Geometry Chapter 2: Geometric Reasoning
... Ex: #1 Is each conclusion a result of inductive or deductive reasoning? Explain your answer. A. There is a myth that you can balance an egg on B. There is a myth that the Great Wall of China is its end only on the spring equinox. A person was the only man-made object visible from the Moon. able to b ...
... Ex: #1 Is each conclusion a result of inductive or deductive reasoning? Explain your answer. A. There is a myth that you can balance an egg on B. There is a myth that the Great Wall of China is its end only on the spring equinox. A person was the only man-made object visible from the Moon. able to b ...
Inductive Reasoning
... observed pattern will continue. This may or may not be true. Ex: x = x • x This is true only for x = 0 and x = 1 ...
... observed pattern will continue. This may or may not be true. Ex: x = x • x This is true only for x = 0 and x = 1 ...
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... numbers x and y such that x2 + y 2 + z 2 = 3xyz, but there is only set of values of x and y satisfying the inequality z > y ≥ x. The conjecture is easy enough to check for small values. For example, for z = 5, we could even test x and y we know not to be Markov numbers with minimum increase in compu ...
... numbers x and y such that x2 + y 2 + z 2 = 3xyz, but there is only set of values of x and y satisfying the inequality z > y ≥ x. The conjecture is easy enough to check for small values. For example, for z = 5, we could even test x and y we know not to be Markov numbers with minimum increase in compu ...
Poincaré conjecture
In mathematics, the Poincaré conjecture (/pwɛn.kɑːˈreɪ/ pwen-kar-AY; French: [pwɛ̃kaʁe]) is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space. The conjecture states: Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere. An equivalent form of the conjecture involves a coarser form of equivalence than homeomorphism called homotopy equivalence: if a 3-manifold is homotopy equivalent to the 3-sphere, then it is necessarily homeomorphic to it.Originally conjectured by Henri Poincaré, the theorem concerns a space that locally looks like ordinary three-dimensional space but is connected, finite in size, and lacks any boundary (a closed 3-manifold). The Poincaré conjecture claims that if such a space has the additional property that each loop in the space can be continuously tightened to a point, then it is necessarily a three-dimensional sphere. The analogous conjectures for all higher dimensions had already been proven.After nearly a century of effort by mathematicians, Grigori Perelman presented a proof of the conjecture in three papers made available in 2002 and 2003 on arXiv. The proof built upon the program of Richard Hamilton to use the Ricci flow to attempt to solve the problem. Hamilton later introduced a modification of the standard Ricci flow, called Ricci flow with surgery to systematically excise singular regions as they develop, in a controlled way, but was unable to prove this method ""converged"" in three dimensions. Perelman completed this portion of the proof. Several teams of mathematicians verified that Perelman's proof was correct.The Poincaré conjecture, before being proven, was one of the most important open questions in topology. In 2000, it was named one of the seven Millennium Prize Problems, for which the Clay Mathematics Institute offered a $1,000,000 prize for the first correct solution. Perelman's work survived review and was confirmed in 2006, leading to his being offered a Fields Medal, which he declined. Perelman was awarded the Millennium Prize on March 18, 2010. On July 1, 2010, he turned down the prize saying that he believed his contribution in proving the Poincaré conjecture was no greater than Hamilton's (who first suggested using the Ricci flow for the solution). As of 2015, the Poincaré conjecture is the only solved Millennium problem.On December 22, 2006, the journal Science honored Perelman's proof of the Poincaré conjecture as the scientific ""Breakthrough of the Year"", the first time this honor was bestowed in the area of mathematics.