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Topic: Normal number

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	Normal number - Wikipedia, the free encyclopedia
		This theorem established the existence of normal numbers, but Waclaw Sierpinski was the first to give an example of one.
		No rational number is normal to any base, since the digit sequences of rational numbers are eventually periodic.
		It is extremely hard to prove the normality of numbers which were not explicitly constructed for the purpose.
	en.wikipedia.org /wiki/Normal_number (591 words)

	Normal number (computing) - Wikipedia, the free encyclopedia
		In computing, a normal number is a non-zero number in a floating-point representation which is within the balanced range supported by a given floating-point format.
		The magnitude of the smallest normal number in a format is given by b
		Non-zero numbers smaller in magnitude than the smallest normal number are called denormalized numbers or subnormal numbers.
	en.wikipedia.org /wiki/Normal_number_(computing) (204 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-11-06)
		I've some questions concerning "normal numbers." Let's say that a real number r is normal in base b if, writing r as a "decimal" number in base b, the frequencies of the b digits (0, 1,..., b-1) approach 1/b when you take more and more digits.
		A rational number q = n/m (where n, m are integers) cannot be normal (as it is not normal in base m), but not all the irrational numbers are normal - for instance, 0.03003300033300003333...
		Every proof I have seen relies on counting the normal numbers between 0 and 1 and counting all the numbers between 0 and 1, and showing that the measure of the former is 0, and the measure of the latter is 1.
	mathforum.org /library/drmath/view/55933.html (784 words)

	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.
		The first one was constructed by Sierpinski in 1916, and a related construction led to a computable absolutely normal number in 2002.
		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.
	planetmath.org /encyclopedia/Normal6.html (313 words)

	Normal number: Definition and Links by Encyclopedian.com - All about Normal number (Site not responding. Last check: 2007-11-06)
		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)

	Is Pi normal? (Site not responding. Last check: 2007-11-06)
		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)

	Normal number (Site not responding. Last check: 2007-11-06)
		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)

	CHROMOSOMAL PATTERNS OF INHERITANCE
		is the normal number of chromosomes in the mature egg and sperm (23 or n).
		The zygote starts with the normal complement of 46 chromosomes then, sometime in early embryonic development, a misdivision occurs in a cell leading to two daughter cells; one with 47 chromosomes (trisomy) and one with 45 chromosomes (monosomy).
		It is the number of genes (increased or decreased) that is abnormal.
	www.usd.edu /med/som/genetics/curriculum/1DCHROM2.htm (1619 words)

	Is e Normal? (Site not responding. Last check: 2007-11-06)
		A number is called "normal" with respect to a given base if, when the number is expressed in that base, the asymptotic frequencies of occurrence of each distinct string of k digits are equal, and this applies to every positive integer k.
		For example, if a number is normal in the base 10, the asymptotic frequency of occurrence of each of the decimal numerals 0,1,..,9 is precisely 1/10, and the asymptotic frequency of each two-digit strings 00,01,02,...,99 is exactly 1/100, and so on.
		Therefore, if a number is normal in the base 10, the asymptotic frequency of 10- digit strings comprising permutations of all 10 numerals must be 10!/10^10, which equals roughly 1 / 2755.73.
	www.mathpages.com /home/kmath519.htm (1771 words)

	Ivars Peterson's MathTrek - Absolutely Abnormal (Site not responding. Last check: 2007-11-06)
		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)

	Normal Numbers (Site not responding. Last check: 2007-11-06)
		A normal number is a real number for which the representation in any base has the property that any block of consecutive digits is equally likely to occur as we move along the digit expansion of the number.
		It is a little surprising at first that almost all real numbers have this property; however, it is quite a chore to come up with even one specific example.
		The project would be to summarize why almost all numbers are normal and to work through a proof that a specific number is normal.
	www.math.okstate.edu /~wrightd/4713/papers/node6.html (147 words)

