Re: [math-fun] A transcendental puzzle
"Adam P. Goucher" <apgoucher@gmx.com> wrote:
I had a simpler example in mind, namely one that follows from *only* the following facts about algebraic numbers:
-- The algebraic numbers form a field of characteristic zero; -- x and exp(x) cannot both be algebraic.
As did I. The solution I was thinking of is the unique real solution to x = exp(-x). Approximately 0.5671432. Decades ago, by the same reasoning, I realized that the unique real solution to x = cos(x) must be transcendental. Of course that means cos(cos(x)), cos(cos(cos(x))), cos(cos(cos(cos(x)))), etc., are all also transcendental, since they're all the same number. Approximately 0.7390851. (Puzzle: Is the solution to x = cos(x) still transcendental if you do it in degrees instead of radians?) Dan Asimov <asimov@msri.org> wrote:
Why did God make it so hard to prove transcendence?
That's nothing compared to normality. Again "nearly all" real numbers are normal, but it's not easy to prove that any specific number is normal. Pi, for instance, was proven irrational in the 18th century and transcendental in the 19th, but the 20th didn't accomplish anything with it except increase the number of known digits from a few hundred to a few hundred billion. And the 21st has done nothing except increase the number of known digits by another factor of 100 or so. Maybe the 22nd century will prove (or disprove) that it's normal. Or maybe pi's normality is literally unknowable. Are any specific numbers known to be normal? That's not clear to me. Wikipedia says, without references: .... This theorem established the existence of normal numbers. In 1917, Wacl/aw Sierpinski showed that it is possible to specify a particular such number. Becher and Figueira proved in 2002 that there is a computable absolutely normal number, however no digits of their number are known. What is the specification of Sierpinski's number? And what is the algorithm for computing Becher/Figueira's number, and why have no digits been computed? Thanks.
Do you mean normal to a specific base? Given a base b, it's easy to construct uncountably many numbers normal to that base by just making sure that every possible string of N digits occurs, in the limit, the fraction 1/b^N of the time. Normal to every base seems much harder. I'm also interested in normal to factorial base, which seems more natural than normal to base b or even to all bases. —Dan
On Aug 12, 2015, at 8:15 PM, Keith F. Lynch <kfl@KeithLynch.net> wrote:
[Proving transcendence is} nothing compared to normality. Again "nearly all" real numbers are normal, but it's not easy to prove that any specific number is normal. Pi, for instance, was proven irrational in the 18th century and transcendental in the 19th, but the 20th didn't accomplish anything with it except increase the number of known digits from a few hundred to a few hundred billion. And the 21st has done nothing except increase the number of known digits by another factor of 100 or so. Maybe the 22nd century will prove (or disprove) that it's normal. Or maybe pi's normality is literally unknowable.
Are any specific numbers known to be normal? That's not clear to me. Wikipedia says, without references:
.... This theorem established the existence of normal numbers. In 1917, Wacl/aw Sierpinski showed that it is possible to specify a particular such number. Becher and Figueira proved in 2002 that there is a computable absolutely normal number, however no digits of their number are known.
What is the specification of Sierpinski's number? And what is the algorithm for computing Becher/Figueira's number, and why have no digits been computed? Thanks.
On 2015-08-12 20:32, Dan Asimov wrote:
Do you mean normal to a specific base? Given a base b, it's easy to construct uncountably many numbers normal to that base by just making sure that every possible string of N digits occurs, in the limit, the fraction 1/b^N of the time.
Normal to every base seems much harder.
I'm also interested in normal to factorial base, which seems more natural than normal to base b or even to all bases.
—Dan
Mma Doc: ChampernowneNumber RefLink[ChampernowneNumber,paclet:ref/ChampernowneNumber][b] gives the base-b Champernowne number Subscript[C, b]. RefLink[ChampernowneNumber,paclet:ref/ChampernowneNumber][] gives the base-10 Champernowne number. Details Mathematical constants treated as numeric by NumericQ and as constants by D. ChampernowneNumber[b] is a normal transcendental real number whose base-b representation is obtained by concatenating base-b digits of consecutive integers. ChampernowneNumber can be evaluated to arbitrary numerical precision. ChampernowneNumber automatically threads over lists. --rwg
On Aug 12, 2015, at 8:15 PM, Keith F. Lynch <kfl@KeithLynch.net> wrote:
[Proving transcendence is} nothing compared to normality. Again "nearly all" real numbers are normal, but it's not easy to prove that any specific number is normal. Pi, for instance, was proven irrational in the 18th century and transcendental in the 19th, but the 20th didn't accomplish anything with it except increase the number of known digits from a few hundred to a few hundred billion. And the 21st has done nothing except increase the number of known digits by another factor of 100 or so. Maybe the 22nd century will prove (or disprove) that it's normal. Or maybe pi's normality is literally unknowable.
Are any specific numbers known to be normal? That's not clear to me. Wikipedia says, without references:
.... This theorem established the existence of normal numbers. In 1917, Wacl/aw Sierpinski showed that it is possible to specify a particular such number. Becher and Figueira proved in 2002 that there is a computable absolutely normal number, however no digits of their number are known.
What is the specification of Sierpinski's number? And what is the algorithm for computing Becher/Figueira's number, and why have no digits been computed? Thanks.
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To my knowledge the base-b Champernowne number has not been proved normal in bases that aren't closely related to b. (I don't remember exactly what "closely related" means in this context.) On Thu, Aug 13, 2015 at 3:03 PM, rwg <rwg@sdf.org> wrote:
On 2015-08-12 20:32, Dan Asimov wrote:
Do you mean normal to a specific base? Given a base b, it's easy to construct uncountably many numbers normal to that base by just making sure that every possible string of N digits occurs, in the limit, the fraction 1/b^N of the time.
