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	Ideal number - Wikipedia, the free encyclopedia
		In mathematics an ideal number is an algebraic integer which represents an ideal in the ring of integers of a number field; the idea was developed by Kummer, and lead to Dedekind's definition of ideals for rings.
		This means there is an element of the ring of integers of the class field, which is an ideal number, such that all multiples times elements of this ring of integers lying in the ring of integers of the original field define the nonprincipal ideal.
		Kummer's 1844 memoir was in honor of the jubilee celebration of the University of Königsberg and was meant as a tribute to Jacobi.
	en.wikipedia.org /wiki/Ideal_number (724 words)

	Ideal (ring theory) - Wikipedia, the free encyclopedia
		An ideal can be used to construct a factor ring in a similar way as a normal subgroup in group theory can be used to construct a factor group.
		In the ring Z of integers, every ideal can be generated by a single number (so Z is a principal ideal domain), and the ideal determines the number up to its sign.The concepts of "ideal" and "number" are therefore almost identical in Z (and in any principal ideal domain).
		The sum and the intersection of ideals is again an ideal; with these two operations as join and meet, the set of all ideals of a given ring forms a lattice.
	en.wikipedia.org /wiki/Ideal_(ring_theory) (1343 words)

	PlanetMath: ideal
		When the failure of unique factorization in number fields was first noticed, one of the solutions was to work with so-called ``ideal numbers'' in which unique factorization did hold.
		These ``ideal numbers'' were in fact ideals, and in Dedekind domains, unique factorization of ideals does indeed hold.
		This is version 11 of ideal, born on 2001-10-19, modified 2005-01-17.
	planetmath.org /encyclopedia/Ideal.html (162 words)

	Prime ideal -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-29)
		In any ring R, a maximal ideal is an ideal M that is (additional info and facts about maximal) maximal in the set of all proper ideals of R, i.e.
		One use of prime ideals occurs in (additional info and facts about algebraic geometry) algebraic geometry, where varieties are defined as the zero sets of ideals in polynomial rings.
		An ideal such that ab in P implies that a or b is in P is called a completely prime ideal.
	www.absoluteastronomy.com /encyclopedia/p/pr/prime_ideal.htm (836 words)

	Prime ideal
		In mathematics, a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers.
		In any ring R, a maximal ideal is an ideal M that is maximal in the set of all proper ideals of R, i.e.
		One use of prime ideals occurs in algebraic geometry, where varieties are defined as the zero sets of ideals in polynomial rings.
	www.brainyencyclopedia.com /encyclopedia/p/pr/prime_ideal.html (816 words)

	A New Universal Tree of Life (Site not responding. Last check: 2007-10-29)
		Six is the common denominator of all the numbers being used when reduced; that is when all the digit numbers of the paths and spheres are added together and reduced down to a single digit.
		The numbers of the paths are derived from the 22 numbers of the Tarot, 0 through 21, the 22 Hebrew letters, and the 10 Planets and 12 Signs of the Zodiac (10+12=22).
		Zero adds no quantity that is not already intrinsically present and doesn't change the nature of the number it is added to; however it is the qualifying source for the value all the quantitative numbers.
	mysticalkeys.com /nutol/nutol_numbers.htm (253 words)

	Complex number - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-29)
		The existence of complex numbers was not completely accepted until the geometrical interpretation (see below) 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.
		Möbius must also be mentioned for his numerous memoirs on the geometric applications of complex numbers, and Dirichlet for the expansion of the theory to include primes, congruences, reciprocity, etc., as in the case of real numbers.
		The conjugate of the complex number z corresponds to the transformation which rotates through the same angle as z but in the opposite direction, and scales in the same manner as z; this can be described by the transpose of the matrix corresponding to z.
	xahlee.org /_p/wiki/Complex_Number.html (2592 words)

