 but you cannot have a infinite number of combinations of a sequence of a fixed length.
Re: Re:
You are assuming a distribution within an infinite sequence that may not exist. As a previous commenter pointed out, an infinite sequence of non-repeating numbers could completely omit the number 9 and then would not contain ANY finite sequence that had a 9 in it.
Pi may very well contain every finite sequence, but you have certainly not proven it. I do encourage you to continue to try — if you succeed, there may be practical applications in cryptography or other areas of math.
I will leave it to professional mathematicians to critique your future publications….
Re: Re: Re:
your infinite sequence of non-repeating (RANDOM) numbers that does not contain a 9 would still work as long as your sequence also played by THE SAME RULES, and itself did not contain 9’s.
sure, you could also when trying to ‘make up’ such a silly theory include extra things and call them what you like.
does not change the facts, and changing the rules still does not work.
and the fact is with a very very very long string of random numbers you will get all possible sequences by virtue of the fact that it is a very long non-repeating string of numbers.
we can all work out that the digits 0 thru 9 DO REPEAT and the statistics of any one digit being either 0,1,2,3,4,5,6,7,8,9 is 10%, it’s going to be that set of numbers REPEATED AT RANDOM, and an even random distribution means that ANY 1 digit sequence WILL APPEAR in that random distribution with a specific statistical probability.
And you certainly do not need an infinite length of the original string to achieve it.
If you do happen to have that infinite string of numbers, then your sequence, no matter how long will appear within that string an infinite amount of times.
I have nothing further to prove to you, it is simply up to you to gain some basic understanding, sorry.
I am not responsible for your ignorance.
The solution to this problem is to use what’s known as the Poisson approximation to the binomial, when the numbers are large. We can actually approximate the above formula as:
Odds(finding string of length k in N digits of pi) = 1 – 1/e(N*0.1d).
http://www.angio.net/pi/whynotpi.html
if you have positions that can only contain numbers 0 to 9 with a random distribution, you have a 10% chance of ANY ONE of those positions being filled with a particular number.
that’s for a 1 digit size sequence of Decimal numbers (no cheating and removing some).
so you have a 100% chance of finding your 1 digit sequence in that 10 digit number.
(assume your sequence number is 2) for you to get a 100% chance of finding a 2 digit sized sequence you will need to ensure that every to has following it a 0 to 9 number, so you will need at least 10 2’s in your random number, enough 2’s for there to be one of every possible digit (0 thru 9).
you will also need 10 1’s 3’s … up to 10 9’s all followed by a 0 to 9 number.
to get a 3 number sequence you will have to get all the numbers you already have from the first two numbers and add a 0 to 9 after each of them, and so on, as long as they are random with enough of them you have 100% chance of matching your sequence.
With small sequence numbers the original number does not have to be very large. As the probability of any 1 digit being the right on is 10%. you can get away with a 10 digit number.
with a tiny size of 100 million digits you can get a 99.995 chance of matching a 7 digit number, and 100 million is tiny compared to infinity.
but as someone said earlier trying to omit a digit is just stupid..
I find it amazing you have so much trouble with very big and infinite.
And that you don’t even need an infinite length to find ANY length of numbers within it, you can calculate (well I can) how many (or how long) a random number had to be until a string of ANY length can be found within a string of random numbers.
tell me how long your sting is and ill give you the number of random digits required before you have a 100% chance of that string occurring.
ALL FINITE NUMBERS TOO.. (and not leaving some out !!!)..
Re: Re: Re:
As a previous commenter pointed out, an infinite sequence of non-repeating numbers could completely omit the number 9 and then would not contain ANY finite sequence that had a 9 in it.
SO SAD.. why did you forget to mention NON-REPEATING RANDOMLY DISTRIBUTED NUMBERS.
you cannot omit a number (like 9) and them claim it’s a random distribution.
and if your argument is if you make it non-random and leave out a number, but INCLUDE that number in your search string you will not get a match. that’s the most stupid things I have heard you say to date.
and if you base your argument that you are right and I am wrong based on such a outright stupid statement you must KNOW YOU have lost..
also the odds of the sequence being 88888888 or 12345678 coming out of a large distributed list of random numbers IS EXACTLY THE SAME.. and are the same for any 8 digit combination coming out.
no, I don’t expect you to comprehend that either, so ill just accept your ignorance and move on..
missing the point of the article
pies are not just desserts…
I shudder to think how you would cope with imaginary and complex numbers!!
if you have a random distribution of 100 numbers (0 to 9) each of those 100 positions has a 10% chance of containing any of those positions. I is possible (not probably) that you will roll 100 random numbers (0 to 9) and never have the number 1 occur. But generally with a random distribution you would expect about 10 1’s to occur (10% of 100).
so make your start number 1000 numbers in length, and you can expect on average 100 1’s 2’s .. 0’s to occur approx. 100 times each.
Yes, it is still possible to roll your 0-9 dice 1000 times and never get a 1, but probably not.
out of those 100 1’s or 2’s or 3’s you get from your list of 1000 numbers you have a 10% chance that the following number will be 0 to 9. so with 100 occurrences of any 1 number you have 100 x 10% chance of getting and complete 2 digit sequence.
so therefore sooner or later in a very long sequence of randomly distributed numbers you will get any one specific sequence. What is necessary is your initial number length is enough.
what is surprising to me is how SMALL a string of randomly distributed numbers are to achieve an extremely high to 100% probability of a sequence to occur.
like what are the odd’s of rolling your 10 dice and getting 9 ‘8s’ in a row !!!.
But it occurs in pi at around 50 million decimal places !!
you get 0 to 9 dice and roll them 50 million times and you might roll 8 ‘0s’ too.
everything does and will occur if you do it long enough.
Seriously, who gives a fiddler’s fart?
Re: Re:
a fiddler !!!