	Neurogenetics at UT Health Science Center
		Variation in the numbers of neurons in a nucleus or ganglion of a group of animals belonging to the same species is often regarded as an experimental nuisance, and in many cases it is not possible to separate genuine intraspecific diversity from technical artifact (Konigsmark 1970, Williams and Rakic, 1986, 1988 a,b).
		Their interpretation is that the small number of founder cells of each organ is variable, but once they are committed as founders, a stereotypic series of cell divisions invariably ensues—six asymmetric divisions, each leading to one non-mitotic cell and one new stem cell, followed by one symmetric division that produces two non-mitotic cells.
		This suggests that number as well as form is a property that is ascribed to the early neuroblasts at the beginning of brain development—precisely the same conclusion that was suggested by the analysis of single gene effects.
	www.nervenet.org /papers/NUMBER_REV_1988.html (13478 words)

	Newran - random number generator library
		It is particularly appropriate for the situation where one requires sequences of identically distributed random numbers since the set up time for each type of distribution is relatively long but it is efficient when generating each new random number.
		The library includes classes for generating random numbers from a number of distributions and is easily extended to be able to generate random numbers from almost any of the standard distributions.
		The numbers are generated by generating a pair of numbers uniformly distributed over these rectangles and then accepting the X coordinate as the next random number if the pair corresponds to a point below the density function.
	www.robertnz.net /nr02doc.htm (2794 words)

	[No title]
		Copeland and Erdos proved in 1945 that this number is normal (in all bases), and it is therefore called the Copland-Erdos constant.
		Copeland and Erdos proved in 1945 > that this number is normal (in all bases), and it is therefore called the > Copland-Erdos constant.
		251 they say, "an explicit example of a number that is normal in all bases is still lacking." I hate to say this, but I'm starting to suspect that you may have misstated the result.
	www.math.niu.edu /~rusin/known-math/99/copeland (818 words)

	AE4451 Propulsion
		The “strength” of each shock depends on the “normal Mach number” upstream of each shock, and through each such shock, there is a loss in stagnation pressure – which translates to drag.
		The geometry of the inlet may be variable so that the normal Mach number upstream of each shock is kept the same – the first oblique shock, which occurs at the flight Mach number, is at a shallow slope, the next a bit closer to normal, etc.
		In this case, the Mach number is brought down from the flight Mach number to 2.0 through 3 oblique shocks, each having the same normal Mach number.
	www.adl.gatech.edu /classes/ae4451/w02a1.html (1407 words)

	Magic Squares
		normal magic square is an arrangement of the numbers 1, 2, 3,...
		When this upward-right movement would result in a location outside the boundaries of the square, we place the new number at the opposite end of the row or column that would contain the new number, if the rows and columns were not bounded.
		The number of normal magic squares increases dramatically as the size of the square increases.
	www.jcu.edu /math/vignettes/magicsquares.htm (742 words)

	Xtreme Visual Basic Talk - Std. Normal Random Number Generation
		The numbers generated should be normally distributed with an expoected value of 0 and standard deviation of 1.
		Thanks, but what I am looking for is, the numbers generated by the system should be normally distributed with an expected value of zero and a standard deviation of 1.
		We are looking for a random number that is 'normally distributed with an expected value of zero and a deviation of 1'.
	www.visualbasicforum.com /showthread.php?t=2718 (764 words)

	Random Number Research Bibliography
		Battaglia, F. Reproduction of autocorrelations in the generation of pseudorandom normal numbers.
		Nakamura, M. Obtaining a normal sequence with prescribed covariances from an independent uniform distribution of random variables.
		Whittlesey, J. A comparison of the correlational behaviour of random number generators for the I.B.M. Burr, I. A useful approximation to the normal distribution function, with application to simulation.
	randomnumber.org /bibliography.htm (6745 words)

	Uniform and Normal Random Number Generators (Site not responding. Last check: 2007-11-06)
		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)

