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Topic: Proof of Leibniz formula

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	The Logic of Leibniz, Chapter 3
		Leibniz adds a personal remark to this copy in which he maintains that if the establishment of this language depends on the true philosophy, it does not depend on its completion or perfection; in other words, it rests on the first principles of the sciences but does not presuppose their completion.
		Thus, when in 1680 Leibniz planned to present a fragment of his philosophical language by applying it to geometry, he announced that he would employ the inflections, particles, and constructions of Latin and would be content to invent new names to express the generation of figures and hence their construction or definition.
		Leibniz was therefore led by the very progress of his project for a universal language, or rather by the development of the idea that was its principle, to surpass it.
	philosophy2.ucsd.edu /~rutherford/Leibniz/ch3.htm (9454 words)

	2. Prelude
		Leibniz's rationalism appears in his principle of sufficient reason, which in effect states that everything which is true can be proven, you just might have to cast around a bit to find the necessary premises, and indeed you might find the proof beyond you (but it won't be beyond God).
		Leibniz regards all propositions as having subject predicate form, in the final analysis, and in all true propositions the truth is owing to the containment of the subject in the predicate.
		Leibniz envisaged that any problem could be solved by translating the problem into the universal language and then calculating the solution using the calculus of reasoning, which he sought to automate as a calculating machine.
	www.rbjones.com /rbjpub/www/books/philrev/node3.htm (3076 words)

	Pi - the free encyclopedia (Site not responding. Last check: 2007-10-29)
		In physics, appearance of π in formulas is usually only a matter ofconvention and normalization.
		The Slovene mathematician Jurij Vega in 1789 calculated the first 140 decimal places for π of which the first 137 were correct and held the world recordfor 52 years until 1841, when William Rutherford calculated208 decimal places of which the first 152 were correct.
		In non-Euclidean geometry the sum of the angles of atriangle may be more or less than π radians, and the ratio of a circle's circumference to its diameter may also differ fromπ.
	www.free-web-encyclopedia.com /?t=Pi (2145 words)

	Leibniz formula for pi - Wikipedia, the free encyclopedia
		See Leibniz formula for other formulas known under the same name.
		Practically speaking, Leibniz' formula is very inefficient for either mechanical or computer-assisted π calculation, as it requires an enormous number of steps to be performed to achieve noticeable precision.
		The formula is a special case of the Boole summation formula for alternating series.
	en.wikipedia.org /wiki/Leibniz_formula_for_pi (375 words)

	Leibniz's Predicate-in-Notion Principle and Some of its Alleged Consequences
		Leibniz says that this seems to him to be self-evident when he considers what is meant by a proposition being true.
		Leibniz says that it is important to distinguish between the complete notion of a species, e.
		I suppose that Leibniz may have had some such considerations as these in his mind when he assumed that the phrase 'every predicate of an individual which refers to any moment in its history' denotes a genuine collection which is complete at every moment.
	www.ditext.com /broad/leib.html (5582 words)

	E.W.Dijkstra Archive: The notational conventions I adopted, and why (EWD 1300)
		There is a mathematical style in which proofs are presented as strings of unmotivated tricks that miraculously do the job, but we found greater intellectual satisfaction in showing how each next step in the argument, if not actually forced, is at least something sweetly reasonable to try.
		When he seems to derive one formula from another, the transformations he allows are those that seem to be true in the universe he has in mind.
		The formalist, however, prefers to manipulate his formulae, temporarily ignoring all interpretations they might admit, the rules for the permissible symbol manipulations being formulated in terms of those symbols: the formalist calculates with uninterpreted formulae.
	www.cs.utexas.edu /users/EWD/transcriptions/EWD13xx/EWD1300.html (3042 words)

	Fundamental Theorem of Algebra (Site not responding. Last check: 2007-10-29)
		D'Alembert in 1746 and Euler in 1749 attempted proofs of the FTA.
		Laplace attempted proofs, but also failed, as they were assuming the existence of roots.
		Gauss is credited with the first proof of the FTA in his doctoral thesis of 1799.
	www.und.edu /dept/math/history/fundalg.htm (451 words)