	Kummer's Objection (Site not responding. Last check: 2007-10-29)
		The numbers Lame' was talking about for a given prime p are called the "p-cyclotomic integers", and have as their basic "units" the pth roots of 1.
		Kummer soon published his theory of "ideal numbers" which enabled him to recover unique factorization (up to units) for cyclotomic integers, and this, in turn, led to a genuine proof of Fermat's Last Theorem for a large set of prime exponents, although not all.
		Specifically, Kummer's ideal numbers (the forerunners of Dedekind's "ideals") and his work on the higher reciprocity laws enabled him to prove Fermat's Last Theorem for all exponents p such that the class number h(p) of the cyclotomic integers is not divisible by p itself.
	www.mathpages.com /home/kmath447.htm (575 words)

	Middle Platonism [Internet Encyclopedia of Philosophy] (Site not responding. Last check: 2007-10-29)
		Eudorus brought together the apparently opposing views of Xenocrates and Crantor regarding the origin of numbers; the former stating that they are produced by the One and the Dyad, the latter that they are produced in the mind of the World-Soul as he contemplates the Forms.
		Eudorus taught that number was generated simultaneously with the World-Soul, who was responsible for translating the smallest multiplicity (the number three) into solid bodies (the number four).
		Albinus anticipated Plotinus in the prime role he allotted to contemplation in the ideal existence of the soul, and Origen in his doctrine of the intellectual generation of souls by the godhead.
	www.iep.utm.edu /m/midplato.htm (8719 words)

	Xenocrates biography .ms (Site not responding. Last check: 2007-10-29)
		The idea, inasmuch as it is a law of universal mind, which in particular minds produces aggregates of sensations called things, is a "determinant", and as such is styled "quantity" and perhaps "number"; but the ideal numbers are distinct from arithmetical numbers.
		Xenocrates, however, failing, as it would seem, to grasp the idealism which was the metaphysical foundation of Plato's theory of natural kinds, took for his principles arithmetical unity and plurality, and accordingly identified ideal numbers with arithmetical numbers.
		Soul is a self-moving number, derived from the two fundamental principles, unity and plurality, whence it obtains its powers of rest and motion.
	xenocrates.biography.ms (1124 words)

	Mu
		Numbers: their generation and use, v, vi, a21-1093b29 [The context is somewhat different here in M and N than in Alpha.
		numbers are either separate from things, or are not separate but in sensibles (albeit not like we considered at first [chapter ii, 1076a38-b11] as sensibles being made of numbers [the Pythagorean doctrine]), or neither or both, a37-b4 5.
		Number is from the One and the Infinite Dyad, the principles and elements of number.
	www.morec.com /classics/mu.htm (9599 words)

	SPEUSIPPUS - LoveToKnow Article on SPEUSIPPUS (Site not responding. Last check: 2007-10-29)
		When they, the immediate successors of Plato, rejected their masters ontology and proposed to themselves as ends mere classificatory sciences which with him had been means, they bartered their hope of philosophic certainty for the tentative and provisional results of scientific experience.
		Xenocrates indeed, identifying ideal, and mathematical numbers, sought to shelter himself under the authority of Plato; but, as the Xeno cratean numbers, though professedly ideal as well as mathematical, were in fact mathematical only, this return to the Platonic terminology was no more than an empty form.
		It would seem, then, that Academic scepticism began with those who had been reared by Plato himself, having its origin in their acceptance of the scientific element of his teaching apart from the ontology which had been its basis.
	www.1911encyclopedia.org /S/SP/SPEUSIPPUS.htm (1524 words)

	Rh122: Ideal-family-size and Sex-composition Preferences among Wives and Husbands in Nepal (Site not responding. Last check: 2007-10-29)
		Therefore, at issue are both the ideal number of children and the ideal number of sons (or daughters) wives desire in comparison with the ideal numbers that their husbands desire, and the extent to which women and men are willing to exceed their family-size preference to obtain their desired number of sons (or daughters).
		Figure 2 describes the percentage distributions of ideal family size, ideal number of sons, and ideal number of daughters among coresident wives and husbands.
		One way to observe the prevalence of desires for certain numbers of children of one sex is to compare the percentage distributions of ideal number of sons and ideal number of daughters.
	www.hsph.harvard.edu /grhf-asia/suchana/0317/stash.html (7185 words)

