
	Where results make sense

Topic: Absolutely normal number

Related Topics

Essential thrombocytopenia

John Ripley

Sandy Point, Chile

Independence class aircraft carrier

American Convention on Human Rights

George Passant

Tordesilhas Line

Upper Pohatcong Mountain

Viatcheslav Ekimov

Jackie Earle Haley

Mount Kembla

Kazusa Murai

In the News (Fri 17 Feb 12)

EXECUTIVE POWER

Iran

ANALYSIS

Pakistan

Family compact

	PlanetMath: absolutely normal
		Borel also proved that almost all real numbers are absolutely normal, in the sense that the numbers that are not absolutely normal form a set with Lebesgue measure zero.
		For instance, it has been conjectured that all irrational algebraic numbers are absolutely normal since no counterexamples are known; on the other hand, not a single irrational algebraic number has been proven normal in any base.
		This is version 5 of absolutely normal, born on 2003-04-05, modified 2005-03-08.
	planetmath.org /encyclopedia/Normal7.html (313 words)

	Normal number -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-09)
		In (A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement) mathematics, a normal number is, roughly speaking, a real number whose digits show a random distribution with all digits being equally likely.
		The set of non-normal numbers is known to be (additional info and facts about uncountable) uncountable; this follows easily by simply omitting one digit from each real number.
		No (An integer or a fraction) rational number is normal to any base, since the digit sequences of rational numbers are eventually periodic.
	www.absoluteastronomy.com /encyclopedia/N/No/Normal_number.htm (665 words)

	Normal number: Definition and Links by Encyclopedian.com - All about Normal number
		However, the set of non-normal numbers is uncountable.
		A computable normal number was constructed by Becher and Figuera; an example of an uncomputable normal number is given by Chaitin's constant Ω.
		Bailey and Crandall conjectured in 2001 that every irrational algebraic number is normal; while no counterexamples are known, not a single irrational algebraic number has ever been proven normal in any base.
	www.encyclopedian.com /no/Normal-number.html (509 words)

	Encyclopedia: Normal number
		In mathematics, a prime number, or prime for short, is a natural number greater than one and whose only distinct positive divisors are one and itself.
		In mathematics, a rational number (or informally fraction) is a ratio of two integers, usually written as the vulgar fraction a/b, where b is not zero.
		In mathematics, an algebraic number relative to a field F is any element x of a given field K containing F such that x is a solution of a polynomial equation of the form anxn + an−1xn−1 + ··· + a1x + a0 = 0 where n is a positive integer called the degree...
	www.nationmaster.com /encyclopedia/Normal-number (1075 words)

	Normal number (Site not responding. Last check: 2007-10-09)
		In mathematics, a normal number is, roughly speaking, a realnumber whose digits show a random distribution with all digits being equally likely.
		No rational number is normal to any base, since the digitsequences of rational numbers are eventually periodic.
		A computable normal number was constructed by Verónica Becher andSantiago Figueira; an example of an uncomputable normal number is given by Chaitin's constant Ω.
	www.therfcc.org /normal-number-211888.html (536 words)

	Is Pi normal?
		The idea of normality, first introduced by E. Borel in 1909, is an attempt to formalize the notion of a real number being random.
		A number that is normal in all bases is called normal.
		That normality is not sufficient follows from the observation that a truly random sequence of digits ought to be normal when only digits in positions corresponding to perfect squares are examined.
	pi314.at /math/normal.html (668 words)

	Wacław Sierpiński - Wikipedia, the free encyclopedia
		He was known for outstanding contributions to set theory (research on the axiom of choice and the continuum hypothesis), number theory, theory of functions and topology.
		Three well-known fractals are named after him (the Sierpinski triangle, the Sierpinski carpet and the Sierpinski curve), as are Sierpinski numbers and the associated Sierpiński problem.
		In 1916, Sierpiński gave the first example of an absolutely normal number.
	en.wikipedia.org /wiki/Waclaw_Sierpinski (907 words)