	Logic terms and concepts : Thomas Alspaugh : UCI
		The material biconditional is a logical formula, not to be confused with the meta-logical relation of logical equivalence.
		The material implication is a logical formula, not to be confused with the meta-logical relation of implication.
		A quantifier is used to make an assertion about the application of a formula to every object in the domain: either that the formula is true for every object in the domain (a universal quantification) or that it is true for some object in the domain (an existential quantification).
	www.ics.uci.edu /~alspaugh/logic/logicConcepts.html (4121 words)

	God's secret Formula - The Deciphering of the Riddle of the Universe and the Prime Number Code (Site not responding. Last check: 2007-10-29)
		His studies in chemistry, physics and biology led him, after decades of theoretical study, to a final proof that this world is based on a hitherto hidden, diving plan of construction based on simple numbers.
		The progressive drop in frequency of the prime numbers and their distribution throughout the number system is the most perplexing puzzle of numbers theory in mathematics, a puzzle that was investigated in all early civilisations.
		The progressive drop in frequency of the prime numbers and their distribution throughout the number system is the perplexing puzzle of numbers theory in mathematics, a puzzle that was investigated in all early developed civilisations.
	www.plichta.de /english/e_b_gods_secret_formula.php (800 words)

	Pi - Encyclopedia, History, Geography and Biography
		The formulae often given for calculating the digits of π have desirable mathematical properties, but are often hard to understand without a background in trigonometry and calculus.
		In 1789, the Slovene mathematician Jurij Vega improved John Machin's formula from 1706 and calculated the first 140 decimal places for π of which the first 126 were correct [1] and held the world record for 52 years until 1841, when William Rutherford calculated 208 decimal places of which the first 152 were correct.
		This formula is most easily verified using polar coordinates of complex numbers, starting with
	www.arikah.com /encyclopedia/Pi (4998 words)

	SingaporeMoms - Parenting Encyclopedia - Pi
		In physics, appearance of π in formulas is usually only a matter of convention and normalization.
		The current record (December 2002) stands at 1,241,100,000,000 digits, which were computed in September 2002 on a 64-node Hitachi supercomputer with 1 terabyte of main memory, which carries out 2 trillion operations per second, nearly twice as many as the computer used for the previous record (206 billion digits).
		In non-Euclidean geometry the sum of the angles of a triangle may be more or less than π radians, and the ratio of a circle's circumference to its diameter may also differ from π.
	www.singaporemoms.com /parenting/Pi (2349 words)

	Teaching Logic as a tool
		Warford pressed the point that proofs were previously the difficult and hated part of conventional discrete-math courses, while the development of proofs was readily accepted by students with the new approach.
		The emphasis in proofs is on substitution of equals for equals instead of modus ponens.
		In teaching students about proofs and their development, it helps to be able to demonstrate principles and strategies for developing proofs.
	www.cs.cornell.edu /Info/People/gries/logicthetool.html (2912 words)

	Clinton Goveas :: Wikipedia Reference (Site not responding. Last check: 2007-10-29)
		Most formulas given for calculating the digits of π have desirable mathematical properties, but may be difficult to understand without a background in trigonometry and calculus.
		It is said that he was so proud of this accomplishment that he had them inscribed on his tombstone.
		All of these formulae are a consequence of the formula for circumference.
	www.clintongoveas.com /wikipedia/?title=Pi (5108 words)

	Pi - TvWiki, the free encyclopedia (Site not responding. Last check: 2007-10-29)
		Formulae of this kind are known as Machin-like formulae.
		The current record (December 2002) by Yasumasa Kanada of Tokyo University stands at 1,241,100,000,000 digits, which were computed in September 2002 on a 64-node Hitachi supercomputer with 1 terabyte of main memory, which carries out 2 trillion operations per second, nearly twice as many as the computer used for the previous record (206 billion digits).
		Bailey and Crandall showed in 2000 that the existence of the above mentioned Bailey-Borwein-Plouffe formula and similar formulae imply that the normality in base 2 of π and various other constants can be reduced to a plausible conjecture of chaos theory.
	www.tvwiki.tv /wiki/Pi (3354 words)

