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Topic: Transcendental number

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	PlanetMath: transcendental number
		A transcendental number is a complex number that is not an algebraic number.
		Cantor showed that, in a sense, “almost all” numbers are transcendental, because the algebraic numbers are countable, whereas the transcendental numbers are not.
		This is version 7 of transcendental number, born on 2001-11-04, modified 2005-02-28.
	planetmath.org /encyclopedia/TranscedentalNumber.html (94 words)

	ooBdoo
		The arithmetical operations of numbers, such as addition, subtraction, multiplication and division, are generalized in the branch of mathematics called abstract algebra, the study of abstract number systems such as groups, rings and fields.
		In the base-ten number system, they are written as a string of digits, with a period (decimal point) (in, for example, the US and UK) or a comma (in, for example, continental Europe) to the right of the ones place; negative real numbers are written with a preceding minus sign.
		The existence of complex numbers was not completely accepted until the geometrical interpretation had been described by Caspar Wessel in 1799; it was rediscovered several years later and popularized by Carl Friedrich Gauss, and as a result the theory of complex numbers received a notable expansion.
	www.oobdoo.com /wikipedia/?title=Number (3888 words)

	number. The Columbia Encyclopedia, Sixth Edition. 2001-05 (Site not responding. Last check: 2007-10-21)
		The real numbers are those representable by an infinite decimal expansion, which may be repeating or nonrepeating; they are in a one-to-one correspondence with the points on a straight line and are sometimes referred to as the continuum.
		Numbers of the form z = x + yi, where x and y are real and i = [radical]-1, such as 8 + 7i (or 8 + 7[radical]-1), are called complex numbers; x is called the real part of z and yi the imaginary part.
		The complex numbers are in a one-to-one correspondence with the points of a plane, with one axis defining the real parts of the numbers and one axis defining the imaginary parts.
	www.bartleby.com /65/nu/number.html (513 words)

	Transcendenteel getal -Transcendental Number
		A transcendental number is a number that is not the root of any integer polynomial, meaning that it is not an algebraic number of any degree.
		Transcendental numbers are important in the history of mathematics because their investigation provided the first proof that circle squaring, one of the geometric problems of antiquity that had baffled mathematicians for more than 2000 years was, in fact, insoluble.
		Hardy, G. and Wright, E. "Algebraic and Transcendental Numbers," "The Existence of Transcendental Numbers," and "Liouville's Theorem and the Construction of Transcendental Numbers." §11.5-11.6 in An Introduction to the Theory of Numbers, 5th ed.
	users.skynet.be /fa956617/math/topics/TranscendentalNumber.html (666 words)

	PlanetMath: example of transcendental number
		Next we use the theorem to construct a transcendental number.
		"example of transcendental number" is owned by alozano.
		This is version 4 of example of transcendental number, born on 2005-02-16, modified 2005-02-16.
	planetmath.org /encyclopedia/ExampleOfTranscendentalNumber.html (177 words)

	HP CALCULATOR HISTORY - Trascendental Functions
		Transcendental functions are those whose values cannot in general be expressed as solutions to an algebraic equation.
		Transcendental numbers are numbers which are the solutions of transcendental equations - equations which transcend algebraic methods - they cannot be solved purely by algebraic manipulation.
		In this case, this is because measuring C exactly with a straight ruler requires that an infinite number of infinitesimally small pieces of the circumference be measured.
	www.xnumber.com /xnumber/WMJ_transcendental.htm (1613 words)

	Transcendental number - Wikipedia, the free encyclopedia
		In mathematics, a transcendental number is any complex number that is not algebraic, that is, not the solution of a non-zero polynomial equation with integer (or, equivalently, rational) coefficients.
		Because algebraic numbers form an algebraically closed field, this would imply that the roots of the polynomial, a and b, must be algebraic.
		Any Liouville number must have unbounded terms in its continued fraction expression, and so using a counting argument one can show that there exist transcendental numbers which are not Liouville.
	en.wikipedia.org /wiki/Transcendental_number (1249 words)

	The Incredible Herkommer Number (Site not responding. Last check: 2007-10-21)
		Algebraic numbers, such as the square root of 3 or the golden ratio (phi) are representated by perodic infinite simple continued fractions.
		Transcendental numbers are a more complex class of numbers.
		Rationals are the simplest, followed by algebraic numbers, next are transcendental numbers of the first kind (e.g., e), then transcendental numbers of the second kind (e.g., pi).
	www.petrospec-technologies.com /Herkommer/herk_num.htm (317 words)

