One use of calculating Pi!

Well, here's one kinky use of Pi I have, though admitedly it brands me as an incurable geek...

I keep a printout of the first 100,000 decimals of Pi (20 pages) by the computer. You want a random number? Just pick up the printout, select a page, and choose a series of digits. Voilà! Great for passwords.

The idea is that since Pi is an irrational number (i.e., it cannot be represented by a fraction of two integers), this means that all finite series of digits end up somewhere in Pi.

Don't believe me? Check the "Where is your birthday in Pi?" page: www.facade.com/Fun/amiinpi

This may (or may not, for all I know) be true of Pi, it is definitely not true of irrational numbers in general. This number is irrational:
1.101001000100001....1<0**n>1<o**(n+1)>1
(i.e, we keep increasing numbers of zeroes
between ones)

Many finite strings of digits will never occur in this expansion.....

(something funny happened to my posting, the result was not what I entered, perhaps an angle bracket got corrupted. I meant
1.
01
001
0001
00001
000001.....

I keep a printout of the first 100,000 decimals of Pi (20 pages) by the computer. You want a random number? Just pick up the printout, select a page, and choose a series of digits. Voilà! Great for passwords.

Alas, you just reduced the entropy of your password to lg(100000)==16.6 bits -- under three alphanumeric characters' worth.

I have never heard of a theorem that says that an irrational transcendental number must contain all possible finite strings of numbers. I think it is highly unlikely that pi has, anywhere in its expansion, 3333333....333 where the "..." represents five thousand repetitions of the number 3.

Alas, you just reduced the entropy of your password to lg(100000)==16.6 bits -- under three alphanumeric characters' worth.

I use random numbers in conjunction with mixed cases. The purpose is not to get maximum entropy, but to avoid choosing familiar numbers that could later be deduced by a clever social engineer.

This may (or may not, for all I know) be true of Pi, it is definitely not true of irrational numbers in general. This number is irrational:
1.101001000100001....1<0**n>11
(i.e, we keep increasing numbers of zeroes
between ones)

Hmm. You're absolutely right. It's definitely not true of all irrational numbers, but rather of numbers for which no rule is established. Another such number would be e.

That's how pseudo-random numbers are generated on computers, for instance. You think your home computer has a random generator? Think again. The random number is actually a long series of digits, and the computer samples a segment of that series of digits when it wants a random number.

The idea is that this series of digits is long enough that repeating segments will occur with a very small probability, i.e., probably once in a lifetime.

That's the rationale behind my use of Pi as a random number generator. If I point at random in the string of numbers, I'm likely to stumble upon a string of numbers which are more or less random.

However, someone knowing this could search through the digits of Pi to find the string in question. Say, for instance, that he is looking for a string of 10 numbers starting with 27344. The probability of finding the right string of ten digits starting with 27344 is 1/10^5 in a truly random case, but more with the first 100,000 digits of Pi.

But with ALL the digits of Pi (yeah, I know, impossible), we'd be back to 1/10^5.

I have never heard of a theorem that says that an irrational transcendental number must contain all possible finite strings of numbers. I think it is highly unlikely that pi has, anywhere in its expansion, 3333333....333 where the "..." represents five thousand repetitions of the number 3.

Why not? Why is it hard to believe for an infinitenumber of digits which we know have no period?

Since we do not know the rule for individual digits of Pi, we have to consider it as a random series. In essence, Pi is a pseudo-random number. The probability of finding five thousand 3's in Pi is 1/10^4999, which may be astronomical, but pales in comparison to infinity, as with all infinite numbers.

The more digits you compute, the more you're likely to stumble upon strings of digits which seem to make sense; for instance, your birthday, or 123456. They're still random, though; we just took a large enough sample that we stumbled upon a pattern we recognise as sensible, while in actuality it is no such thing.

Five thousand 3's in succession are a pattern to us, but it's still a random string that can naturally occur.

Doing some research, I have found a string of seven 3's in the first 10 million digits of Pi (i.e. '3333333'). The probability of such an event is 1/10^7, and we happen to have 10^7 digits. It's all within the realm of the probable.

So if you have 10^5000 digits (I know, not enough atoms in the Universe by far, but still), why not have a succession of five thousand 3's?