	Ivars Peterson's MathTrek - Absolutely Abnormal
		In 1909, mathematician Émile Borel (1871-1956) introduced the concept of normality as one way to characterize the resemblance between the digits of a mathematical constant such as pi (the ratio of a circle's circumference to its diameter) and a sequence of random numbers.
		A number is said to be "absolutely normal" if its digits are normal not only to base 10 but also to every integer base greater than or equal to 2.
		Liouville had introduced such numbers as examples of transcendental numbers--real numbers that are not roots of polynomial equations with integer coefficients.
	www.maa.org /mathland/mathtrek_11_05_01.html (583 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		Intuitively, $x$ is normal in base $b$ if all digits and digit-blocks in the base-$b$ digit sequence of $x$ occur just as often as would be expected if the sequence had been produced completely randomly.
		Borel also proved that \htmladdnormallink{almost all}{http://planetmath.org/encyclopedia/AlmostSurely.html} \htmladdnormallink{real numbers}{http://planetmath.org/encyclopedia/RealNumber.html} are absolutely normal, in the sense that the numbers that are not absolutely normal form a set with \htmladdnormallink{Lebesgue measure}{http://planetmath.org/encyclopedia/LebesgueMeasure.html} zero.
		For instance, it has been conjectured that all \htmladdnormallink{irrational}{http://planetmath.org/encyclopedia/Irrational.html} \htmladdnormallink{algebraic numbers}{http://planetmath.org/encyclopedia/AlgebraicNumber.html} are absolutely normal since no counterexamples are known; on the other hand, not a single irrational algebraic number has been proven normal in any base.
	www.ma.utexas.edu /~jcorneli/e/work%20folder/massive/AbsolutelyNormal.tex (348 words)

	Encyclopedia: Absolutely normal number
		The number x is called normal in base b if
		A computable normal number was constructed by Ver nica Becher and Santiago Figueira; an example of an uncomputable normal number is given by Chaitin's constant Ω.
		Click for other authoritative sources for this topic (summarised at Factbites.com).
	www.nationmaster.com /encyclopedia/Absolutely-normal-number (608 words)

	Pi à la Mode: Science News Online, Sept. 1, 2001 (Site not responding. Last check: 2007-10-09)
		Pi would be considered normal to base 10 if any single digit appears one-tenth of the time, any two-digit combination one-hundredth of the time, any three-digit combination one-thousandth of the time, and so on.
		Bailey and Crandall have proposed that the normality of log 2 to base 2 is linked to a particular iterative process, or dynamical map, that generates a sequence of numbers between 0 and 1.
		Lagarias, J.C. On the normality of arithmetical constants.
	www.sciencenews.org /20010901/bob9.asp (1861 words)

	Uniform and Normal Random Number Generators (Site not responding. Last check: 2007-10-09)
		The generators use a generalisation of Marsaglia's "xorshift" random number generators [218].
		ranut is a uniform pseudo-random number generator which uses recurrences based on primitive trinomials, or in some cases trinomials which have large primitive factors (over the field GF(2)).
		Most algorithms for generating normally distributed random numbers are slow because they first generate uniformly distributed numbers and then transform them to normally distributed numbers.
	web.comlab.ox.ac.uk /oucl/work/richard.brent/random.html (411 words)

	Sierpinski (Site not responding. Last check: 2007-10-09)
		In 1916, during his time in Moscow, Sierpinski gave the first example of an absolutely normal number, that is a number whose digits occur with equal frequency in whichever base it is written.
		Borel had proved such numbers exist but Sierpinski was the first to give an example.
		Private libraries of all the four professors of mathematics from Warsaw University and also quite a number of manuscripts of their works and handbooks written during the war were burnt too.
	www-history.mcs.st-andrews.ac.uk /history/Mathematicians/Sierpinski.html (1772 words)

