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Topic: Infinitesimal number

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	Fatal Dog Attacks: The Facts Behind the Statistics
		Despite enormous public and political interest in fatal dog attacks, there is no agency or organization that does investigative work (with the exception of this study) into each of the individual cases of fatal dog attacks and records the number and circumstances of fatal dog attacks on a continuous, yearly basis.
		This study is conducted in an attempt to understand the human and canine behaviors that contribute to a fatal dog attack.
		Approximating 20 deaths per year in a dog population of 53 million yields an infinitesimal percent of the dog population (.0000004%) involved in a human fatality.
	www.fataldogattacks.com (1111 words)

	Infinitesimal - Wikipedia, the free encyclopedia
		In mathematics, an infinitesimal, or infinitely small number, is a number that is greater in absolute value than zero yet smaller than any positive real number.
		The first mathematician to make use of infinitesimals was Archimedes, although he did not believe in their existence.
		Thereafter, infinitesimals are numbers which represent limits or relationships between limits, and using them as symbols continues to be convenient for simplifying notation and calculation.
	en.wikipedia.org /wiki/Infinitesimal_number (443 words)

	[No title] (Site not responding. Last check: 2007-11-01)
		This can be shown using the least upper bound axiom of the real numbers: consider whether the least upper bound c of the set of all infinitesimals is or is not an infinitesimal.
		The use of infinitesimals was attacked as incorrect by Bishop Berkeley in his work The analyst: or a discourse addressed to an infidel mathematician.
		Nevertheless, the use of infinitesimals continues to be convenient for simplifying notation and calculation.
	www.informationgenius.com /encyclopedia/i/in/infinitesimal.html (366 words)

	Infinitesimal
		In mathematics, an infinitesimal is a number greater in absolute value than zero yet smaller than any positive real number.
		An infinitesimal is only a notional quantity - there exists no infinitesimal real number.
		Infinitesimals are legitimate quantities in the non-standard analysis of Abraham Robinson.
	www.brainyencyclopedia.com /encyclopedia/i/in/infinitesimal.html (407 words)

	Hyperreal number Information - TextSheet.com (Site not responding. Last check: 2007-11-01)
		The hyperreal numbers or nonstandard reals (usually denoted as R) are an extension of the real numbers* R that adds infinitely large as well as infinitesimal numbers to R.
		The study of these numbers, their functions and properties is called nonstandard analysis which some find more intuitive than standard real analysis.
		However, in the 1960s Abraham Robinson showed how infinitely large and infinitesimal numbers can be rigorously defined and used to develop the field of nonstandard analysis.
	www.digicoaster.sferahost.com /encyclopedia/h/hy/hyperreal_number.html (882 words)

	Hyperreal Number [Definition] (Site not responding. Last check: 2007-11-01)
		The essence of the axiomatic approach is to assert (1) the existence of at least one infinitesimal number, and (2) the validity of the transfer principle.
		The most familiar numbers are the whole numbers {0, 1, 2,...} denoted by W and the natural numbers {1, 2, 3,...} used for counting and denoted by N. If the negative whole numbers are included, one obtains the integers Z. Ratios of integers are called rational numbers or frac...
		The complex numbers contain a number, the imaginary unit, with, i.e., is a square root of.
	www.wikimirror.com /Hyperreal_number (4432 words)

	[No title] (Site not responding. Last check: 2007-11-01)
		Infinitesimally speaking, a circle is actually a polygon with infinitely many infinitesimal sides.
		The infinitesimals and their other nonstandard companions are now coming into their own in many areas of mathematics and physics.
		By making the molecules infinitesimally small, he could use the mathematics of NSA to do the proof once and for all, and he succeeded in proving that the equation is well behaved after all.
	www.chez.com /ouannani01/maple.htm (2189 words)

	Surreal number Information - TextSheet.com (Site not responding. Last check: 2007-11-01)
		The surreal numbers are a class of numbers which includes all of the real numbers, and additional "infinite" numbers which are larger or smaller than any real number.
		They also include "infinitesimal" numbers that are closer to zero than any real number, and each real number is surrounded by surreals that are closer to it than any real number.
		In this, the surreals are similar to the hyperreal numbers, but their construction is very different and the class of surreals is larger and contains the hyperreals as a subset.
	www.rsk.soldat.sferahost.com /encyclopedia/s/su/surreal_number.html (3092 words)

