What are Euclid's axioms and Euclid's Postulates?
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Euclid (also referred to as Euclid of Alexandria) (Greek: Εὐκλείδης) (c. 325–c. 265 BC), a Greek mathematician, who lived in Alexandria, Hellenistic Egypt, almost certainly during the reign of Ptolemy I (323 BC–283 BC), is often considered to be the "father of geometry". His most popular work, Elements, is thought to be one of the most successful textbooks in the history of mathematics. Within it, the properties of geometrical objects are deduced from a small set of axioms, thereby founding the axiomatic method of mathematics.\n\nAlthough best-known for its geometric results, the Elements also includes various results in number theory, such as the connection between perfect numbers and Mersenne primes, the proof of the infinitude of prime numbers, Euclid's lemma on factorization (which lead to the fundamental theorem of arithmetic, on uniqueness of prime factorizations), and the Euclidean algorithm for finding the greatest common divisor of two numbers.\n\nEuclid also wrote works on perspective, conic sections, spherical geometry, and possibly quadric surfaces. Neither the year nor place of his birth have been established, nor the circumstances of his death.\nIn geometry, the parallel postulate, also called Euclid's fifth postulate since it is the fifth postulate in Euclid's Elements, is a distinctive axiom in what is now called Euclidean geometry. It states that:\n\nIf a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.\n\nEuclidean geometry is the study of geometry that satisfies all of Euclid's axioms, including the parallel postulate. A geometry where the parallel postulate cannot hold is known as a non-euclidean geometry. Geometry that is independent of Euclid's fifth postulate (i.e., only assumes the first four postulates) is known as absolute geometry (or, in some places, neutral geometry).
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Try \n1. http://en.wikipedia.org/wiki/Eccentricity_%28mathematics%29\n\n2. http://mathworld.wolfram.com/Eccentricity.html
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Theorytical if the line are formed on a flat gemetrical plane they will not intersect.Or if two tangent in a in the same circle are parallel they will never intersect.\nThe fith postulate of Euclid tackled this question.Later mathematicians, like gauss proved that they could intersect. But this proof was based on a different idea=paralell line having curvatures! Of course they would intersect. So the geometry which based on curvatures was started and was called Non Euclid Geometry.\nActually this could have been proven alot fast with some common sense! In reality it is imposible to construct a perfect circle or a perfect straight line(this is my own postulate).   This is because we are not able to have a perfect resolution of measurement.\nNot matter how you try to measure two parallel line  the results is ether convergence or divergence of the two lines if you make the line long enough!So in practice the 5th postulate of Euclid did have a practical application.
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Think what you are looking for is Euclidian Space:\n\nAround 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called "Euclidean geometry", which is the study of the relationships between angles and distances in space. Euclid first developed "plane geometry" which dealt with the geometry of two-dimensional objects on a flat surface. He then went on to develop "solid geometry" which analyzed the geometry of three-dimensional objects. All of the axioms of Euclid have been encoded into an abstract mathematical space known as a two- or three-dimensional Euclidean Space. These mathematical spaces may be extended to apply to any dimensionality, and such a space is called an n-dimensional Euclidean space or an n-space. This article is concerned with such mathematical spaces.\n\nIn order to develop these higher dimensional Euclidean spaces, the properties of the familiar Euclidean spaces must be very carefully expressed and then extended to an arbitrary dimension. Although the resulting mathematics is rather abstract, it nevertheless captures the essential nature of the Euclidean spaces we are all familiar with.\n\nAn essential property of a Euclidean space is its flatness. Other spaces exist that are not Euclidean. For example, the surface of a sphere is not a Euclidean space, nor is the four-dimensional spacetime described by the theory of relativity when gravity is present.
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Potted Leibnitz\nhttp://www.maths.tcd.ie/pub/HistMath/People/Leibniz/RouseBall/RB_Leibnitz.html\nPotted Descartes\nhttp://www.maths.tcd.ie/pub/HistMath/People/Descartes/RouseBall/RB_Descartes.html\n\nI can't find anything in common between the two except work on curves but everybody does that... maybe you'll have better luck.
