3.4 Complex Zeros and the Fundamental Theorem of
... the Intermediate Value Theorem, Theorem 3.1, the Fundamental Theorem of Algebra guarantees the existence of at least one zero, but gives us no algorithm to use in finding it. In fact, as we mentioned in Section 3.3, there are polynomials whose real zeros, though they exist, cannot be expressed using ...
... the Intermediate Value Theorem, Theorem 3.1, the Fundamental Theorem of Algebra guarantees the existence of at least one zero, but gives us no algorithm to use in finding it. In fact, as we mentioned in Section 3.3, there are polynomials whose real zeros, though they exist, cannot be expressed using ...
1 Proof by Contradiction - Stony Brook Mathematics
... Contradicting Fermat’s Last Theorem. Of course one can adapt Hippasus’ proof as well. Some more easy examples: Theorem 3. The complex numbers C (a field) cannot be given the structure of an ordered field. Proof. Recall that complex numbers are expressions a + bi, a, b ∈ R where i2 = −1. Assume for t ...
... Contradicting Fermat’s Last Theorem. Of course one can adapt Hippasus’ proof as well. Some more easy examples: Theorem 3. The complex numbers C (a field) cannot be given the structure of an ordered field. Proof. Recall that complex numbers are expressions a + bi, a, b ∈ R where i2 = −1. Assume for t ...
Countability - Computer Science
... • Need to handle negative rationals (as in showing Int countable). ...
... • Need to handle negative rationals (as in showing Int countable). ...
SESSION 1: PROOF 1. What is a “proof”
... show that no human can do everything; we assume that there exists a person who can do everything. In particular (s)he can make a sandwich that is too big to be finished by any human. This means that there is a sandwich that (s)he is unable to finish, and so (s)he is unable to eat a sandwich; this co ...
... show that no human can do everything; we assume that there exists a person who can do everything. In particular (s)he can make a sandwich that is too big to be finished by any human. This means that there is a sandwich that (s)he is unable to finish, and so (s)he is unable to eat a sandwich; this co ...
MODERATE DEVIATIONS FOR BOUNDED SUBSEQUENCES
... random variables (p > 2). A certain subseries therein is convergent for the same range of parameters as in the case of martingale difference or i.i.d. sequences. Copyright © 2006 George Stoica. This is an open access article distributed under the Creative Commons Attribution License, which permits un ...
... random variables (p > 2). A certain subseries therein is convergent for the same range of parameters as in the case of martingale difference or i.i.d. sequences. Copyright © 2006 George Stoica. This is an open access article distributed under the Creative Commons Attribution License, which permits un ...
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"".