If pi has an infinite number of zeroes, and if each digit is random, then the probability that ANY finite sequence of digits can be found within pi approaches, but is not equal to 100%. So, 50 BILLION 3's in a row most likely can be found somewhere in pi.

It's just like the monkeys banging on the typewriters and eventually banging out Hamlet. If you give them an eternity to do it, they probably eventually will.

However, there is an infinitessimal probability that they NEVER will.

- Jim

I meant if pi has an infinite number of digits.

[/quote]It's just like the monkeys banging on the typewriters and eventually banging out Hamlet. If you give them an eternity to do it, they probably eventually will.[/quote]
Don't wait up
There is exactly one document that would be the correct Hamlet, and an infinite number of possable documents that would not be Hamlet.
Even given the rest of time, there is exactly one chance out of infinity that the monkeys would type out Hamlet. Of couse, math with infinities is always a questionable venture, and I'm not about to get into an argument over my answer.

OTOH, I heard someone argue that the laws of entropy state that there is a certain threshold beyond which something is too orderly to have been created by a chaotic process. I don't fully understand entropy, but the argument makes sense. Essentially, a monkey typing out Shakespere's plays would be akin to a volcano randomly spewing out a late model Corvette ZR-1 with a 5 disk CD player and the puncture-proof tires.

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"I had a feeling that in Hell there would be mushrooms." -The Secret of Monkey Island

Regarding the speed limit for the expansion of the universe (a point raised by Dr. Neil Basecu in his comment on the pi question): it's not the speed of light. At least according to noted physics popularizer John Gribbin at: http://epunix.biols.susx.ac.uk/Home/John_Gribbin/cosmo.htm
Space expanded faster than the speed of light for at least part of the universe's history, according to most cosmologists. The speed of light is an absolute limit for objects traveling through space. So you may need those 39 decimal places after all.

" Hmm. You're absolutely right. It's
" definitely not true of all irrational
" numbers, but rather of numbers for which
" no rule is established. Another such
" number would be e.

I do not know this has been established.
Certainly it is true if the digits truly have a "random" quality. I had the impression those interested in extremely long expansions of Pi wanted to assess how "random" those digits really were.

" That's how pseudo-random numbers are
" generated on computers, for instance. You
" think your home computer has a random
" generator? Think again. The random number
" is actually a long series of digits, and
" the computer samples a segment of that
" series of digits when it wants a random
" number.

Please, that sounds a little patronizing. You can certainly VISUALIZE the computer as sampling from a series of digits, but far more typical is running an algorithm that would RESULT in a long series of digits. The most common pseudo-random number generator is probably the linear congruential method, in which
X(n+1) = mod(a*X(n) + c, m)
generates a sequence of numbers X(1), X(2), etc. starting with an X(0) known as the "seed". Choice of appropriate constants a, c, and m is very important for getting good qualities of numbers. Knuth, in his "The Art of Computer Programming", dedicates about a third of the second volume to pseudo-random number generators for those interested in methods of generation, as well as methods of evaluating their qualities.

<< If pi has an infinite number of zeroes, and if each digit is random, then the probability that ANY finite sequence of digits can be found within pi approaches, but is not equal to 100%. So, 50 BILLION 3's in a row most likely can be found somewhere in pi. >>

Yeah, but that's the point. The digits in pi are NOT random (in the sense that you're using the term "random.") There are any number of forumlas that are used to caclulate the digits of pi, which makes them not random at all. No more than the .1010010001 ... sequence is random.

Now, any small chain of digits in pi -- like oh, say the first billion digits -- appear to be random to the human eye. And if you want to use them as a source of generating random numbers of two or three or a dozen digits, as Elijah says, that's fine. Close enough for practical purposes, as they used to say.

If you have a truly random infinite sequence of numbers, then, as Elijah comments, there is reasonable probability that the sequence contains a subsequence of a thousand 3's (or whatever). And as someone else commented, that's just like the monkeys banging at the typewriters. Now, if you have to actually FIND that sequence of a thousand 3's, forget it... Just like, you'd probably have to spend your lifetime looking through all the dribble typed the monkeys, to come up with even one coherent speech from Hamlet, let alone a whole act.

You're going to have extreme trouble finding more than 85 3's in a row in pi. There are only about 10^85 electrons in the universe, so it will be hard to represent 10^85 digits in a computer, even if it is very economical in storage (one bit/electron).