	The Green Head - Emporium : Book Shop - Absolutely Normal Chaos (Site not responding. Last check: 2007-10-09)
		I liked the book because it was about a girl that was my age and I could relate to the things that she writes about in her journal.
		I didn't like that her best friend, Beth Ann, left her out and started to ignore her when she got her first boyfriend and when some girls at school started a new club and wanted her to join but they never did include Mary Lou.
		"Absolutely Normal Chaos" is set up like a journal and you can actually understand how the character sees all that is going on in front of her eyes.
	www.thegreenhead.com /emporium/0064406326/Absolutely-Normal-Chaos.html (1123 words)

	Math Trek: Absolutely Abnormal, Science News Online, Nov. 3, 2001 (Site not responding. Last check: 2007-10-09)
		In 1909, mathematician Émile Borel (1871—1956) introduced the concept of normality as one way to characterize the resemblance between the digits of a mathematical constant such as pi (the ratio of a circle's circumference to its diameter) and a sequence of random numbers.
		Mathematician Greg Martin of the University of Toronto recently turned his attention to the opposite extreme—real numbers that are normal to no base whatsoever.
		Liouville had introduced such numbers as examples of transcendental numbers—real numbers that are not roots of polynomial equations with integer coefficients.
	www.sciencenews.org /20011103/mathtrek.asp (680 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		He was attracted by the work of one of his teachers Voronov who worked on the number theory.
		In 1916 he was the first to give an example of an absolutely normal number (number whose digits occur with equal frequency in whichever base it is written).
		But he continued to give seminar on the theory of numbers up to 1967 and to work as editor for several mathematics journals.
	www.enseeiht.fr /hmf/travaux/CD9900/travaux/optmfn/hi/00pa/mfn03/biograp.htm (622 words)

	SignOnSanDiego.com > News > North County -- Camacho's demeanor 'absolutely normal' after shooting, paramedic ...
		VISTA – The demeanor of an ex-con accused of fatally shooting an Oceanside police officer following a traffic stop was "absolutely normal" during an ambulance ride to a hospital, a paramedic testified Tuesday.
		Adrian Camacho "was alert" while being taken to Scripps Memorial Hospital in La Jolla for treatment of a gunshot wound to his leg and cuts to his wrists, said Timothy Huerta, a paramedic with the Oceanside Fire Department.
		A number of witnesses testified that they watched Zeppetella collect paperwork from the defendant, then heard gunshots after the 27-year-old officer stepped away.
	www.signonsandiego.com /news/northcounty/20051101-1419-camacho.html (688 words)

	RE: Dr. Dobb's Python-URL! - weekly Python news and links (Mar 17)
		Last I heard (a few years ago), no. Note two meanings for "normal": the original definition required normality in all bases simultaneously, and that's usually called "absolutely normal" now.
		Also, AFAIK, no absolutely normal number is known (constructed or not).
		This is curious because the set of non-normal reals has measure 0 ("almost all" reals are absolutely normal).
	www.talkaboutprogramming.com /group/comp.lang.python/messages/247054.html (282 words)

	dBforums - RE: Dr. Dobb's Python-URL! - weekly Python news and links (Mar 17)
		> normal number is known (constructed or not).
		to be 10-normal, but not known to be absolutely normal.
		When one talks about a number being b-normal, b is an integer.
	dbforums.com /t725207.html (1107 words)

	Absolutely Normal - ROY, ANN (Site not responding. Last check: 2007-10-09)
		Boston, MA, Houghton Mifflin, 1947, Ed.: 1st, octavo, Ill.: by Ann Roy.
		Pink cloth, hc, Cartooned ep's, a book of uncaptioned cartoons -- in fact, this is the first book I would recommend to illiterates..
		They offer full satisfaction and normal prices - no markups, no hidden costs, no overcharged shipping costs.
	www.antiqbook.com /boox/bok/2081.shtml (84 words)

