SEQUENCES, CONTINUED Definition 3.13. A sequence {sn} of real
... Recall that S R is the set of extended real number subsequential limits of fsn g. Theorem 3.17. (a) s 2 S and s 2 S. (b) For each x > s there exists N 2 N such that sn < x for n N . For each x < s there exists N 2 N such that sn > x for n N . Moreover, s and s are the only numbers with properties (a ...
... Recall that S R is the set of extended real number subsequential limits of fsn g. Theorem 3.17. (a) s 2 S and s 2 S. (b) For each x > s there exists N 2 N such that sn < x for n N . For each x < s there exists N 2 N such that sn > x for n N . Moreover, s and s are the only numbers with properties (a ...
Licensed to: iChapters User
... The numbers that correspond to points to the right of O in Figure 2 are positive real numbers. Numbers that correspond to points to the left of O are negative real numbers. The real number 0 is neither positive nor negative. Note the difference between a negative real number and the negative of a re ...
... The numbers that correspond to points to the right of O in Figure 2 are positive real numbers. Numbers that correspond to points to the left of O are negative real numbers. The real number 0 is neither positive nor negative. Note the difference between a negative real number and the negative of a re ...
Sets
... If P(x) is the statement “x has won a race” where the domain of discourse is all runners, then the universal quantification of P(x) is x P ( x ) , i.e., every runner has won a race. The negation of this statement is “it is not the case that every runner has won a race. Therefore there exists at l ...
... If P(x) is the statement “x has won a race” where the domain of discourse is all runners, then the universal quantification of P(x) is x P ( x ) , i.e., every runner has won a race. The negation of this statement is “it is not the case that every runner has won a race. Therefore there exists at l ...
Floating-Point Numbers
... Normalization causes gap around zero in floatingpoint system If leading digits allowed to be zero, but only when exponent at its minimum value, then gap “filled in” by additional subnormal or denormalized floating-point numbers ...
... Normalization causes gap around zero in floatingpoint system If leading digits allowed to be zero, but only when exponent at its minimum value, then gap “filled in” by additional subnormal or denormalized floating-point numbers ...
Recursion Over Partitions
... the structure of divisibility mod a prime number p which is an extension to the known results of Ramanujan. In our paper we present a simpler use of fractals in the theory of partitions. In the process of creating mathematics we invent new symbols and enrich their meaning by using recursion. In this ...
... the structure of divisibility mod a prime number p which is an extension to the known results of Ramanujan. In our paper we present a simpler use of fractals in the theory of partitions. In the process of creating mathematics we invent new symbols and enrich their meaning by using recursion. In this ...
Notes on Combinatorics - School of Mathematical Sciences
... Subsets and binomial coefficients One of the features of combinatorics is that there are usually several different ways to prove something: typically, by a counting argument, or by analytic methods. There are lots of examples below. If two proofs are given, study them both. Combinatorics is about te ...
... Subsets and binomial coefficients One of the features of combinatorics is that there are usually several different ways to prove something: typically, by a counting argument, or by analytic methods. There are lots of examples below. If two proofs are given, study them both. Combinatorics is about te ...
CHAPTER I: The Origins of the Problem Section 1: Pierre Fermat
... methods more closely resemble our current method, and even though Descartes was reluctant at first to acknowledge Fermat’s work, eventually he did. Fermat also made significant progress on problems that led to the development of the calculus. Around this same time, he worked with Blaise Pascal in fo ...
... methods more closely resemble our current method, and even though Descartes was reluctant at first to acknowledge Fermat’s work, eventually he did. Fermat also made significant progress on problems that led to the development of the calculus. Around this same time, he worked with Blaise Pascal in fo ...
Non-standard analysis
The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta procedures rather than infinitesimals. Non-standard analysis instead reformulates the calculus using a logically rigorous notion of infinitesimal numbers.Non-standard analysis was originated in the early 1960s by the mathematician Abraham Robinson. He wrote:[...] the idea of infinitely small or infinitesimal quantities seems to appeal naturally to our intuition. At any rate, the use of infinitesimals was widespread during the formative stages of the Differential and Integral Calculus. As for the objection [...] that the distance between two distinct real numbers cannot be infinitely small, Gottfried Wilhelm Leibniz argued that the theory of infinitesimals implies the introduction of ideal numbers which might be infinitely small or infinitely large compared with the real numbers but which were to possess the same properties as the latterRobinson argued that this law of continuity of Leibniz's is a precursor of the transfer principle. Robinson continued:However, neither he nor his disciples and successors were able to give a rational development leading up to a system of this sort. As a result, the theory of infinitesimals gradually fell into disrepute and was replaced eventually by the classical theory of limits.Robinson continues:It is shown in this book that Leibniz's ideas can be fully vindicated and that they lead to a novel and fruitful approach to classical Analysis and to many other branches of mathematics. The key to our method is provided by the detailed analysis of the relation between mathematical languages and mathematical structures which lies at the bottom of contemporary model theory.In 1973, intuitionist Arend Heyting praised non-standard analysis as ""a standard model of important mathematical research"".