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outrun
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Re: Mathematica high precision question

April 24th, 2017, 8:27 am

and then could  [$]\pi[$] be is inside [$]e^5[$] (if we ignore decimal points) ? I was once thinking many of these numbers are part of the same random sequence, we just need to find a simple system where each of these numbers have their starting point inside this sequence??
Ah, no! A very simple and intuitive proof is "Cantor's diagonal argument"
great!  I think also K had some objections against every word is inside Pi
Person of Interest
I've never seen that series. Looks good!
Can we find Pi inside Pi (inside Pi, inside Pi,..) ? Would that make it a fraction?
 
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Cuchulainn
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Re: Mathematica high precision question

April 24th, 2017, 9:51 am

It was only a matter of time before someone started on pi :D Such a boring number. A cure is to do Dedekind cuts.

Archimedes, in his Measurement of a Circle, created the first algorithm for the calculation of π based on the idea that the perimeter of any (convex) polygon inscribed in a circle is less than the circumference of the circle, which, in turn, is less than the perimeter of any circumscribed polygon.



Can you compute it to 100 digits. Having done that you can then check the area

http://www.boost.org/doc/libs/1_64_0/li ... g/aos.html
Last edited by Cuchulainn on April 24th, 2017, 10:11 am, edited 1 time in total.
 
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Cuchulainn
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Re: Mathematica high precision question

April 24th, 2017, 10:07 am

"Why, Sir, you find no man, at all intellectual, who is willing to leave [$]e^5[$]. No, Sir, when a man is tired of [$]e^5[$], he is tired of life; for there is in [$]e^5[$] all that life can afford."

— Samuel Johnson
Interesting!  If the deeper digits of [$]e^5[$] are random , then wouldn't everything ever written be somewhere in them?
and then could  [$]\pi[$] be is inside [$]e^5[$] (if we ignore decimal points) ? I was once thinking many of these numbers are part of the same random sequence, we just need to find a simple system where each of these numbers have their starting point inside the random sequence?? We need to find the point where the head is biting the tail.
I would say not. [$]e[$] is just a crazy number but [$]\pi[$] is geometry. Oil and water?
 
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outrun
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Re: Mathematica high precision question

April 24th, 2017, 10:30 am

For one number B to be inside the number A you are saying that (iff)

A = (I + B)/10^n

With I some integer. If A and B don't relate to eachother like this then you can't find one embedded in the other.
 
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Cuchulainn
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Re: Mathematica high precision question

April 24th, 2017, 10:56 am

For one number B to be inside the number A you are saying that (iff)

A =  (I + B)/10^n

With I some integer. If A and B don't relate to eachother like this then you can't find one embedded in the other.
Do you have an example? what's n?
 
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outrun
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Re: Mathematica high precision question

April 24th, 2017, 11:03 am

Like this, 0.159265... Inside pi:
3.14159265... = (314 + .159265...)/100

314/100 is the trailing bit of A you put in front of B

Some general note:
The Cantor diagonal argument says that there are more A and B numbers than integers I.

Pi and e are irrational, a number can only be found inside itself if it's repetitive, which means it's rational.
 
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Cuchulainn
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Re: Mathematica high precision question

April 24th, 2017, 11:21 am

Actually, these numbers are transcendental, so they cannot be found by solving equations, in contrast to good old [$]\sqrt2[$]

What is the 239th digit of e? 
 
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outrun
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Re: Mathematica high precision question

April 24th, 2017, 12:21 pm

2
 
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outrun
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Re: Mathematica high precision question

April 24th, 2017, 12:28 pm

What's cool is that even though pi and e are transedental, you can very efficiently compute the n-th digit using spigot algorithms without computer the digits before it!
https://en.wikipedia.org/wiki/Bailey%E2 ... fe_formula
 
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Traden4Alpha
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Re: Mathematica high precision question

April 24th, 2017, 12:53 pm

Like this, 0.159265... Inside pi:
3.14159265... = (314 + .159265...)/100

314/100 is the trailing bit of A you put in front of B

Some general note:
The Cantor diagonal argument says that there are more A and B numbers than integers I.

Pi and e are irrational, a number can only be found inside itself if it's repetitive, which means it's rational.
That's proof that there are no turducken numbers among the transcendentals but not proof that pi (or the tail of pi) does not appear in e.

Or, more generally, are the distributions of digit sequences the same for pi and e?
 
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Traden4Alpha
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Re: Mathematica high precision question

April 24th, 2017, 12:55 pm

What's cool is that even though pi and e are transedental, you can very efficiently compute the n-th digit using spigot algorithms without computer the digits before it!
https://en.wikipedia.org/wiki/Bailey%E2 ... fe_formula
A cryptographer would conclude from that fact that the digits of pi and e are highly non-random.
 
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Re: Mathematica high precision question

April 24th, 2017, 1:04 pm

What's cool is that even though pi and e are transedental, you can very efficiently compute the n-th digit using spigot algorithms without computer the digits before it!
https://en.wikipedia.org/wiki/Bailey%E2 ... fe_formula
A cryptographer would conclude from that fact that the digits of pi and e are highly non-random.
The last digit of Pi is random! I will tell you what it is, just follow the sound:
Image
 
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outrun
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Re: Mathematica high precision question

April 24th, 2017, 1:26 pm

Like this, 0.159265... Inside pi:
3.14159265... = (314 + .159265...)/100

314/100 is the trailing bit of A you put in front of B

Some general note:
The Cantor diagonal argument says that there are more A and B numbers than integers I.

Pi and e are irrational, a number can only be found inside itself if it's repetitive, which means it's rational.
That's proof that there are no turducken numbers among the transcendentals but not proof that pi (or the tail of pi) does not appear in e.

Or, more generally, are the distributions of digit sequences the same for pi and e?
Indeed!
It proves that e is not in e, that pi in not in pi, but *not* that no-one has ever noticed that pi = e + 0.7 !
 
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outrun
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Re: Mathematica high precision question

April 24th, 2017, 1:32 pm

What's cool is that even though pi and e are transedental, you can very efficiently compute the n-th digit using spigot algorithms without computer the digits before it!
https://en.wikipedia.org/wiki/Bailey%E2 ... fe_formula
A cryptographer would conclude from that fact that the digits of pi and e are highly non-random.
Computable and random are not the same thing. Lots of pseudorandom generators are efficiently computable (of course) but the generated numbers pass all statistical test for randomness. The NSA once proposed (managed to turn it into an ISO standard, bribed RSA to use it in their token keychain) that everyone should use their random number generator algorithm, but it has a backdoor that allowed (only) them to compute the next digits once it had observed only a dozen numbers: https://en.wikipedia.org/wiki/Dual_EC_DRBG
 
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ExSan
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Re: Mathematica high precision question

July 27th, 2017, 2:21 pm

Image
I just discovered this particular feature of Visual Studio 2015
the C compiler can not achieve an exact representation of the value 0.7
I would have expected more accuracy
ps. sorry not precisely related to Mathematica
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