	Springer Online Reference Works
		Cantor [2], after discovering the countability of the set of all algebraic numbers and the uncountability of the set of all real numbers, thus proved that the transcendental real numbers form a set of the cardinality of the continuum.
		It was later found that Liouville transcendental numbers form an everywhere-dense subset of the real axis, having the cardinality of the continuum and zero Lebesgue measure.
		The development of methods of the theory of transcendental numbers has proved to have a strong influence on new studies in Diophantine equations [10], [11].
	eom.springer.de /t/t093640.htm (537 words)

	The Prime Glossary: algebraic number (Site not responding. Last check: 2007-10-21)
		A real number is an algebraic number if it is a zero of a polynomial with integer coefficients; and its degree is the least of the degrees of the polynomials with it as a zero.
		For example, the rational number a/b (with a, b and non-zero integers) is an algebraic number of degree one, because it is a zero of bx-a.
		In fact, almost all real numbers are transcendental because the set of algebraic numbers is countable.
	primes.utm.edu /glossary/page.php?sort=AlgebraicNumber (123 words)

	Numbers
		A perfect number is a natural number that is also equal to the sum of all of its divisors.
		Mersenne prime numbers have their own web site (Mersenne.org) which is dedicated to a netwide search for Mersenne prime numbers and related tasks.
		Numbers related to 1 in 10^n are commonly used in areas such as QoS (Quality of Service), 6 sigma, defining miracles, and such.
	www.georgehernandez.com /h/xzMisc/Math/Numbers.htm (2156 words)

	What's a number?
		Given the difficulty of establishing whether a given number is algebraic or not, this was one of Cantor's early surprising results.
		The rest of the complex numbers could also be defined by adding this new number i to the set of reals and postulating that usual arithmetic operations (addition, subtraction, multiplication) apply to the expanded set and all the laws known to hold for these operations hold for the new set as well.
		The rest of surreal numbers (included are the numbers we discussed so far and myriads of numbers some of which I have a difficulty imagining.) are formed starting with 0 and applying just two simple rules.
	www.cut-the-knot.org /do_you_know/numbers.shtml (3745 words)

	KryssTal : Introduction to Numbers
		The sum of the natural numbers is the value obtained when a selection of the numbers (beginning from 1) are added together.
		Square Numbers are integers that are the square of smaller integers.
		Skewes' Number is far, far larger than the number of all the particles in the observable Universe.
	www.krysstal.com /numbers.html (1871 words)

	Transcendental Numbers (Site not responding. Last check: 2007-10-21)
		It should be noted that the term "transcendental" in mainstream mathematics requires that any transcendental number not be the root of a polynomial equation with integral coefficients.
		Inasmuch as it is not within the purview of this website to accept such undue limitations, transcendental numbers will be viewed as those which cannot be precisely identified to the point where the next number in the sequence can be known purely from the previous numbers in the sequence.
		The most well known of the transcendental numbers is p, which is typically defined as the ratio of the circumference of a circle to its diameter.
	www.halexandria.org /dward089.htm (2341 words)

	Composite number for pi satisfies Euler's equation.
		numbers on a number line using a specific unit length from the start to the first point, and the same unit length from point to point.
		number system; but, when they are subtracted from each other then the difference is not included in the system, e.g., subtracting five from three (3 – 5 = – 2) and subtracting four from four (4 – 4 = 0).
		The set of whole numbers that includes the number zero and all the negatives, is 0, – 1, – 2, – 3, – 4,....
	members.ispwest.com /r-logan/narrative.html (2193 words)

	Math Lair - Transcendental Numbers (Site not responding. Last check: 2007-10-21)
		On the other hand, numbers such as log 2 and π are transcendental.
		The existence of transcendental numbers was not proved until 1840, when Joseph Liouville proved that the number 0.1100010000000000000000010000..., where 1's appear in the n!
		Showing that π was transcendental also proved that one of the three famous construction problems of antiquity was impossible.
	www.stormloader.com /ajy/transcendental.html (303 words)

	Mathematical Constants
		Not many transcendental numbers are known (e and Liouville's number are another two examples) but in fact in 1874 Cantor showed that almost all real numbers are transcendental.
		It is also intimately connected to the Fibonnaci numbers Fibonacci Numbers and the Golden Section 1, 1, 2, 3, 5, 8, 13, 21, 34 where each term is the sum of the previous two.
		It is actually an estimate for a number that occurs in combinatorics and the joke is that the number is believed to be 6, so Graham's number must rank as one of the worst estimates ever.
	www.mayer.dial.pipex.com /maths/constant.htm (2115 words)