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http://www.citlink.net/~math/geometry/ch1/1-4_notes.pdf#search='segment%20addition%20postulate'
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A transfinite number is generally associated with an ordinal or a cardinal.  From your description, it appears that you're asking a question about cardinality (i.e. where ordering is not important).\n\nNow to clarify your question, what you said about points on a line being the next smallest transfinite number is only true if you believe in the Continuum Hypothesis, which says that the powerset (set of subsets) of the Natural numbers, P(N) is the ``next smallest cardinal''.  However, this axiom is independent of standard ZFC set theory meaning that both it and its negation are consistent with ZFC so that you can't hope to prove it or its negation true in ZFC assuming ZFC is consistent.\n\nIn the case that you do believe in the continuum hypothesis (generally denoted CH), then since it can readily be shown that the powerset of the Natural numbers can be put in a 1-1 correspondence with the points on a line, that part of your assertion is correct.\n\nHowever, the infinity you claim to be the next smallest uses something even stronger than CH, mainly P(P(N)) is the next smallest cardinal after P(N).  Since P(N) is equinumerous (equipotent or equipollent) to the points on a line (that number usually being denoted c for continuum), we can then show that there is a one-one correspondence between P(P(N)) and the set of functions f: R -> R or even the set of functions f: R -> {0,1}!  Therefore the next part of your assertion becomes correct.\n\nAgain, we still can't prove anything about what the next smallest cardinal looks like, but if we assume in general, the Generalized Continuum Hypothesis (GCH), then we simply assume that for any cardinal, the powerset of that set is the ``next smallest infinity''.\n\nIn that case, the answer to your question is simply aleph_3 = cardinality of P(P(P(N))) = cardinality of set of subsets of functions of form f: R -> R.
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From the times of Euclides, it does two thousand two hundred years already, the mathematicians have tried to divide of certain called statements "axioms" and to deduce after them all class of useful conclusions.\nIn certain aspects it is almost like a game, with two rules. In the first place, the axioms must be less possible. Secondly, the axioms must be consistent. It must be impossible to deduce two conclusions that are contradicted mutually.\nAny book of baccalaureate geometry begins with a set of axioms: by two points nobodies a straight line can only be drawn up; the total is the sum of the parts, etc. During long time one assumed that the axioms of Euclides were the unique ones that could constitute a consistent geometry and that for that reason were "true".\nBut in century XIX it was demonstrated that modifying of certain way the axioms of Euclides different geometries could be constructed, "noneuclidianas". Each one of these geometries differed from the others, but all of them were consistent. From then it did not have sense already to ask which of them was "true". In place, of it it was necessary to ask which was useful.\nIn fact, the sets of axioms are many from which a consistent mathematical system could be constructed: all of them different and all of them consistent ones.\nIn no of those mathematical systems it would have to be possible to deduce, from its axioms, that something is simultaneously thus and not thus, because then the mathematics would not be consistent, it would be necessary to reject them. But what happens if we establish a statement and we verified that we cannot demonstrate that is or thus or not thus?\nLet us suppose that I say: "the statement that I am doing is false."\nIt is false? If it is false, then it is false that it is saying something false and it must be saying something true. But if I am saying something true, then it is certain that I am saying something false and would be truth that I am saying something false. It could be going of a side for another one indefinitely. It is impossible to demonstrate that what I have said is or thus or not thus.\nLet us suppose that we fit the axioms of the logic in order to eliminate the possibility of making statements of that type. We could find another way to make statements of the type "neither thus nor not thus"?\n In 1931 the Austrian mathematician Kurt Gödel presented/displayed a valid demonstration that for any set of axioms always it is possible to make statements that, from those axioms, it cannot demonstrate nor that they are thus nor that are not thus. In that sense, it is impossible to never elaborate a set of axioms from which a complete mathematical system can be deduced.\n It means this that never we will be able to find the "truth"? Nor to speak!\n First: the one that a mathematical system is not complete does not mean that what contains it is "false". The system can continue being very useful, whenever we do not try to use it beyond its limits.\n Second: the theorem of Gödel is only applied to deductive systems of the type that is used in mathematics. But the deduction is not the only way to discover the "truth". There are no axioms that allow to deduce the dimensions us of the Solar System. These last ones were obtained by means of observations and measures - another way fairy the "truth".
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It was believed that he saw the shadows of the pillars on the tiled temple floor, that looked like a right angled triangle with squares attached. \nFrom there he developed his theory that the square drawn on the hypotenuse of a right angled triangle, is equal in area to the sum of the squares drawn on the other two sides.\nPythagoras put forward a proof, but said proof contained errors to do with transient numbers, however it seemed the theory was sound.\nSome two centuries later the Great Euclid himself proved it, with his brilliant proof using congruent triangles, true poetry of geometry.
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I need to do a drawing to answer this. Anyway, here are some usefull sites:\n\nhttp://www.bymath.com/studyguide/geo/sec/geo14a.htm\n\nThe parallel postulate. If parallel lines are cut by a transversal, then corresponding angles are congruent.