	Pi (Site not responding. Last check: 2007-10-29)
		In physics, appearance of π in formulae is usually only a matter of convention and normalization.
		Vega improved John Machin's formula from 1706 and his method is still mentioned today.
		So, in particular, π is not affected by the shape of the universe; it is not a physical constant but a mathematical constant defined independently of any physical measurements.
	www.apawn.com /search.php?title=Pi (2730 words)

	Peirce's Rules of Inference
		This formula, which has four implications, may be read "If p implies r and q implies s, then p and q implies r and s." Each of the four implications maps to a pair of a shaded and an unshaded area in Figure 7.
		The remainder of the proof would iterate subgraphs from the shaded area into the inner unshaded area by Rule 2i and apply more rules to transform the result into the desired conclusion.
		One of those axioms was redundant, but the proof of its redundancy was not discovered by the authors or by any of their readers for another 16 years.
	www.jfsowa.com /peirce/infrules.htm (3390 words)

	Welcome to the prime numbers formula
		This formula is one of the on-to generating functions for the prime numbers that for every natural "m" it generates all the prime numbers (3,5,7,2,11,13,2,17,19,...)
		I have discovered the formula of the prime numbers after 20 years of research and injury.
		We look for proofs that are short, concise, clear, and if possible that combine previous disparate concepts or teach you something new.
	www.primenumbersformula.com (1395 words)

	Frege's Logic, Theorem, and Foundations for Arithmetic (Stanford Encyclopedia of Philosophy)
		The reason is that the rules govern not only his graphical notation for molecular and quantified formulas, but also his special purpose symbols, such as certain lowercase letters used as placeholders, certain Gothic letters and letters used as bound variables, and various other signs of his system we have not yet mentioned.
		The proofs of these facts, in each case, require the identification of a relation that is a witness to the relevant equinumerosity claim.
		We leave the proof of the Equinumerosity Lemma ‘Converse’ and the proof that Predecessor is a function as exercises for the reader.
	plato.stanford.edu /entries/frege-logic (15082 words)

	Calculating Pi
		The graph in the middle is the approximation from Vieta's formula, which calculates the area of a regular polygon of
		To derive Vieta's formula we need to combine several results from calculus and trigonometry.
		In comparison, the number of correct digits in iteration formulas grows quadratically with the iteration number.
	documents.wolfram.com /v5/Demos/Notebooks/CalculatingPi.html (613 words)

	Mathematical Background
		A proof is a sequence of formulas, each of which is derived from axioms, definitions, or previous formulas in the sequence by applying some rule of inference.
		If the first line of a proof is a formula p, called a hypothesis, the conclusion of the proof must be true whenever the hypothesis p and the starting axioms are true.
		If a formula C was derived from formulas A and B by combining them with Boolean operators, then all occurrences of variables that are free in A and B are also free in C.
	www.jfsowa.com /logic/math.htm (14436 words)

	PI - meaning of word
		The current record (December 2002) stands at 1,241,100,000,000 digits, which were computed in September 2002 on a 64-node Hitachi (company) supercomputer with 1 terabyte of main memory, which carries out 2 trillion operations per second, nearly twice as many as the computer used for the previous record (206 billion digits).
		When different derivations are made from the theory sometimes a factor of pi will appear in a formula, and sometimes not, for detailed mathematical reasons (eg the inversion of a fourier transform).
		The π in (almost?) all physics formulae is not 3.14159...
	wordsonline.org /Pi (10142 words)

	An article (Site not responding. Last check: 2007-10-29)
		A New Formula for Pi One of the charms of mathematics is that it is possible to make elementary discoveries about objects that have been studied for millenia.
		The fourth, which we mention to show that base-10 formulas do exist, is also a Maclaurin expansion; this one was used by BBP to compute the 5,000,000,065th decimal digit of log 9/10, a world record for decimal digits.
		The proof is identical to the proof in section 1.
	www-2.cs.cmu.edu /~adamchik/articles/pi/pi.htm (4403 words)

