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Everything contained in the ratio of Pi

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cr1t
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cr1t
« on: April 08, 2013, 11:38:01 AM »

I came across this, this morning, I suppose it's true much like infinite monkey is true.

What do you think?

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Mefiante
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« Reply #1 on: April 08, 2013, 12:01:25 PM »

What do you think?
It is necessarily true that the decimal expansion (or binary or hexadecimal or any radix of your choice) of π must somewhere contain an encoding of anything and everything that can be symbolically and finitely represented.  Moreover, the representation will occur infinitely often.  However, the same is also true of all other transcendental numbers of which there exists an uncountable infinitude.  The tricky bit is finding the sequence that does it.

Infinity in mathematics is anything but simple.

'Luthon64
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brianvds
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« Reply #2 on: April 08, 2013, 12:49:29 PM »

Pi will not contain not only everything exists, but indeed anything that we can imagine: Beethoven's tenth symphony, plays Shakespeare never wrote but better than anything he actually did write, recipes for heavenly foods we have never conceived of, designs for weapons and machines we'll never build, and jokes so funny they may be dangerous to your health. On the down side, also an infinite number of Barbara Cartland novels and political speeches.

I'm sure there is a neat SF movie to be made about an eccentric mathematician that works out a way of extracting all of this from Pi. Of course, the script for the movie itself is also in there somewhere... :-)

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BoogieMonster
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« Reply #3 on: April 08, 2013, 13:21:35 PM »

Quote from: brianvds
I'm sure there is a neat SF movie to be made about an eccentric mathematician that works out a way of extracting all of this from Pi

More likely someone will write a bestseller about how you use the digits in Pi to select letters from the bible.... I think I can stop there.
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Mefiante
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« Reply #4 on: April 08, 2013, 14:28:51 PM »

More likely someone will write a bestseller about how you use the digits in Pi to select letters from the bible.... I think I can stop there.
As if there wasn’t already enough circular reasoning about religion… Grin

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Brian
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« Reply #5 on: April 08, 2013, 14:44:50 PM »

Old Hat: the Bible Code has done it I believe:
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BoogieMonster
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« Reply #6 on: April 08, 2013, 14:50:22 PM »

Hmm, if it was in the "Bible Code" they just didn't look far enough, there should be a line in there somewhere that says: "The bible is a lie".
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brianvds
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« Reply #7 on: April 08, 2013, 15:01:19 PM »

More likely someone will write a bestseller about how you use the digits in Pi to select letters from the bible.... I think I can stop there.
As if there wasn’t already enough circular reasoning about religion… Grin

'Luthon64

Yes, it's completely irrational, but what can you do? When it comes to such things, everyone has a finger in the pie.

And here we go again with another round of punning. :-)
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Mefiante
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« Reply #8 on: April 08, 2013, 15:33:44 PM »

Yes, it's completely irrational…
In fact, π is a transcendental number.  That’s the chief reason why it resonates with woo-woos of various stripes.  Number theory dictates that it is a necessary condition, but not a sufficient one, that a number must be completely irrational for it to have any hope of being transcendental.

When it comes to such things, everyone has a finger in the pie.
Yes, but instead of encasing their finger in a French letter, they use a Greek one instead.

'Luthon64
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cr1t
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« Reply #9 on: April 09, 2013, 10:26:25 AM »

LOL all the answers are very good.

The issue here is you would need to have an infinite amount of memory to store Pi.

But you could say that an infinite combination of {0,1} contain everything.
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Mefiante
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« Reply #10 on: April 09, 2013, 12:46:02 PM »

The issue here is you would need to have an infinite amount of memory to store Pi.
Actually, no.  Given sufficient resources, π can be calculated to arbitrary precision and you’d only need to inspect a finite subset of its expansion at a time, discarding it if it doesn’t encode anything worthwhile.  One would likely use a sliding window of variable length that starts at successive digits to parse the string, appending new digits as needed if at any point the window shows anything promising, otherwise discarding the leading digit and moving to the next one.  (As an aside, there’s an ongoing informal competition among those in the mathematical sciences to develop algorithms that converge on π faster than all the previous ones.)