	Hyperreal number - free-definition (Site not responding. Last check: 2007-11-01)
		The application of hyperreal numbers and in particular the transfer principle to problems of analysis is called nonstandard analysis; some find it more intuitive than standard real analysis.
		Nonetheless these concepts were from the beginning seen as suspect, notably by Berkeley, and when in the 1800s calculus was put on a firm footing through the development of the epsilon-delta definition of a limit by Cauchy, Weierstrass and others, they were largely abandoned.
		Hence we have a homomorphic mapping, st(x), from F to R whose kernel conisists of the infinitesimals and which sends every element x of F to a unique real number whose difference from x is in S; which is to say, is infinitesimal.
	www.free-definition.com /Hyperreal-number.html (1645 words)

	Kids.net.au - Encyclopedia Surreal number -
		The surreal numbers are a class of numbers which includes all of the real numbers, and additional "infinite" numbers which are larger than any real number.
		These operations can be shown to be well-defined for surreal numbers, i.e., if they are applied to well-defined surreal numbers then the result will again be a well-defined surreal number, i.e., the left set of the result will be "smaller" than then the right set.
		Lots of numbers can be generated this way; in fact so many that no set can hold them all.
	www.kids.net.au /encyclopedia-wiki/su/Surreal_number (2876 words)

	The Infinitesimal
		The 'infinitesimal' as it is known, is practically the opposite of infinity.
		Archimedes was one of the first to think of this 'number'.
		In this case, the infinitesimal couldn't possibly exist, as you could always divide it.
	stewy6.tripod.com /infinitesimal.htm (241 words)

	The Genesis Flood
		The infinitesimal light-clocks are used as a means of incorporating within a theory the most basic propagation properties of electromagnetic radiation without relying upon any of the controversial scenarios relative to the composition of such radiation.
		In particular, when so infinitesimalized, it is the linear light path as modeled by the infinitesimal light-clock arm that is considered as being measured by a positive infinitesimal number.
		For each specific case, where the measures are made by infinitesimal light-clocks, it is an analysis of expressions such as (8) and (9) that leads to predictions for the alterations in physical behavior.
	www.serve.com /herrmann/pp7.htm (9345 words)

	Untitled Document
		They have a number of seemingly advantageous properties -- they are self-erecting, in that tensioning the final cable transforms them from a compact group of members into a large three-dimensional volume, and they are predominately tension systems, in that they can function as a VGT manipulator while actuating members only in tension.
		However, they have a number of properties which make them seemingly inappropriate for use -- they are not conventionally rigid, they exist only under specific conditions of geometry with a corresponding prestress state, and the governing equations that do exist include singular (non-invertible) matrices.
		The number of prestress states and the number of (infinitesimal) mechanisms is found from the rank of [A]; in a tensegrity geometry that matrix is seen to be rank-deficient.
	www.ri.cmu.edu /~isarc/abstract_htmls/Oppenheim-tensegrityPrisms.html (1727 words)

	Articles - Non-standard analysis (Site not responding. Last check: 2007-11-01)
		Much of the earliest development of the infinitesimal calculus by Newton and Leibniz was formulated using expressions such as infinitesimal number and vanishing quantity.
		As noted in the article on hyperreal numbers, these formulations were widely criticized by Bishop Berkeley and others.
		One way of thinking of the standard part of a hyperreal, is in terms of Dedekind cuts; any bounded hyperreal s defines a cut by considering the pair of sets (L,U) where L is the set of standard rationals a less than s and U is the set of standard rationals b greater than s.
	oldion.com /articles/Non-standard_analysis (2035 words)

	Surreal number (Site not responding. Last check: 2007-11-01)
		The surreal numbers are a class of numbers in mathematics that includes all of the real numbers and additional "infinite" numbers that are larger or smaller than any real number.
		This book is a mathematical novelette, and is notable as one of the rare cases where a new mathematical idea was first presented in a work of fiction.
		We will first start with the numbers that can be created by applying the rule a finite number of times.
	www.sciencedaily.com /encyclopedia/surreal_number (3106 words)

	Math Forum: Ask Dr. Math: A Mathematical Essay
		In his work, Euler freely used the notion of infinitesimals and tacitly assumed that almost all of the usual laws of arithmetic carried over without modification, plus a few extra properties that somehow seemed natural to him (and were later formalized and proven correct in an accepted manner).
		And thus, circa 1872, infinitesimals were purged from the accepted mathematical literature and epsilon-delta limits were used to define integrals, derivatives, convergent sequences, and all similar notions.
		When comparing two hyperreal numbers, a and b, we can form three disjoint sets: the agreement set (set of indices of corresponding equal terms of the sequence) the "a-greater set" (set of indices of corresponding terms greater in a than b), and the "b-greater set" (whatever is left over).
	mathforum.org /dr.math/faq/analysis_hyperreals.html (9021 words)

