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.
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?
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:
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:
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
Hi Steinz,
I’m not an expert in number theory, but the patterns you describe would be significant. I don’t think anyone has found that every trillionth digit is, say, 7. It would be surprising if this happened.
There’s no theorem I’m aware of that says no such patterns exist. I think this is an open problem.
It seems a lot of people in this thread think that all digits (0-9) must appear infinitely often in pi, and that all patterns are locatable in the infinity of digits. This isn’t demonstrably true–there is no reason why, for instance, 100 3’s must appear consecutively somewhere. There’s no reason that, say, 9 can’t stop appearing at some point. Also, it’s conceivable that at some point only, say, 1’s 3’s and 7’s appear randomly.
There is simply not a lot of proven facts about specific transcendental numbers. No one knows, for example, if the sum of two transcendental numbers (eliminating obvious things like pi plus (minus) pi = 0) is ever non-transcendental.
Bill
Let me get back to the topic of creating random numbers and ways to accomplish that.
Years back, on an old PET (commodore’s first desktop upgradeable to 16K of RAM), I wrote some cheesy BASIC programs. Creating random numbers was kind of a problem and frequently I ran across programs that asked the user to pick a number between 1 and ‘a large number’ to use as a ‘seed’ for random number generation. I didn’t like that because I trust members of the human race i.e. they will always (trust me - always) try to beat the system. My solution: ask the user to type a name and store the last digit of the CPU clock (converted to decimal) as the user struck the keys typing in a name. Even the old PET had a fast enough clock to make this pretty random.
Since I switched majors to English not long after, I don’t know whether current software uses my trick or not. I wrote an article for Commodore Magazine about it but it was never printed.
I do think that a long stored string of digits such as a piece of pi would be useful in producing apparently random numbers from a seed (that’s what you want right? apparently random numbers?).
If ‘meese’ was the plural of ‘moose’ then what would the plural of ‘mice’ be? 
You should not use the digits of Pi to generate pseudo random numbers. There are much better ways to do this; from the computational effort point of view and also methods that can be theorethically justified.
Can you be sure that there is no effective algorithm that can determine the first occurance of some pattern (e.g. 999999) in the digits of Pi (for the example, starting at position 762)
Well, eveyone keeps asking for proofs of things reating to Pi, and it just isn’t going to happen. So, I did some practical (and simple) statistical testing. I downloaded lists of digits in Pi, which had 10,000, 1 million, and 10 million digits. Using simple tests to determine if each of the digits 0-9 occur with probability .1 determined that they probably do. Of course, running thirty tests with certainty level .05 did “suggest” that some digits did not occur ecactly 10% of the time, but of course this statistical significance has no practical meaning. So, it does indeed apprear that, as long as your list of digits is not longer than 10 million, the digits are uniformly distributed. Those who have more free time than I could also look for repeating patterns, etc. to see if what is true on a more global scale is normally true in all “regions”.
I read in a recent magazine (PC Magazine, maybe) that a chip company (Intel, I think) created a random number generator that is truly random instead of psuedo-random as all are today. It uses the heat sensors on the motherboard to measure the heat enrgy leaving the chip. Supposedly quantum mechanics says that this is a random occurance. If anybody else read this or knows more than me perhaps you could explain this better.
Look, Pi isn’t random. At all. It just looks random. It’s completely predictable. Pi is the ratio of a circle’s circumference to its diameter. That’s a pattern. Pi is one big pattern. 4(1 - 1/3 + 1/5 …) is a completely predictable equation. Everyone who calculates Pi (correctly) gets the exact same number. The fifth digit of Pi after the decimal is 9 no matter how many times I look at it. That’s not random.
In fact, if Pi is indeed “normal” (and I think it is), that would mean that every finite string of numbers appears in Pi somewhere. There’s a pattern.
This obviously isn’t true for all irrational numbers. Imagine an irrational number that begins 0.4869934120341… and goes on forever without repeating, but has no 7s. Nowhere in it is there a 7, ever. It’s irrational, sure. But the pattern ‘777’ will never be found in it. That isn’t true of Pi (assuming normality).
Regarding monkeys, there are a finite number of documents the same size of Hamlet that aren’t Hamlet. So an infinite number of monkeys would certainly succeed, assuming those monkeys didn’t have an aversion to typing an ‘e’ after a ‘b’. (That would be analogous to an irrational number without 7s.) For a more humourous look at monkey-typing, check out http://www.brunching.com/features/feature-randommonkeys.html.
-Quadell
Ahem. Same size as Hamlet. I guess I’m no Cecil.
-Quadell