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You are thinking of the book 'Principia Mathematica' by (Bertrand) Russell and Whitehead, written in 1913. The book was actually dedicated to putting mathematics on a logical foundation. It uses rather antiquated notation and really didn't accomplish what it intended to do, but it was a massive effort. The statement that 1+1=2 came in the middle of the second volume of the three volume set. There is a paperback version for about the first half of the first volume that is still available.
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Euclid! Come and help us please!!
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I'm sorry, but I don't think that you have actually asked a question here, or perhaps you have left off some very pertinent information. Unless you just wanted the above statement to be explained.\n\nIn this instance, the universe of discourse just means all the points contained in the set S = (p(x,y)), which is all the possible combinations of x = 1,2,3 and y = 1,2,3. For example, (1,3) and (2,2) and (3,1) would all be points contained in the universe of discourse for p(x,y).
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This again is a direct theorem from the field axioms: http://mathworld.wolfram.com/FieldAxioms.html . \nSuppose a*b = 0. If a is non-zero, multiply both sides by a^(-1) to get a^(-1)ab = a^(-1)0, and thus b = 0, since the product of 0 and any number is 0.\nIf b is non-zero, multiply both sides by b^(-1) to get the similar result a = 0. Thus, either b or a must be 0. \nThe final case is that both a and b are 0, which is trivial.\nFurther discussion of this theorem can be found in the source.\nThis theorem can be used to solve equations by getting 0 on one side of the equation and factoring the other side completely, then using the theorem to imply that at least one of the factors must be zero. For example, x^2 = -3x - 2 can be rearranged to be x^2 + 3x + 2 = 0. We factor the left side to read (x + 1)(x + 2) = 0. From the theorem we know that either x + 1 = 0 or x + 2 = 0 or both. Thus we have two solutions, x = -1 or x = -2. It is obvious that in this case, only one factor could be 0.
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The algebra of sets develops and describes the basic properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the relations of set equality (mathematics) and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.
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Rene Descartes (the Cartesian coordinate plane plus many other contributions)\n\nPierre de Fermat\n\nBlaise Pascal\n\nLook them up. That can get you started. Personally, I'm going to bed.
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weezy is on the right track, that glow1989 is on the right track.  For everyone who thinks that writing 1+1=2 is so obvious that it constitutes a self-contained proof of itself(!)... if you undertake a serious study of mathematical proof, you might find that 357 pages is a good length for a quick introduction to the ideas that are involved in considering questions in this area.  Of course, I am talking about actually UNDERSTANDING something - very different to simply passing an exam, regardless of the 'level' of maths you are enrolled in.\n\nYes, as you mentioned, this is a hard question.  'Proving' that 1+1=2 is not as simple as it sounds, because of the inherent question of what we mean by 'prove'.  You might have noticed that some proofs you have used depend on other proofs, which in turn depend on other proofs, ... which ultimately rest on a handful of axioms.  It is the nature of operations that are allowed by any axiomatic system, that determine the characteristics of mathematical problems that can be said to be SOLVABLE.  It is no small task to understand the axioms that underlie our systems of mathematics.  For example, maths students learn: that division by zero is undefined; that any object raised to a null power is equal to unity; some differences between rational and irrational numbers; what we mean when we consider objects to exist in 'sets'; and so on.  Those who consider themselves to be excellent students are usually able to list these rules in rapid-fire style.  But if we are asked to 'prove' why those rules are the way they are, the questions start to seem fuzzy, and you typically run out of convenient formulas to rely on.  You could easily write 357 pages just to explain what you actually MEAN by the numbers 1 or 2, without worrying about the definition of the addition operation!  As usual, people might do better to stop and consider what the question is asking before booting in a smug answer.\n\nLook into Hilbert's Problem (David Hilbert - German mathematician), which was a more workable and compact form of a proposal by Russell and Whitehead to formalise mathematics on a general basis, according to well-established axioms and procedures.\n\nOther notable mathematicians to work in this area were Alan Turing, Alonzo Church (look for Church's Lambda Calculus), Emil Post, and the brilliant Kurt Gödel.  In more recent decades, look for Gandy (1988) and of course many others.\n\nLol, molawby is spot-on.  I wish he'd posted b4 me... would've saved me a bit of typing.  If all you need to do is learn to sing the right song, to pass your exams and get the job that stipulates that you need to have completed such-and-such a course, then you don't have to worry about any of this.  Of course, that doesn't mean you have to dismiss it either.  But if you are chasing a deeper understanding, it probably does take the 20 yrs, and you won't necessarily get paid for it, heheheh.\n\nFor bedside reading, I suggest Roger Penrose etc...