	Waclaw Sierpinski (Site not responding. Last check: 2007-10-09)
		In 1916, during his time in Moscow, Sierpinski gave the first example of an absolutely normal number.
		Borel had proved such numbers exit but Sierpinski was the first to give an example.
		In 1919, he was promoted to professor at Warsaw, and he spent the rest of his life there.
	www.stetson.edu /~efriedma/periodictable/html/Si.html (611 words)

	Dr. Dobb's Python-URL! - weekly Python news and links (Mar 17) (Site not responding. Last check: 2007-10-09)
		Tim Peters wrote: > Last I heard (a few years ago), no. Note two meanings for "normal": > the > original definition required normality in all bases simultaneously, > and > that's usually called "absolutely normal" now.
		True, although if mentioned without reference to a base, "normal" often means "absolutely normal." (i.e., "It's normal in base b" obviously doesn't mean absolutely normal, but "pi is thought to be normal" probably does.) > Also, AFAIK, no absolutely > normal number is known (constructed or not).
		Champernowne and Copeland-Erdos are known to be 10-normal, but not known to be absolutely normal.
	mail.python.org /pipermail/python-list/2003-March/154048.html (220 words)

	Absolutely Normal Chaos - Sharon Creech book review summary
		I read the story Absolutely Normal Chaos, by Sharon Creech.
		Mary Lou Finney is a thirteen year old girl, who has had a huge crush on Alex Cheevey for a long time.
		Beth Ann is a fourteen year old girl that is boy crazy.
	www.buildingrainbows.com /review/reviewid/1109 (275 words)

	Sharon Creech Box Set: Absolutely Normal Chaos, Walk Two Moons, Chasing Redbird
		Absolutely normal Mary Lou is all set for a boring summer, trying to find something, anything, to say in her school-assigned journal.
		Three girls, three stories--Sharon Creech is the well-loved and award-winning author of these three novels, pulled together in a handy boxed set for her many fans who won't want to wait to get their hands on the next Creech title.
		Absolutely Normal Chaos, Walk Two Moons (a Newbery Medal winner), and Chasing Redbird are each sensitively drawn, witty, complex, and unexpected looks at young girls in that strange and bumpy land of adolescence.
	www.literacyconnections.com /0_0064410080.html (249 words)

	Absolutely Normal Chaos \| People Blurb (Site not responding. Last check: 2007-10-09)
		The Absolutely Normal Chaos is part of our discount Book catalog.
		Used Absolutely Normal Chaos are in stock for only $0.50.
		Discount pricing is subject to change, in order to get the Book Absolutely Normal Chaos at this reduced price, you must buy now!
	peopleblurb.com /amazon/asin.0064406326.Book_Absolutely_Normal_Chaos.html (528 words)

	Chronology for 1910 to 1920
		Sierpinski gives the first example of an absolutely normal number, that is a number whose digits occur with equal frequency in whichever base it is written.
		Hausdorff introduces the notion of "Hausdorff dimension", which is a real number lying between the topological dimension of an object and 3.
		It is used to study objects such as Koch's curve.
	www-history.mcs.st-andrews.ac.uk /history/Chronology/1910_1920.html (318 words)

	List of number theory topics Article, Listnumbertheorytopics Information (Site not responding. Last check: 2007-10-09)
		(listnumbertheorytopics,list number theory topics,listnumber theory topics,list numbertheory topics,list number theorytopics)
		Proof that the sum of the reciprocals of the primesdiverges
		NB Listing here is not pejorative, since many famoustopics in number theory have origins in challenging problems posed purely for their own sake.
	www.anoca.org /prime/theorem/list_of_number_theory_topics.html (114 words)

	Cross Curriculum Lesson Plans, Glencoe Health 2003
		Explain how strong communication skills are a key ingredient in the health of the family.
		Ask a few volunteers to describe the composition of their households (i.e., the number of and relationship among adults and children who live in their house).
		If any student mentions a relative that would make the family an extended one—such as grandparents, aunts, uncles, or cousins—write these titles on the board.
	www.glencoe.com /sec/health/teachres/lessonplans/ccchaos.shtml (492 words)

Try your search on: Qwika (all wikis)