	Is Ideal cut over rated?
		So apparently "ideal" did not mean "ideal" or "as good as it can get" because now we have cuts that are claimed to be even better than "perfect".
		I looked at many many diamonds that were ideal per the numbers but this one stone that I purchased was the one that jumped at me. I swear it grabbed me by the shoulders and shook me up and down.
		Ideal is currently too broad of a term to be enough by itself.
	www.pricescope.com /diamonds/x29915.htm (3111 words)

	PART I, SECTION II, CHAPTER IV - SOME THINGS DEFINED (Site not responding. Last check: 2007-10-29)
		The numbers that we built up using the pattern observed in the formation of the cycles as given in the Brahmanical Tables happen to be the most important ones, some of which are identical to those in the table of the angles of the regular polyhedra.
		The number of degrees in this figure is 15,120, or the number of years related to 7 Signs of the Zodiac, whereas the Cube, which represents fully embodied Man, has 2,160 degrees, or the same as the number of years in the Messianic cycle, associated with 1 Sign of the Zodiac.
		We shall now construct a table showing the numbers of revolutions of the planets during the important cyclical periods that we found previously, to wit: 60, 2, 360, 2,160 and 25,920 years respectively, and we shall see the continual recurrence of our important numbers as well as some others which are related to them.
	www.wisdomtraditions.com /geometry/cm_p1_s2_c4.html (4309 words)

	Amazon.com: The Queen of Mathematics : An Historically Motivated Guide to Number Theory: Books (Site not responding. Last check: 2007-10-29)
		Number Theory, or the Higher Arithmetic, is the study of those properties of integers and rational numbers which go beyond the ordinary manipulations of everyday arithmetic.
		The Queen Of Mathematics: A Historically Motivated Guide To Number Theory by Jay R. Goldman (School of Mathematics, University of Minnesota) is a college-level mathematical text that scrutinizes number theory as it was developed through the 17th, 18th, and 19th centuries.
		The result is that the reader gets a very broad picture of number theory, the "big picture", seeing how number theory isn't some static piece of knowledge sitting somewhere in space, but a body of concepts, ideas, and techniques which naturally developed over the past 400 years.
	www.amazon.com /exec/obidos/tg/detail/-/1568810067?v=glance (1296 words)

	Ideal (ring theory) (Site not responding. Last check: 2007-10-29)
		In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers like "even number" or "multiple of 3".
		If R is commutative, then R is a field iff it has precisely two ideals, and R.
		The most extreme examples of factor rings are provided by modding out by the most extreme ideals, and R itself.
	www.worldhistory.com /wiki/I/Ideal-(ring-theory).htm (1329 words)

	Articles - Complex number (Site not responding. Last check: 2007-10-29)
		The system of complex numbers, in contrast to the real numbers, is algebraically closed, that is, all non-constant polynomials with complex coefficients have roots in the complex numbers.
		In mathematics, the term "complex" when used as an adjective means that the field of complex numbers is the underlying number field considered, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra.
		Given a complex number (a + ib) which is to be divided by another complex number (c + id) whose magnitude is non-zero, there are two ways to do this; in either case it is the same as multiplying the first by the multiplicative inverse of the second.
	www.regrey.com /articles/Complex_numbers (3010 words)

	Adams Market Research Alcohol Beverage Industry
		However, pinpointing an ideal number of new employees to hire is difficult because it's always a moving figure.
		But if you pre-determine the average number of shifts you want your staff working, you'll be able to begin hiring only those who fit the shift profile, and you'll know you're hiring the number of people you really need.
		Your ideal staffing level for bartenders, rounded to the nearest whole number, is five, but you could even get away with six.
	www.beveragenet.net /cheers/2001/0109/0901staf.asp (1294 words)

	Hair Tissue Mineral Analysis (TMA)
		Your TMA numbers from the lab are also provided at the end of this report.
		Since the ideal levels for each nutrient mineral are different from each other, I then convert your test result numbers to a numerical scale on which the number "1" represents the ideal level for each nutrient mineral.
		This is an "inversion" of the ideal Na/K ratio of 2.4/1.
	www.iahf.com /balance/tma.html (1375 words)

