Irrational numbers digits density

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Philosophy Explorer
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Irrational numbers digits density

Post by Philosophy Explorer »

I had speculated about the density of digits 0-9 and guessed that they're evenly distributed among the decimal representation (including the numbers before the period) for π, e and the square root of 2. As it turned out I was right and someone alerted me that the Wolfram website had that information listed. Ironically there are other irrational numbers (such as 1.01001000100001... which is also transcendental) with uneven distribution of the decimal representation of its digits.

My next thread will discuss prime numbers among the decimal representations of the irrational numbers.

PhilX
wtf
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Re: Irrational numbers digits density

Post by wtf »

Philosophy Explorer wrote:I had speculated about the density of digits 0-9 and guessed that they're evenly distributed among the decimal representation (including the numbers before the period) for π, e and the square root of 2. As it turned out I was right and someone alerted me that the Wolfram website had that information listed. Ironically there are other irrational numbers (such as 1.01001000100001... which is also transcendental) with uneven distribution of the decimal representation of its digits.
I've been meaning to ask you about this. Why do you claim 1.01001000100001... is transcendental? Got a proof?
Philosophy Explorer
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Re: Irrational numbers digits density

Post by Philosophy Explorer »

wtf wrote:
Philosophy Explorer wrote:I had speculated about the density of digits 0-9 and guessed that they're evenly distributed among the decimal representation (including the numbers before the period) for π, e and the square root of 2. As it turned out I was right and someone alerted me that the Wolfram website had that information listed. Ironically there are other irrational numbers (such as 1.01001000100001... which is also transcendental) with uneven distribution of the decimal representation of its digits.
I've been meaning to ask you about this. Why do you claim 1.01001000100001... is transcendental? Got a proof?
A correction. I probably got the sequence wrong. Check this link for Liouville's number (No. 5) which is known to be transcendental:

http://sprott.physics.wisc.edu/pickover/trans.html

where the number one is positioned according to the factorial after the decimal point.

PhilX
wtf
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Re: Irrational numbers digits density

Post by wtf »

Philosophy Explorer wrote:
wtf wrote:
Philosophy Explorer wrote:I had speculated about the density of digits 0-9 and guessed that they're evenly distributed among the decimal representation (including the numbers before the period) for π, e and the square root of 2. As it turned out I was right and someone alerted me that the Wolfram website had that information listed. Ironically there are other irrational numbers (such as 1.01001000100001... which is also transcendental) with uneven distribution of the decimal representation of its digits.
I've been meaning to ask you about this. Why do you claim 1.01001000100001... is transcendental? Got a proof?
A correction. I probably got the sequence wrong. Check this link for Liouville's number (No. 5) which is known to be transcendental:

http://sprott.physics.wisc.edu/pickover/trans.html

where the number one is positioned according to the factorial after the decimal point.

PhilX
I know Liouville's number. That's what I thought you were thinking of.
Scott Mayers
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Re: Irrational numbers digits density

Post by Scott Mayers »

Philosophy Explorer wrote:I had speculated about the density of digits 0-9 and guessed that they're evenly distributed among the decimal representation (including the numbers before the period) for π, e and the square root of 2. As it turned out I was right and someone alerted me that the Wolfram website had that information listed. Ironically there are other irrational numbers (such as 1.01001000100001... which is also transcendental) with uneven distribution of the decimal representation of its digits.

My next thread will discuss prime numbers among the decimal representations of the irrational numbers.

PhilX
Interesting. I was trying to explain just this idea with respect to generating random set of numbers for the Monty Hall Problem on the skepticforum.com site to which it might come in handy. I was trying to argue how true randomness become 'unfair' if we could interpret it as this idea of what you call its "density". In the game many people were arguing for using experiments involving large samples to which I was pointing out that if it becomes distributed evenly like this, it approaches the set of numbers as "fair" in this sense of equal density [I didn't use this word though.]

I felt that given any one unique game, the 'unfairness' of nature itself dictates that whatever we get we get and that once it is determined, looking back on it as a probability no longer applies because it becomes certain as 1 or 0 as a probability. This was how I argued in the single game that the odds have to be interpreted as 1/2 and not 2/3.

Then I argued that the requirement to 'test' would have to at least only allow nature to decide upon the given combinations. I didn't expand on this there but will be doing so. To continue, if their are 9 possible combinations, then one has to test the set in groups of 9 to see if the result either confirms or denies the 2/3. This is not possible because nature would only precisely indicate the odds of any combination as 4/9 if all groups are given fair representation. You might step this up to include the groups of all possible combinations of combinations and keep doing this for (combinations)^infinity and it would and should approach that 1/2 again.