1. While the digits of pi aren't 'truly' random in that they're predetermined by the
umpteen million infinite series that converge to fractions of pi, they're close enough to defeat any outside test of randomness. For instance, knowing a string of 100,000 consecutive digits of pi won't give you any edge in guessing the 100,001st. Since that's generally the sort of thing we mean by 'random', why argue the point?

2. Dr. Askey's sterling character reference for Gregory Chudnovsky has no bearing on the question of the usefulness of taking the expansion of pi out to ridiculous lengths. And, c'mon, what is the sequence of
digits in the expansion of pi going to tell you about either randomness itself, or about any of the properties of transcendental numbers, that the formulas for generating those digits won't tell you more directly? I don't buy it for a second.

'PI' is the name of a new film. It's great too.

It shows another reason for calculating these numbers.

Not to mention since it won first prize, it came with $1M prize money :-)

Alright, lemme try to reply to a few threads all in one post...

RTFirefle:

1. While the digits of pi aren't 'truly' random in that they're predetermined by the umpteen million infinite series that converge to fractions of pi, they're close enough to defeat any outside test of randomness. For instance, knowing a string of 100,000 consecutive digits of pi won't give you any edge in guessing the 100,001st. Since that's generally the sort of thing we mean by 'random', why argue the point?

Thank you. That's exactly my point, and you express it well. That's the same reason we can safely assume the digits generated by a computer are pseudo-random, since, although there is an algorithm, there is no clear rule for extrapolating the digits following a given series of digits.

jens:

Please, that sounds a little patronizing. You can certainly VISUALIZE the computer as sampling from a series of digits, but far more typical is running an algorithm that would RESULT in a long series of digits.

Why is it patronising? Using an algorithm to select a long series of digits is different from the (simplified, granted) explanation that you select a series of digits from a greater series.

Let me put it this way: if you use the following algorithm to select a number:

a_n = n^2 + 1

You can say you use the above algorithm to select a given number, or you can say you merely select a number from a series of number, in that case,

1, 2, 5, ..., (n^2+1), ...

So, what's the difference? Or are you merely saying I oversimplified?

George:

You're going to have extreme trouble finding more than 85 3's in a row in pi. There are only about 10^85 electrons in the universe, so it will be hard to represent 10^85 digits in a computer, even if it is very economical in storage (one bit/electron).

Yep. But the idea is to take a sufficiently large number that it can be approximated to an infinite series. It all depends how precise you want to be, but you'll certainly agree that 10^3 << 10^10.

Hey, I'm not trying to argue Pi is a random number; I'm saying Pi is a pseudo-random number because there is no apparent correlation between digit k and digit k+1. That's good enough for passwords, no?

Maybe we could deemphasize the word "random", since it means a lot of different things.

One thing people are talking about is whether pi is a "normal number", in whose digits every subsequence occurs with the expected probability, and consequently every finite string appears eventually with probability 1. This is widely believed, but not known. http://www.astro.virginia.edu/~eww6n...malNumber.html

Whether something makes a good password doesn't depend on its passing statistical tests. It depends on how big the search space is.

A random 8-digit number is one of 10^8 possibilities. An 8-digit number picked randomly from the first million digits of pi has only 10^6 possibilities. It's the same reason you shouldn't pick passwords from other well-known sources: Monty Python lines, science fiction characters' names, song lyrics, etc.

Although I can't really advance the level of discourse here, which is admittedly impressive, I thought I'd throw in a couple of definitions for anyone who's reading this and scratching their heads.

"Irrational" refers to a nonterminating decimal, and nothing more. 2/3, for example, translates to 0.66666666666... forever. That's "irrational."

"Transcendental" refers to an irrational number that cannot be expressed using a polynomial equation with rational coefficients -- or, in English, a simple (terminating) equation with regular numbers. (In other words, the well-known 1/2 + 1/3 + 1/4 + 1/5 +... 1/n doesn't count.) 0.66666... is therefore not a transcendental number, because it can be expressed as 2/3.

The two best-known transcendental numbers are e and pi, although some dude once proved that there are an infinite number of them. Oh, by the way, seeing that pi is transcendental -- if you learned in school that 22/7 equals pi, be aware it's an approximation that's certainly close enough for most real-world purposes, but it ain't pi; it diverges after just a few decimals.