	Untitled Document
		For the single format, the difference between a normal number and a subnormal number is that the leading bit of the significand (the bit to left of the binary point) of a normal number is 1, whereas the leading bit of the significand of a subnormal number is 0.
		The maximum positive normal number is the largest finite number representable in IEEE single format.
		The minimum positive normal number is often referred to as the underflow threshold.
	www.cis.rit.edu /class/simg726/notes/ieee754/ieee754.html (423 words)

	Math Trek: Absolutely Abnormal, Science News Online, Nov. 3, 2001 (Site not responding. Last check: 2007-11-06)
		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 (708 words)

	Energy Citations Database (ECD) - Energy and Energy-Related Bibliographic Citations
		Normal number of CGG repeats in the FMR-1 gene and abnormal incorporation of fibrillin into the extracellular matrix in Lujan Syndrome
		Cells from the affected individual showed normal synthesis and secretion of fibrillin when compared to control cells, but the fibrillin was not incorporated into the extracellular matrix.
		This may be important not only in normal body morphogenesis but also in the development/function of the brain.
	www.osti.gov /energycitations/product.biblio.jsp?osti_id=133616 (359 words)

	Is It Normal? - Number and Word Play (Site not responding. Last check: 2007-11-06)
		Ask yourself "is it normal?" and choose Yes or No in the green box.
		By correct I mean that the sum of the numbers equal a part of the number or they all add up to an even number like 40, 20, etc.
		All letters of the alphabet have at least 1 corresponding number to me, mostly under 10.
	isitnormal.com /normal-story-193.htm (377 words)

	Is It Normal? - Number and Word Play - Comments (Site not responding. Last check: 2007-11-06)
		When I was younger I used to see those patterns in the numbers, but can hardly do those quick calculations anymore.
		Some of the things OCD makes you do are usefull, but if it gets too much, and your finding it gets in the way, speak to your GP who will put you in touch with someone.
		What "correct" numbers are and why you insist on finding them is beyond me. you should consult a mental health professional, if you are really curious.
	isitnormal.com /comments-story-193.htm (532 words)

	Chromosome Number, Her2 Status, and Oncogenes
		Doctors also use ploidy, a measurement of the number of chromosomes present in cancer cells, to determine how fast a tumor may be growing:
		Diploid cancer cells have a normal number of chromosomes.
		Aneuploid cells have more chromosomes than normal, because they are growing faster than normal cells and need extra genetic material to pass on to the new cells they make.
	www.breastcancer.org /dia_pict_chromo.html (427 words)

	Mucus in stool Each person's body finds its own normal number of bowel movements.
		A double-blind trial found that chronic constipation and problems associated with it can be triggered by intolerance to cow’s milk in two thirds of constipated infants studied.
		The GIT breaks down foods by first using mechanical means such as chewing and then by the application of a host of complex chemical processes.
		Nutritional supplements that may be helpful: Chlorophyll, the substance responsible for the green colour in plants, may be useful for a number of gastrointestinal problems.
	digestion.virtualave.net /ibs/coeliac_disease_information5.html (569 words)

	Encyclopedia: Absolutely normal number
		Updated 355 days 13 hours 4 minutes ago.
		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 here to add NationMaster.com to your list of search engines (requires Mozilla or Firefox)
	www.nationmaster.com /encyclopedia/Absolutely-normal-number (771 words)

	PREGNANCY-BIRTH Messages for January, 2002: Re: Gestational Diabetes testing at 28 weeks- What is considered a normal ...
		I dont' know what my number was the first or second time, but it wasn't a big deal.
		I have never been diagnosed with diabetes before, have only >gained 22 pounds thus far, and am measuring at 28 cm which I've been >told is normal for this stage in the pregnancy.
		Also, what is considered a normal blood >sugar number at 28 weeks of pregnancy??
	forums.obgyn.net /pregnancy-birth/P-B.0201/1360.html (359 words)

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