	Unique and nonunique factorization (Site not responding. Last check: 2007-10-29)
		It's easier to study algebraic numbers as part of a larger structure than on their own.
		Theorem 4 (Minkowski) The number of equivalence classes of ideals in a ring of integers is finite.
		This number is called the class number of the number field.
	mathcircle.berkeley.edu /BMC3/bamc/node2.html (647 words)

	IDEALISM-The Idealist hypothesis
		It is based on the fact that the cause of all the numbers is the idea of unity and the cause of all the relations is the relation to itself that is once again the idea of unity.
		However, it was difficult to establish a bridge between the numbers with their homogeneous units and the particles of matter that show different and specific properties.
		Once again, the hurdle was to define ideal numbers qualitatively different one from another in order to launch a bridge toward the material properties of the particles of matter.
	www.freeworldacademy.com /spiritualodyssey/global.htm (5796 words)

	Phones
		Ideal for users who often call the same customers, suppliers or accounts.
		Ideal Locations: Receptionist desk, Secretaries desk and any other area that deals with heavy call volume.
		Ideal Locations: Receptionist desk, Secretaries desk, or anywhere with heavy call volume.
	www.telcommservices.com /Phones.htm (388 words)

	Prime ideal (Site not responding. Last check: 2007-10-29)
		P is not equal to the whole ring R This generalizes the following property of prime numbers: if p is a prime number and if p divides a product ab of two integers, then p divides a or p divides b.
		A commutative ring is an integral domain if and only if is a prime ideal.
		A commutative ring is a field if and only if is its only prime ideal, or equivalently, if and only if is a maximal ideal.
	www.worldhistory.com /wiki/P/Prime-ideal.htm (835 words)

	Dedekind (Site not responding. Last check: 2007-10-29)
		Excited by Kummer's great discovery [ideal numbers in the cyclotomic integers], I had previously worked for many years on this subject.
		The one is that the investigation of a domain of algebraic integers is initially based on the consideration of a definite [algebraic] number and...
		the corresponding equation, which is treated as [the modulus of] a congruence; and that the definition of ideal numbers (or rather, of divisibility by ideal numbers) so obtained does not allow one to recognize the invariance these concepts in fact have from the outset.
	www.contrib.andrew.cmu.edu /~avigad/Teaching/Seminar/dedekind.html (266 words)

	Random Number Generator: Really Random Numbers Generates High Quality Easy-to-Use Random Numbers for Windows. (Site not responding. Last check: 2007-10-29)
		The result is superior random data that passes virtually all statistical tests of randomness and improves significantly over biased and predictable pseudo-random number generators.
		Really Random Numbers is ideal for use with Monte Carlo simulation, VAR models, audit sampling, lottery selection and other applications where quality random data or sequences are required.
		The Evaluation Edition of Really Random Numbers is licensed only for academic use by students and teachers, personal recreation or hobby use by individuals, or evaluation use in a commercial setting.
	www.sunny-beach.net /random_numbers (618 words)

	Math Forum Discussions
		theory of ideal prime numbers, and from it to the modern theory of numbers.
		led to the introduction of ideal numbers and to the discovery of the law of
		ideal prime numbers was the higher reciprocity law, which he considered to
	mathforum.org /kb/thread.jspa?threadID=1148098&tstart=0 (402 words)

	Ernst Eduard Kummer -- Encyclopædia Britannica (Site not responding. Last check: 2007-10-29)
		German mathematician whose introduction of ideal numbers, which are defined as a special subgroup of a ring, extended the fundamental theorem of arithmetic (unique factorization of every integer into a product of primes) to complex number fields.
		Some other fundamental concepts of modern algebra also had their origin in 19th-century work on number theory, particularly in connection with attempts to generalize the theorem of (unique) prime factorization beyond the natural numbers.
		Nonetheless, during the 19th century the algebraic theory of numbers grew from being a minority interest to its present central importance in pure...
	www.britannica.com /eb/article-9046414 (690 words)

	Corral Forums - Dave King, a little help (Site not responding. Last check: 2007-10-29)
		Ideal ET and MPH numbers for that combo are 10.9@121.
		A 4.30 gear is almost ideal for most blown street combo cars like yours.
		But if the numbers you gave are close, you'll be fine with the 4.30's.
	www.corral.net /forums/showthread.php?t=545829 (1014 words)

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