Anyway, that's my meager contribution to a fascinating thread. That's what I love about the SDMB -- whether the topic is math theory or toejam, the discussion will be enlightened and entertaining. It's a beacon of rationality amid the sea of pseudointellectual detritus that comprises the rest of the 'net...

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"I'm not an actor, but I play one on TV."

Although I can't really advance the level of discourse here, which is admittedly impressive, I thought I'd throw in a couple of definitions for anyone who's reading this and scratching their heads.

You know, you just did. At the very least, you cleared up a misconception I created... When I spoke of numbers who may contain all finite series of numbers, I said these were irrational numbers. Fact is, I spoke of transcendental numbers.

I realise now, also, that there is no such law stating that all finite series of numbers are contained in a transcendental number. Is the following number transcendental?

(In this example, I write Pi_n to say, 'the nth digit of Pi', and c_n to say, 'the nth digit of our supposedly transcendental number'.)

| Pi_n if Pi_n <> 9
c_n = |
| 0 if Pi_n = 9

What do people think? This number seems like it's transcendental to me, even though there is a rule deducing its digits; it's the fact that these digits are deduced from a transcendental number.

Clearly, if the above example is transcendental (and I really don't know if it is), then it does not include all finite series, because any series of digits with a 9 in it is not included.

Maybe we could deemphasize the word "random", since it means a lot of different things.

I use a quite practical definition of "random", which may be a failing: by random, I mean that there is no human means currently available to deduce one digit based on the preceding ones.

In that sense, a coin's toss is random, even though it depends largely on the initial position of the coin, the force with which it is tossed in the air, the viscosity of the tosser's skin, his altitude, atmospheric pressure, etc.

If all these factors were known with enough precision, it can be argued that we could predict the outcome of the coin's toss. Then it would no longer be random. It is our incertitude that makes things appear random, and generating a true random number is more difficult than it seems. Like I mentioned in a previous post, even numbers generated by most computers are not random, but pseudo-random, i.e., it's actually impossible on a practical level to make a difference. (Alright, so in numerical simulations in grad school Chaos theory it was't, but hey. Who said grad school was practical? )

Is the following number random?

837640916027365

It may seem like it is, and establishing a correlation between two digits is gonna be a tough job... I typed it at random on my numpad, and you'd have to know the position of my fingers on the keyboard, the type of keyboard I have, my current mood, whether there was noise distracting and influencing my typing, etc. to determine the actual series of events that took place and be able to extrapolate one digit from the preceding ones.

In my book, that's random enough... Alright, pseudo-random.

Although I can't really advance the level of discourse here, which is admittedly impressive, I thought I'd throw in a couple of definitions for anyone who's reading this and scratching their heads.
"Irrational" refers to a nonterminating decimal, and nothing more. 2/3, for example, translates to 0.66666666666... forever. That's "irrational." -- Cervaise

Well, I'm scratching my head real hard right now!

Cervaise, 2/3 is a rational number. Rational numbers are those that can be expressed as the quotient of two integers (including the integers themselves). They happen to be just those whose mantissas (the part right of the decimal point) are either finite or periodic. All this is obviously true of 2/3 = 0.66666....

Irrational number are... well, the others, provided they can (in principle) be written in decimal form. They have infinite (thus "in principle"), non-periodic mantissas and cannot be expressed as a quotient of integers. The square root of 2 is a classic example.

Rationals and irrationals together are called real numbers. Fancy numbers like the complex ones, which can't be expressed in decimal form at all, are not real and thus neither rational nor irrational.

To be exact, complex numbers also include the real ones. In the previous paragraph, I meant the "really complex" ones, which are called irreal (I think; not sure in English) whose imaginary component is non-zero. (Complex numbers have a real component and an "imaginary" one, the latter being a (real) multiple of the "imaginary unit" i. i is defined as the square root of -1, a number that is beyond most people's comprehension, including mine. But they work, and they are useful.)

Your definition of transcendental numbers seems to be okay.

Confused enough? Summary:
All complex numbers are either real or irreal.
If real, they can be transcendental or not.
If non-transcendental, they can be rational or irrational.
If rational, they can be integer or not.
If integer, they can be natural (i.e. positive) or not.

Holger

Exactly right. The reason they are called
rational numbers is that they can be
expressed as the *RATIO* of two intergers,
i.e. as a fraction. So 2/3 is quite definitely a rational number.