What you would need are enormous amounts of time and energy because statistically the information content (as Claude Shannon defined it) of a string of digits is proportional to its length, whereas the chance of finding that string in an infinite jumble of digits, decreases exponentially with its length.  A stark illustration of this is that if you captured every Joule of energy the Sun ever emitted and will ever emit until it burns out, and harnessed it 100% efficiently, that would be the minimum energy required to generate all possible 45-letter strings drawn from a 26-letter alphabet.  If you generated them at a rate of 1,000,000,000,000,000,000 (= 1018, a billion billion) a second, to do them all would take about 10,000,000,000,000,000,000,000,000,000 (= 1028, ten billion billion billion) times the present age of the universe.

And these are just some of the simpler quirks that bedevil efforts with large numbers.



But you could say that an infinite combination of {0,1} contain everything.
What is “an infinite combination of {0,1}”?  If you mean an infinitely long string of effectively random binary digits (0’s and 1’s) then that is saying the same as what was said about π because we can just as well use its binary expansion instead of its decimal one.  If on the other hand “{0,1}” indicates the interval of all real numbers between zero and one, then all you need to do is pick any one of the infinitely many transcendental numbers in that interval, and use its expansion as with π.

'Luthon64
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cr1t
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« Reply #11 on: April 09, 2013, 14:10:09 PM »

A stark illustration of this is that if you captured every Joule of energy the Sun ever emitted and will ever emit until it burns out, and harnessed it 100% efficiently, that would be the minimum energy required to generate all possible 45-letter strings drawn from a 26-letter alphabet.  If you generated them at a rate of 1,000,000,000,000,000,000 (= 1018, a billion billion) a second, to do them all would take about 10,000,000,000,000,000,000,000,000,000 (= 1028, ten billion billion billion) times the present age of the universe.

And these are just some of the simpler quirks that bedevil efforts with large numbers.
Quote

Hectic well that puts it in perspective.

What is “an infinite combination of {0,1}”?  If you mean an infinitely long string of effectively random binary digits (0’s and 1’s) then that is saying the same as what was said about π because we can just as well use its binary expansion instead of its decimal one.

Yes that was my point, that you could do it with anything that is infint and really there is nothing special about Pi in this sense they are using it.
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BoogieMonster
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« Reply #12 on: April 09, 2013, 14:35:25 PM »

Hmm, what about irrational numbers containing other irrational numbers?

Can anything be said about one irrational number at some point containing another irrational number? (iow: Can it be shown to be impossible?)
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Mefiante
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« Reply #13 on: April 09, 2013, 15:35:49 PM »

Hmm, what about irrational numbers containing other irrational numbers?
An irrational number does not have a finite expansion in any number system with a natural number radix.  This means that, e.g., √3 (which is irrational but not transcendental) has a non-terminating, non-repeating decimal expansion.  If you keep the decimal point and do something like keeping only every fourth digit, or dropping every second one, or chopping out a finite chunk of digits, or substituting every digit with its double, or any other systematic operation with the digits, the resulting number is still irrational.  It has to be because the result is clearly non-terminating and non-repeating, just as the irrational number is that you started with.  That the result is non-terminating should be easy enough to see.  It is non-repeating because if it wasn’t, the systematic digit operation you applied would refute the original number’s non-repeating character and hence the original number could not be irrational.

Since some of these operations leave behind sub-sequences of the original number’s expansion, it should also be clear that an irrational number “contains” other irrational numbers and is in its own turn “contained” in others.

'Luthon64
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brianvds
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« Reply #14 on: April 09, 2013, 15:46:20 PM »

Hmm, what about irrational numbers containing other irrational numbers?
An irrational number does not have a finite expansion in any number system with a natural number radix.  This means that, e.g., √3 (which is irrational but not transcendental) has a non-terminating, non-repeating decimal expansion.  If you keep the decimal point and do something like keeping only every fourth digit, or dropping every second one, or chopping out a finite chunk of digits, or substituting every digit with its double, or any other systematic operation with the digits, the resulting number is still irrational.  It has to be because the result is clearly non-terminating and non-repeating, just as the irrational number is that you started with.  That the result is non-terminating should be easy enough to see.  It is non-repeating because if it wasn’t, the systematic digit operation you applied would refute the original number’s non-repeating character and hence the original number could not be irrational.

Since some of these operations leave behind sub-sequences of the original number’s expansion, it should also be clear that an irrational number “contains” other irrational numbers and is in its own turn “contained” in others.

'Luthon64

I wonder if that means that there is in principle a systematic operation by which one could extract, say, "Macbeth" from pi...

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