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May 17th, 2019, 03:04 PM   #1
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Commonality of Normal Numbers?

Disclaimer...I'm artist not a mathematician

By the definition of Normal Numbers I'm left with the question, how large of digit strings are shared by all normal numbers? For example the string 14639 should appear in a normal number and in all normal numbers and should likely or inevitably show up more than once (infinitely?).

So, by extension it seems that an infinitely long string should also obey this and then, if that is the case, how is one normal number different than another? Is only the starting place different? Could 314... Be found somewhere in all normal numbers?

Sorry I don't know enough math to ask this concisely.
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May 17th, 2019, 08:40 PM   #2
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what is a "normal number" ?
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May 17th, 2019, 08:49 PM   #3
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Ha ok, I guess I'm referring to an irrational number that is normal - that has an even distribution
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May 17th, 2019, 10:15 PM   #4
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First, a disjunctive number is a real number whose decimal expansion contains every possible finite sequence of digits. Somewhere in its infinite decimal expansion we can find 42343243243, and the first 11 million digits of pi, the ASCII encoding of the complete works of Shakespeare, and an ASCII description of each minute of your entire life from the moment of your birth to the moment of your death. [ASCII is the standard way of representing letters of the alphabet in binary code].

Such a number is often -- very often, in fact -- mistakenly called a normal number.

A normal number is disjunctive, to start with. But more than that, a normal number's decimal representation contains each possible finite sequence a statistically equal number of times.

In other words each digit occurs on average 1/10 of the time, as you look at more and more digits and take the limit to infinity.

Each 2-digit combination occurs 1/100 of the time; each 3-digit combination occurs 1/100 of the time, and so forth.

It's clear that each finite sequence of digits must recur infinitely many times somewhere in the decimal representation.

It's been proven that a randomly chosen real number is normal with probability 1. That means that there do exist non-normal reals, but they form a set of measure 0. They are statistically very rare.

Strangely, we can easily produce non-normal numbers. 1/3 = .333333333... is obviously one such and you can easily make up similar examples.

Even more strangely, nobody has ever proven that any specific real number is normal. We think but we don't know that e and pi are normal.

To answer the OP's question, of course normal numbers can all be different from each other. Just because they have to statistically contain every finite sequence in equal proportions, they can still be very irregular for a long time out.

For example there could be some normal number that begins .333333333333...3 where there are a million 3's; but still ends up still being normal, because by the time you take the statistical frequencies all the way to the end, every finite sequence ends up occurring with equal frequency.

[What I described is normal to base 10. A number might be normal in one base but not another. The actual definition of a normal number is that it is normal to every base].
Thanks from topsquark and precisionart

Last edited by Maschke; May 17th, 2019 at 10:44 PM.
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May 18th, 2019, 10:36 AM   #5
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Thank you Maschke, your answer was exactly what I was looking for.

My interest in this is based on a book I read 4 years ago: Our Mathematical Universe by Max Tegmark

There are a few conclusions he came to that didn't sit well with me:

1. Given an infinite amount of space and matter "you" exist in this space more than once, or perhaps and infinite amount of times in Level 1 Parallel Universe.

2. The cosmos and its 4 tiers of parallel universes can be mapped to a number.


My Thoughts:

1. If the cosmos and all its parallel universes were mapped to an irrational number, and "you" are a finite number sequence, then "you" only necessarily repeat if the cosmos is normal right? (Or is this still unavoidable? He reaches this conclusion based on the assumptions in #1 and a density limit to matter.) But not all irrational numbers are normal, and therefore this conclusion is not inevitable, or is it?

3. This begs the very question his book intends to solve, namely, why the universe is what it is - with sufficient junk scenarios you are bound to have our universe.

But, since the cosmos is mapped to a specific number, and there must be other numbers, this number is on the same unstable ground. Granted it is infinite in the ways he presents, but since it is not comprehensive it does not explain why not a different number/cosmos. Is a single number robust enough to be a ground for everything? If it is not, then there are numbers outside of this.

Is there a number that includes all irrational and real numbers?

His book not only left me with a desire to find an exception to his conclusion, but also to conceive of a principle that showed the cosmos doesn't pile on junk universes with infinite repetition. This pursuit perhaps comes from an aesthetic and a desire for elegance and is perhaps incorrect as well.

Thanks again for your elegant answer.

Last edited by skipjack; June 13th, 2019 at 04:17 AM.
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