	The 15 Most Famous Transcendental Numbers - Cliff Pickover
		Even so, only a few classes of transcendental numbers are known to humans, and it's very difficult to prove that a particular number is transcendental.
		Transcendental numbers cannot be expressed as the root of any algebraic equation with rational coefficients.
		If the number is terminating, convert it to non-terminating by subtracting one from the last digit, and adding an infinite string of 9's to the end.
	sprott.physics.wisc.edu /pickover/trans.html (1263 words)

	Vladimir Gennadievich Sprindzuk (Site not responding. Last check: 2007-10-21)
		V.G Sprindzuk was a famous authority on the theory of Diophantine equations, Diophantine approximation and transcendental Number Theory.
		During his postgraduate studies V. Sprindzuk became interested in the metric theory of transcendental numbers his PhD thesis was entitled "Metric theorems on Diophantine approximations by algebraic number of bounded degree".
		His detailed studies of the Thue equation in algebraic number fields proved to be useful for the effective solution of a wide class of Diophantine equations and allowed him to study the possibility of effective approximations to algebraic numbers both in archimedean and non-archimedean domains.
	www.numbertheory.org /obituaries/OTHERS/sprindzuk.html (929 words)

	Transcendental Meditations
		Transcendental numbers have a long history, dating back to the ancient Greeks, even though they were not named or truly recognized until much later.
		Transcendental numbers are irrational numbers that are not the roots of algebraic equations.
		This is fairly clear since the rational numbers were denumerable, but the real numbers, made up of the rational numbers and irrational numbers, were nondenumerable.
	www.andrews.edu /~calkins/math/webtexts/numb14.htm (2074 words)

	Line of real numbers-transcendental numbers
		That is, the set of all transcendental numbers is far larger than the set of algebraic numbers (non-transcendental numbers which includes all rational numbers).
		From Gelfond's theorem, a number of the form is transcendental (and therefore irrational) if a is algebraic, 1 and b is irrational and algebraic.
		Note that most real numbers are not computable, in the same sense as most numbers are transcendental, since the computable numbers are countable and the real numbers are not.
	www.physicsforums.com /showthread.php?t=65605 (1204 words)

	[No title]
		In fact, any root of a polynomial in the algebraic numbers is an algebraic number - as a note further on explains, this allows you to generate an enormous number of transcendentals given a single one.
		Since each of these finite number of polynomials has only a finite number of roots, there are only a finite number of numbers generated as the roots of a polynomial with a given value.
		EG 2^sqrt(2) * 3^sqrt(3) is transcendental ---------------------------------------------------------------------------- 11.
	www.math.niu.edu /~rusin/known-math/95/transcend (1239 words)

	EARTH MYSTERIES: Notes on Pi (¼ ) (Site not responding. Last check: 2007-10-21)
		Pi has had various names through the ages, and all of them are either words or abstract symbols, since pi is a number that can't be shown completely and exactly in any finite form of representation.
		A transcendental number is a number but can't be expressed in any finite series of either arithmetical or algebraic operations.
		for the number was William Jones, a Welsh mathematician, who coined it in 1706.
	witcombe.sbc.edu /earthmysteries/EMPi.html (462 words)

	transcendental number - HighBeam Encyclopedia (Site not responding. Last check: 2007-10-21)
		Most hazily remember it from our schooldays, here Steve Connor charts its history and celebrates a number that is irrational, transcendental...
		Transcendental Meditation Lowers Blood Pressure in Black Adolescents.
		Was Mallarme a transcendental philosopher?: the place of literature in the 'Divagations.' (poet Stephane Mallarme)
	www.encyclopedia.com /doc/1E1-x-trnscdnum.html (188 words)

	James Gosling: on the Java Road
		Therefore, in that range the performance of the JDK's transcendental functions should be nearly the same as the performance of the transcendental functions in C, C++, etc. that are using those same fsin/fcos instructions.
		Since the period of sin/cos is pi and pi is transcendental, this amounts to having to compute a remainder from the division by a transcendental number, which is non-obvious.
		In the excellent paper The K5 transcendental functions by T. Lynch, A. Ahmed, M. Schulte, T. Callaway, and R. Tisdale a technique is described for doing argument reduction as if you had an infinitely precise value for pi.
	blogs.sun.com /jag/entry/transcendental_meditation (2007 words)

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