PPT-The Fundamental Theorem of Arithmetic

By Jess Barak Lindsay Mullen Ashley Reynolds and Abby Yinger The concept of unique factorization stretches right back to Greek arithmetic and yet it plays an important role in modern commutative ring theory Basically unique factorization consists of two properties existence and uniqueness Existence means that an element is representable as a finite product of . But actual computation with real numbers is not very practical because it involves limits and in64257 n i t i e s Instead M A T L A B and most other technical computing environments use o a t i n g p o i n t arithmetic which involves a 64257nite s But actual computation with real numbers is not very practical because it involves limits and in n i t i e s Instead M A T L A B and most other technical computing environments use o a t i n g p o i n t arithmetic which involves a 64257nite set of 3 Theorem 1 Theorem Let be a discrete valuation ring with 64257eld of fractions and let be a smooth group scheme of 64257nite type over Let sh be a strict Henselisation of and let sh be its 64257eld of fractions Then admits a N57524eron model over Gödel and . His. . T. heorems. Naassih . Gopee. What I’ll take . about. Small events during his lifetime. Completeness theorem. First incompleteness theorem. Second incompleteness theore. m. His life…. Rules of Matrix Arithmetic. Properties of Matrix Operations. For real numbers a and b ,we always have . ab. =. ba. , which is called the . commutative law for multiplication. . For matrices, however, AB and BA need not be equal.. Fundamental theorem of calculus. Deriving the Theorem. Let. Apply the definition of the derivative:. Rule for Integrals!. Deriving the Theorem. This is average value of . f. from. x. to . x. + . h. IIIS, Tsinghua University. Logic Conference, Tsinghua Oct. 2013. From . Classical Proof Theory . to. . P . vs.. NP. Complexity Theory. P = PTIME. : . Efficiently computable problems;. . . Algorithms of polynomial run-time. Vesta Coufal. Gonzaga University Philosophy Club. March 16, 2011. Fractal: Mandelbrot Set. http://. math.youngzones.org. /Fractal webpages/. Julia_set.html. Escher: Poincare Disk Model of Hyperbolic Geometry. Robert “Dr. Bob” Gardner. Based on Hungerford’s . Appendix to Section V.3 . in . Algebra. , Springer-. Verlag. (1974). The field of complex numbers, . , is algebraically closed..  . Lemma . V.3.17. Robert “Dr. Bob” Gardner. Based on Hungerford’s . Appendix to Section V.3 . in . Algebra. , Springer-. Verlag. (1974). The field of complex numbers, . , is algebraically closed..  . Lemma . V.3.17. Complex Numbers. Standard form of a complex number is: . a bi.. Every complex polynomial function of degree 1 or larger (no negative integers as exponents) has at least one complex zero.. a . and. b . CMPS 2433 Chapter 8 Counting Techniques Midwestern State University Review From P revious Chapters 2.6 Binary Search For an ordered list of 2 n items, @most n+1 comparisons are needed to find an item CMPS 2433 Chapter 8 Counting Techniques Midwestern State University Dr. Ranette Halverson Review From P revious Chapters 2.6 Binary Search For an ordered list of 2 n items, @most n+1 comparisons are needed to find an item Lecture 11. https://abstrusegoose.com/353. Announcements. Lots of folks sounded concerned about English proofs in sections.. THAT’S NORMAL. English proofs aren’t easy the first few times (or the next few times…sometimes not even after a decade…) .