". For
instance, knowing a string of 100,000 consecutive
digits of pi won't give you any edge in guessing the
100,001st. Since that's generally the sort of thing we
mean by 'random'."


Sure it would.
There is a one in ten chance that it's a digit from one to ten.

". For instance, knowing a string of 100,000 consecutive digits of pi won't give you any edge in guessing the 100,001st. Since that's generally the sort of thing we
mean by 'random'."

Sure it would. There is a one in ten chance that it's a digit from one to ten.

Oh, you're being pedantic. I said give you an edge. 1/10 chances is exactly what you get for a random number. That's not what I call an edge.

A few years ago, an expression for pi was discovered that lets you calculate any digit of it without having to calculate the intervening ones (so long you're using base 16). (Thus, ditto if your base is any power of 2).

This property gives circumstantial evidence aginst pi's digits being random.

A few years ago, an expression for pi was discovered that lets you calculate any digit of it without having to calculate the intervening ones (so long you're using base 16). (Thus, ditto if your base is any power of 2).

This property gives circumstantial evidence aginst pi's digits being random.

I'm sorry, but I'll have to see it before I believe it. People are still flaunting how they manage to calculate Pi to the 1,000,000,000th digit.

Cause otherwise, you could just calculate all digits of Pi separately in base 16 (why base 16????) and convert the whole thing to base 10, which is much easier than calculating 1,000,000 digits to get the 1,000,001st one.

I'm sorry to sound so doubtful, but that's because I am.

To Holger et al: I stand corrected. Serves me right for poking my head into something that interests me, but to which I can't really rationally contribute.

I do have an explanation, though. I was reading this thread, fascinated but with wrinkled brow; I went off to an online encyclopedia to check some terminology, and found myself with the classic "Oooohhhhh!" reaction as the light bulb went on. I figured I'd share what I found, but as it turns out, what really went on was Christmas lights, and although they blink attractively, they don't illuminate the room.

So shame on me for trying to take a concise-to-the-point-of-meaningless definition and translate and expand it for other readers, in the process mangling it beyond recognition.

Oh well...

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"I'm not an actor, but I play one on TV."

Elijah, it's called the Bailey-Borwein-Plouffe pi algorithm & I read about it at http://www.maa.org/mathland/mathland_3_11.html

It works for base 16 because they found a formula that works for base 16. At the time I read about it, it wasn't known whether there's a similar base-10 algorithm.

If I've read this thread correctly (and I may not be), there's some confusion about transcendental numbers. Forgive me for quoting some definitions:

An algebraic number is a complex number that is a root of an algebraic equation

f(x) = a_o*x^m + a_1*x^(m-1) + ... + a_m

where the a_i's are all rational numbers and m is a finite positive integer. All integers, rational numbers, and a subset of irrational numbers are algebraic:
for instance, the square root of 2 is algebraic.

Transcendental numbers are all the numbers "left out" of the algebraics. For instance, pi cannot be the root of an algebraic equation as defined above.

It can be shown that "most" (real or complex) numbers are algebraic, but proving that involves a branch of math called measure theory.



------------------
Bill

I realise now, also, that there is no such law stating that all finite series of numbers
are contained in a transcendental number.

Right, a transcendental number with that property is called "normal" -- see URL above.
Only a few contrived numbers, not including pi, are provably normal.

I use a quite practical definition of "random", which may be a failing: by random, I mean that there is no human means currently available to deduce one digit based on the preceding ones.

One way to get the next digit is to search for the known string. Take this random-looking number: 8831311722359377438741462183. I bet you can guess the next digit. (That string could occur twice and cause ambiguity, but practically speaking it can't.)

For password/cryptographic purposes, even a sequence of coin flips is not random once it's been made public. Size of size space is IMO the least tricky way to look at matters.

This definition, by the way, does not imply "normal" in the above sense. Say I generate successive digits to be 1 or 7 by flipping a coin: that's unpredictable, but will never contain "123".

A deep and rather satisfying definition of randomness (for infinite sequences -- I think there's no good definition for finite ones) is Martin-Lof randomness: basically, any program to generate the sequence has to get about as long as the sequence itself. Pi is obviously not this sort of random.

It can be shown that "most" (real or complex) numbers are algebraic, but proving that involves a branch of math called measure theory.