This is because of the fact that over a period of infinite trials, even using a computer to generate random numbers makes it impossible to ever act like transcendental non-balanced 'densities' as it is the nature of computers only to be pseudo-random only.

What do you think?

Thanks PhilX!
Philosophy Explorer
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Re: Irrational numbers digits density

Post by Philosophy Explorer »

Scott,

It's an open question for me whether our universe is based on randomness. I have a thread running asking where patterns come from and I suspect they're not so random, but to prove it would require very deep concepts.

A case in point are the binomial equations, (x - y)^n. Now binomial equations have existed before I was born. Recently I was looking into the anagram numbers and I discovered that I can break up the equations into three parts, the first part I have just listed is known to be true for all anagram numbers (divisibility by nine). The second part has been partially proven for the anagram numbers and the third part is mostly a conjecture. Now the fact there is this relationship is a mystery to me. Nobody planned for the equations to be associated with anagram numbers in the properties they exhibit and I didn't have binomial equations in mind when I explored anagram numbers. Yet they're naturally involved with each other.

There are many patterns in math. Humans discovered them, but did not create them. So I ask what are the source of those patterns? Are they deterministic in nature or random? (I lean towards deterministic, but can't prove it)

PhilX
Scott Mayers
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Re: Irrational numbers digits density

Post by Scott Mayers »

Philosophy Explorer wrote:Scott,

It's an open question for me whether our universe is based on randomness. I have a thread running asking where patterns come from and I suspect they're not so random, but to prove it would require very deep concepts.

A case in point are the binomial equations, (x - y)^n. Now binomial equations have existed before I was born. Recently I was looking into the anagram numbers and I discovered that I can break up the equations into three parts, the first part I have just listed is known to be true for all anagram numbers (divisibility by nine). The second part has been partially proven for the anagram numbers and the third part is mostly a conjecture. Now the fact there is this relationship is a mystery to me. Nobody planned for the equations to be associated with anagram numbers in the properties they exhibit and I didn't have binomial equations in mind when I explored anagram numbers. Yet they're naturally involved with each other.

There are many patterns in math. Humans discovered them, but did not create them. So I ask what are the source of those patterns? Are they deterministic in nature or random? (I lean towards deterministic, but can't prove it)

PhilX
I like that you are seeking this. I personally interpret that if randomness is sincere in any one world, it would be like the digits of transcendent numbers which could never likely show the form of equal distribution of all digits in number. But given a multiverse interpretation, then I believe this would work to account for all possible worlds should one universe not permit this. It would be interesting to find out for sure though. I appreciate your attempts at this.
Philosophy Explorer
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Re: Irrational numbers digits density

Post by Philosophy Explorer »

Hi Scott,

Scale is a factor in randomness (to what extent is to be determined). Read up on the Schrodinger's cat portion of this article which may stimulate your thinking further:

http://m.phys.org/news/2015-10-physicis ... r-cat.html

PhilX
Scott Mayers
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Re: Irrational numbers digits density

Post by Scott Mayers »

Philosophy Explorer wrote:Hi Scott,

Scale is a factor in randomness (to what extent is to be determined). Read up on the Schrodinger's cat portion of this article which may stimulate your thinking further:

http://m.phys.org/news/2015-10-physicis ... r-cat.html

PhilX
Funny coincidence again! I just finished breaking ice (a crack at least) on the Monty Hall problem with respect to others at the skepticforum.com thread above. I demonstrated how the perspective changes how we see it akin to the Uncertainty Principle and even used Schrodinger's cat to relate this! Is this what lead you to mention this here? I'll look it up now...

Not there yet. ..Okay, now...
Philosophy Explorer
Posts: 5621
Joined: Sun Aug 31, 2014 7:39 am

Re: Irrational numbers digits density

Post by Philosophy Explorer »

Scott Mayers wrote:
Philosophy Explorer wrote:Hi Scott,

Scale is a factor in randomness (to what extent is to be determined). Read up on the Schrodinger's cat portion of this article which may stimulate your thinking further:

http://m.phys.org/news/2015-10-physicis ... r-cat.html

PhilX
Funny coincidence again! I just finished breaking ice (a crack at least) on the Monty Hall problem with respect to others at the skepticforum.com thread above. I demonstrated how the perspective changes how we see it akin to the Uncertainty Principle and even used Schrodinger's cat to relate this! Is this what lead you to mention this here? I'll look it up now...

Not there yet. ..Okay, now...
Another coincidence.

PhilX
Scott Mayers
Posts: 2446
Joined: Wed Jul 08, 2015 1:53 am

Re: Irrational numbers digits density

Post by Scott Mayers »

Philosophy Explorer wrote:
Another coincidence.

PhilX
Oooh.....I guess God just told us He ain't tossin' no dice today!? :P
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