	Pi - The real meaning from Timesharetalk wikipedia (Site not responding. Last check: 2007-10-29)
		Most formulas given for calculating the digits of p have desirable mathematical properties, but may be difficult to understand without a background in trigonometry and calculus.
		In 1789, the Slovene mathematician Jurij Vega improved John Machin's formula from 1706 and calculated the first 140 decimal places for p of which the first 126 were correct [1] and held the world record for 52 years until 1841, when William Rutherford calculated 208 decimal places of which the first 152 were correct.
		Bailey and Crandall showed in 2000 that the existence of the above mentioned Bailey-Borwein-Plouffe formula and similar formulae imply that the normality in base 2 of p and various other constants can be reduced to a plausible conjecture of chaos theory.
	www.timesharetalk.co.uk /wiki.asp?k=Pi (5151 words)

	Andrews. To Truth through Proof. (Site not responding. Last check: 2007-10-29)
		Soundness (that is, every theorem is a tautology) is proven by induction on the length of proofs (that is, by checking that the axioms are tautologies and that modus ponens preserves the property of being a tautology.
		The proof in one direction reduces to the case of a finite set of clauses S by compactness.
		Representability (of naturals, functions on naturals, and relations on naturals) is defined (as the existence of a closed formula such that Q0-inf proves it satisfies the appropriate equations on numerals to represent the function or relation).
	www.andrew.cmu.edu /user/cebrown/notes/tttp.html#5 (6301 words)

	Common Errors in College Math
		In a direct proof, you start with the hypotheses, and you generate consequences -- i.e., you start making sentences, where each sentence is either a hypothesis of the theorem, an axiom (if you're using an abstract theory), or a result deduced from some earlier theorem using sentences you've already generated in the proof.
		This kind of proof is harder to read, but it is actually easier to discover and to write: we have more hypotheses (not only A, but also ~B), so it is easier to generate consequences.
		A formula or notation may work properly in one context, but some students try to apply it in a wider context, where it may not work properly at all.
	www.math.vanderbilt.edu /~schectex/commerrs (13343 words)

	[No title]
		Also states the % `fundamental transformation formula for divided differences', something % that reduces to the formula, % \sum_{j=1}^n \psi_{1,j-1}\dvd{x_1,\ldots,x_j} + % \sum_{j>n} \psi+_{j-n+1,j-1}(x_j-x_{j-n})\dvd{x_{j-n},\ldots,x_j} % with \psi^+_{r,s}:= (\cdot-x_r)_+\cdots(\cdot-x_s)_+, for the interpolant % at the sequence x_1< x_2< \cdots that agrees, on [x_j\fromto x_{j+1}], with % the polynomial interpolant at x_{j-n+1},\ldots,x_j.
		Evaluation of this % formula at x_i provides a unique description of the linear functional [x_i] % in terms of the linear functionals \dvd{x_{max(1,j-n+1)},\ldots,x_{j-1}}, % j=1,2,\ldots, hence Popoviciu's name `transformation formula' for the % latter.
		Also proves the `mean-value formula for divided differences': % \dvd{t_0,\ldots,t_n} = \sum_j a_j(t,s)\dvd{s_j,\ldots,s_{j+n}} % for any monotone refinement s of monotone t, with a_j(t,s)\ge0 and summing % to 1.
	www.cs.wisc.edu /~deboor/bib/P (405 words)

	Pi
		While the original Greek letter for pi was phonetically equivalent to the English letter p, it has now evolved to be pronounced like the word pie in most circles.
		This formula is most easily verified using polar coordinates of complex numbers, starting with :
		The most pressing open question about π is whether it is a normal number, i.e.
	www.keywordmage.net /pi/pi.html (2395 words)

	Analysis, Convergence, Series, Complex Analysis - Numericana
		Since the magnitudes of such terms tend to zero, partial sums tend toward S. S is therefore the sum of the rebuilt series.
		Jacob Bernoulli, who was first in a long list of notorious mathematicians (including Leibniz) who failed to discover the above solution.
		Use Euler's formulas to compute the Fourier coefficients of f(x).
	home.att.net /~numericana/answer/analysis.htm (4095 words)

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