Most are not algebraic, that is. Just because each algebraic corresponds to a finite series of integers, and those are countably infinite (because pairs are countable and by induction); whereas the reals are uncountably infinite, so infinitely more numerous.

Then of course if you want to say anything about algebraics being "of measure zero" you're going to have to drag in measure theory after all. :-)

I know this is a stupid question, and you'll hate me for asking, but why would anyone care to find out the 10 millionth digit of pi? Is there any practical use of that digit? If not, why not stop at 200 digits? Isn't that small enough?

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A common mistake people make when trying to design something completely foolproof is to underestimate the ingenuity of complete fools. -Douglas Adams

Regarding pi--- I like apple-with vanilla ice cream if you've got it. The finer things in life are not found on mathematical tables, but on kitchen tables. :}

Ooops!!! I meant to say most numbers are transcendental. Thanks for picking up this mistake, Eli.


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Bill

I've always used a practical definition of "random" - it is the quality we attribute to events whose outcome we cannot, due to lack of information or time, acccurately predict before they occur.
A coin flip is often portrayed as "random". Certainly such an event is subject to the laws of physics and is therefore not "truly random". If we knew enough about the wind, force applied, distance to the landing surface, construction of the coin, etc., and had enough resources we could predict coin flips with 100% accuracy. Do we therefore call a a coin flip "pseudo-random"? Certainly not!
Many things we once considered random can now be predicted with reliable accuracy due to increased knowledge. Birth defects/genetics comes to mind.
Conclusion: Randomness is a matter of perception that is inversely proportional to knowledge. We will never be omniscient, and as a result there will always be the appearance of randomness.

Attention, everyone who enjoys looking for patterns within a long series of apparently random digits --- I strongly recommend reading Carl Sagan's novel "Contact". I think you'll love it!

[The movie was okay, but Patterns Within Random Digits was a major aspect of the novel, whereas the movie version put the focus on the more general-interest action-type stuff.]

Many things we once considered random can now be predicted with reliable accuracy due to increased knowledge. Birth defects/genetics comes to mind. Conclusion: Randomness is a matter of perception that is inversely proportional to knowledge. We will never be omniscient, and as a result there will always be the appearance of randomness.

It may be true of classical reality, but if you consider quantum mechanics, true randomness exists. The wave of probability of an electron, for instance, is a true random function, and only by considering things globally, that is, at a large scale, do they become predictable again.

quote:
-----------
You're going to have extreme trouble finding more than 85 3's in a row in pi. There are only about 10^85 electrons in the universe, so it will be hard to represent 10^85 digits in a computer, even if it is very economical in storage (one bit/electron).
-----------

Hang on, George. I don't need to save 10^85 digits to find 85 3's in a row. All I have to save is enough values to save the state of my calculation (typically this doesn't take too much storage) and the previous 84 digits. In the 1 - 1/3 + 1/5 ... series that Cecil mentioned, I'd only need to save enough digits that I could adjust the previous digits in case I needed to round up or down (for example, ...26599 + ...00001 would result in ...26600), and the last integer that I inverted and added to the series. The sequence of digits you need to save to do the addition without error might be long, but it's unlikely to be as long as the sequence itself.

Having said all that, if it is indeed possible to get a sequence of 10^85 9's (or 0's) in a row, then you would have to save them all to be able to continue the calculation. So far, though, that hasn't been a limiting factor for anyone.

Yikes. No sooner had I posted my previous message than I realized that my example had a serious flaw:

Even if you only have to save a small number of digits to the left of the MSB of the value that you're currently adding to the total, you have to save ALL the digits to the right. So, at the very first step, -1/3 = -0.33333.... Obviously, you have to save all those digits to the right in order to add something to them later.

Ugh. Bad example. Humble apologies.

Elijah, it's called the Bailey-Borwein-Plouffe pi algorithm & I read about it at http://www.maa.org/mathland/mathland_3_11.html

I'm speechless. They've calculated the 5 trillionth hexadecimal digit of Pi with this algorithm, and the only reason this algorithm is used instead of the decimal formula is for speed, given afterwards, converting a string of digits from hex to decimal is trivial.

This is a major advance, and the actual algorithm is relatively simple...

Wow.