	A - Glossary of chemical terms
		Absorptance (α) is ratio of the radiant or luminous flux in a given spectral interval absorbed in a medium to that of the incident radiation.
		Absorption coefficient (a) is the relative decrease in the intensity of a collimated beam of electromagnetic radiation, as a result of absorption by a medium, during traversal of an infinitesimal layer of the medium, divided by the length traversed.
		Atomic number (Z) is a characteristic property of an element, equal to the number of protons in the nucleus.
	www.ktf-split.hr /periodni/en/abc/a.html (2092 words)

	sciforums.com - Points Have Size
		The problem with the ideas of infinity and infinitesimal are that they are not practically realisable.
		If there are an infinite number of points in a line segement, then this will go on for eternity and make any movement immpossible.
		As for the infinite number of infinitesimal steps, I do believe that might be the way integration works...the number of steps reach infinity and the size of the steps (remember Riemann sums?) reach zero.
	www.sciforums.com /showthread.php?t=10688 (1176 words)

	[No title]
		From that you subtract the number of independent elements which is half those not including the diagonal.
		In particle physics, that is the number of bosons that mediate the force.
		The number of Casimir operators is equal to the number of diagonal matrices.
	www.geocities.com /jefferywinkler/standardmodel.html (4258 words)

	What's a number?
		However, there are numbers which are neither rational or irrational (for example, infinitesimal numbers are neither rational nor irrational).
		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 (3757 words)

	Def. of Integral (Site not responding. Last check: 2007-11-01)
		An Integral is, loosely speaking, the sum of an infinite number of infinitesimal numbers.
		The integral from a to b of f(x) dx is the limit as n goes to infinity of the sum of n numbers, where each number is (1/n) times the value of the function in the i
		This relationship between the limit of a sum and the corresponding integral was described by Georg Friedrich Bernhard Riemann (1826-1866).
	mcraeclan.com /mathhelp/CalculusTheoremDefinitionOfIntegral.htm (650 words)

	The Number of Species of Birds, by Ernst Mayr
		In the subsequent 150 years a number of additional counts were published, each one to be quickly superseded by a newer one.
		To give an estimate of the number of genera is very difficult in view of the current revolution of the generic concept.
		A comparison of the number of species of birds with that of other groups of animals is appended.
	www.wku.edu /~smithch/biogeog/MAYR1946.htm (1136 words)

	On Put Bets and Precision Shooting - A Roundtable with Heavy and MP
		That reduces the house-edge to an infinitesimal number.
		With a $1 Put bet on the number 9, backed up with $100 in odds, the payout is $151.
		You simply make a mental note of the player and any numbers he is repeating as a result of his setting and throwing.
	www.dicesetter.com /hh/heavy13putbets.htm (1124 words)

	Citizens of Another Universe \| by: Dr. Noah H. Kersey, Ph.D.
		Back in the late 1960’s a considerable number of orphans, upon reaching their late teens, were asked to drop out of school and join the military.
		Maybe an unknown number of orphans were able to struggle long enough to finish high school.
		Perhaps an infinitesimal number even graduated from college and went on to successful jobs or careers.
	family-marriage-counseling.com /mentalhealth/citizens-universe.htm (765 words)

	Surreal number - free-definition (Site not responding. Last check: 2007-11-01)
		The surreal numbers are an example of what is sometimes called a Field, meaning a proper class on which there is defined an addition, multiplication and multiplicative inverse which satisfy all of the axioms of a field except for the fact that the elements form a proper class but not a set.
		With that caveat they are an ordered field containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number, and therefore the surreals are algebraically similiar to superreal numbers and hyperreal numbers.
		They were introduced in Donald Knuth's 1974 book Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness.
	www.free-definition.com /Surreal-number.html (3248 words)

	Chapter 11
		an infinitely large non-standard number, can be said to exceed a simple by what one may call "infinity times infinity", meaning that as every finite number exceeds any infinitesimal number, so also does every infinitely large number exceed any finite number.
		In such a system, the infinitely large and infinitely small numbers are inverses of one another, in the manner that the number 2 is the inverse of the number 1/2 (the product of these is the number 1).
		There are a number of verses in the Psalms which suggest an intimate connection between God and light.
	www.gfisher.org /ch_1_celestial.htm (15404 words)

	NIGHT OWL MK. II -- Philosophy/Abortion (Reply #15)
		According to your claim, we should expect an infinitesimal number of accidental pregnancies.
		Of course, if you increase the number of sexually-active women to its real value, if you increase their rate of sexual activity to more likely figures, and reduce the effectiveness of Depo-provera to its real-world values, you will greatly increase the number of accidental pregnancies.
		In the real world, parents only want a limited number of children and wish for all of them to be ideal in their eyes.
	www.bakkster.com /r_abrt15.htm (3712 words)

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