Normal to every base seems much harder.
I'm also interested in normal to factorial base, which seems more natural than normal to base b or even to all bases.
—Dan
Mma Doc: ChampernowneNumber
RefLink[ChampernowneNumber,paclet:ref/ChampernowneNumber][b] gives the base-b Champernowne number Subscript[C, b]. RefLink[ChampernowneNumber,paclet:ref/ChampernowneNumber][] gives the base-10 Champernowne number.
Details
Mathematical constants treated as numeric by NumericQ and as constants by D.
ChampernowneNumber[b] is a normal transcendental real number whose base-b representation is obtained by concatenating base-b digits of consecutive integers.
ChampernowneNumber can be evaluated to arbitrary numerical precision.
ChampernowneNumber automatically threads over lists. --rwg
On Aug 12, 2015, at 8:15 PM, Keith F. Lynch <kfl@KeithLynch.net> wrote:
[Proving transcendence is} nothing compared to normality. Again "nearly
all" real numbers are normal, but it's not easy to prove that any specific number is normal. Pi, for instance, was proven irrational in the 18th century and transcendental in the 19th, but the 20th didn't accomplish anything with it except increase the number of known digits from a few hundred to a few hundred billion. And the 21st has done nothing except increase the number of known digits by another factor of 100 or so. Maybe the 22nd century will prove (or disprove) that it's normal. Or maybe pi's normality is literally unknowable.
Are any specific numbers known to be normal? That's not clear to me. Wikipedia says, without references:
.... This theorem established the existence of normal numbers. In 1917, Wacl/aw Sierpinski showed that it is possible to specify a particular such number. Becher and Figueira proved in 2002 that there is a computable absolutely normal number, however no digits of their number are known.
What is the specification of Sierpinski's number? And what is the algorithm for computing Becher/Figueira's number, and why have no digits been computed? Thanks.
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How do you define normal to factorial base? - Scott On Wed, Aug 12, 2015 at 8:32 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Do you mean normal to a specific base? Given a base b, it's easy to construct uncountably many numbers normal to that base by just making sure that every possible string of N digits occurs, in the limit, the fraction 1/b^N of the time.
Normal to every base seems much harder.
I'm also interested in normal to factorial base, which seems more natural than normal to base b or even to all bases.
—Dan
On Aug 12, 2015, at 8:15 PM, Keith F. Lynch <kfl@KeithLynch.net> wrote:
[Proving transcendence is} nothing compared to normality. Again "nearly all" real numbers are normal, but it's not easy to prove that any specific number is normal. Pi, for instance, was proven irrational in the 18th century and transcendental in the 19th, but the 20th didn't accomplish anything with it except increase the number of known digits from a few hundred to a few hundred billion. And the 21st has done nothing except increase the number of known digits by another factor of 100 or so. Maybe the 22nd century will prove (or disprove) that it's normal. Or maybe pi's normality is literally unknowable.
Are any specific numbers known to be normal? That's not clear to me. Wikipedia says, without references:
.... This theorem established the existence of normal numbers. In 1917, Wacl/aw Sierpinski showed that it is possible to specify a particular such number. Becher and Figueira proved in 2002 that there is a computable absolutely normal number, however no digits of their number are known.
What is the specification of Sierpinski's number? And what is the algorithm for computing Becher/Figueira's number, and why have no digits been computed? Thanks.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
So now I wonder if there's an x such that the countable sequence
x, e^x, e^(e^x), e^(e^(e^x)), ... are *all* transcendental.
Chaitin's constant is an explicit example.
"Adam P. Goucher" <apgoucher@gmx.com> wrote:
I had a simpler example in mind, namely one that follows from *only* the following facts about algebraic numbers:
-- The algebraic numbers form a field of characteristic zero; -- x and exp(x) cannot both be algebraic.
As did I. The solution I was thinking of is the unique real solution to x = exp(-x). Approximately 0.5671432.
Very nice example. It does, however, make the (perfectly valid, but unnecessary) assumption that the reals are (analytically) complete. An analysis-free solution to the problem is simply: 2 + log(2) whose image under exp is: 2 * exp(2) both of which are transcendental.
Decades ago, by the same reasoning, I realized that the unique real solution to x = cos(x) must be transcendental. Of course that means cos(cos(x)), cos(cos(cos(x))), cos(cos(cos(cos(x)))), etc., are all also transcendental, since they're all the same number. Approximately 0.7390851.
(Puzzle: Is the solution to x = cos(x) still transcendental if you do it in degrees instead of radians?)
Transcendental, by Gelfond-Schneider. If it were algebraic, then x = cos(pi x / 180) = Re(exp(pi i / 180) ^ x) would be transcendental. Sincerely, Adam P. Goucher
On 13/08/2015 04:15, Keith F. Lynch wrote:
"Adam P. Goucher" <apgoucher@gmx.com> wrote:
I had a simpler example in mind, namely one that follows from *only* the following facts about algebraic numbers:
-- The algebraic numbers form a field of characteristic zero; -- x and exp(x) cannot both be algebraic.
As did I. The solution I was thinking of is the unique real solution to x = exp(-x). Approximately 0.5671432.
I suspect Adam was thinking of something more like 2 + log 2 -> 2 exp 2, which feels like a much more elementary (and explicit) example to me. -- g
I wrote:
I suspect Adam was thinking of something more like 2 + log 2 -> 2 exp 2, which feels like a much more elementary (and explicit) example to me.
When I posted that, I hadn't yet read Adam's clarification that he had exactly that example in mind. My apologies for the redundancy. -- g
participants (7)
-
Adam P. Goucher -
Allan Wechsler -
Dan Asimov -
Gareth McCaughan -
Keith F. Lynch -
rwg -
Scott Huddleston