(The actual Bailey-Borwein-Plouffe algorithm can be found at: http://www.mathsoft.com/asolve/plouffe/plouffe.html . This is truly groundbreaking work. And yeah, it's still totally useless. )

Your argument may have been flawed, but given the result mentioned earlier in the discussion (that there is an algorithm to compute the nth digit of pi (at least in a hexidecimal expansion) without computing any of the other digits) the conclusion might still be correct.

I don't know the algorithm myself, but provided the algorithm is actually constant space usage without regard to the value of n , your conclusion stands. You probably won't live to see the result of the calculation, but if there is a result the only thing standing in the way of eventually reaching a solution is the inevitable heat-death of the universe, or armageddon, depending on your religious persuasion.

Hi,

Look, this is, granted, arguing a kinda moot point.... but..

pi is a 'transcendental' number... so it's infinitely long... so EVERY number will eventually show up... i know in your example, practially speaking, ...333...333 (where '...' represents "any number of threes") will probably never be found, but even probabilisticily speaking, such a number does exist, and pi would contain it, would it not?

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uh. i don't know what UBB Code is. so i don't care.

| All complex numbers are either real or irreal.

I hate to nit-pick...well, I love to nitpick, but...some numbers are real, some numbers are imaginary, and some numbers are neither.

5 is real.
5i ("i" means the square-root of -1) is imaginary
5 + 5i is complex.

| If real, they can be transcendental or not.
As someone else pointed out, "not transcendental" == "algebraic"

M

mlorton wrote:

| All complex numbers are either real or irreal.

I hate to nit-pick...well, I love to nitpick, but...some numbers are real, some numbers are imaginary, and some numbers are neither.

5 is real.
5i ("i" means the square-root of -1) is imaginary
5 + 5i is complex.

Yeah, let's nitpick!! Courtesy of http://www.m-w.com/ :

Main Entry: imaginary number
Function: noun
Date: circa 1911
: a complex number (as 2 + 3i) in which the coefficient of the imaginary unit is not zero -- called also imaginary; compare PURE IMAGINARY

Main Entry: pure imaginary
Function: noun
Date: 1947
: a complex number that is the product of a real number other than zero and the imaginary unit

Thus, all of your examples are complex; 5i and 5+5i are imaginary; 5+5i is pure imaginary.

I love nitpicking too!!

Holger

And don't you start nitpicking about my messed-up quotation! Next try:

mlorton wrote:

| All complex numbers are either real or irreal.

I hate to nit-pick...well, I love to nitpick, but...some numbers are real, some numbers are imaginary, and some numbers are neither.

5 is real.
5i ("i" means the square-root of -1) is imaginary
5 + 5i is complex.

Yeah, let's nitpick!! Courtesy of http://www.m-w.com/ :

Main Entry: imaginary number
Function: noun
Date: circa 1911
: a complex number (as 2 + 3i) in which the coefficient of the imaginary unit is not zero -- called also imaginary; compare PURE IMAGINARY

Main Entry: pure imaginary
Function: noun
Date: 1947
: a complex number that is the product of a real number other than zero and the imaginary unit

Thus, all of your examples are complex; 5i and 5+5i are imaginary; 5+5i is pure imaginary.

I love nitpicking too!!

Holger

I can't believe this! Of course I meant that 5i is pure imaginary. 5+5i is not. But both are imaginary and irreal. 5 is real and not imaginary but complex. The other two are complex as well. More coffee please. PLEASE!

Holger

This isn't a reply to anyone in particular, it is more of a yes or no question regarding pi. Would it be true to say (or has it been proven that) there is absolutely no pattern to be found in PI? Given that it is a non-repeating decimal, this alone does not seem to exclude some possible pattern. For example, what if every billionth number of Pi was an odd number (any odd number)--that would be a pattern. I don't mean that every billionth number in the sequence might be 5, but that after one billion numbers there would be a 3, and then after another billion a 9, and then after another billion a 3, off into infinity so that every billionth digit would be some odd number.
Even better: if someone could prove that every trillionth number in the sequence was not an 8, that too would be a pattern. A very weak pattern, but it would be a pattern nonetheless. Again, is there some theorem which proves that such patterns don't exist--so that one could say that pi is absolutely pattern-less?

I memorized the first 200 decimal digits of pi in high school. They're still there in my head, available for retrieval at any time.

What in heaven's name was I thinking?!

I also memorized 100 digits of pi years ago; I just can't remember their right order.

Holger