Algebraic Numbers - Département de Mathématiques d`Orsay
... where Q is a nonzero polynomial, with integer coefficients, a root of which is α). Accordingly, two numbers can be algebraically independent only if both are transcendental. As pointed out in Section 1.3, it is an open problem to prove that no polynomial P (X, Y ) of the shape qX + qY − p (with nonz ...
... where Q is a nonzero polynomial, with integer coefficients, a root of which is α). Accordingly, two numbers can be algebraically independent only if both are transcendental. As pointed out in Section 1.3, it is an open problem to prove that no polynomial P (X, Y ) of the shape qX + qY − p (with nonz ...
Chapter 3 Finite and infinite sets
... matched up with the set N of natural numbers. In other words, A is countable if we can label its elements a1 , a2 , . . . , without ever coming to an end, so that every element of A is labelled an for some natural number n, and every natural number labels an element of A. This amounts to the same th ...
... matched up with the set N of natural numbers. In other words, A is countable if we can label its elements a1 , a2 , . . . , without ever coming to an end, so that every element of A is labelled an for some natural number n, and every natural number labels an element of A. This amounts to the same th ...
The Cantor Expansion of Real Numbers
... because not all primes would cancel with the prime factors of q. Thus, formula (2) would give 1 = A -pB whence, in view of A + (a/s) = pn (B + (b/s)) it would follow that pn= (a +s)/b <2s, in contradiction to (i). It is well known (see, e.g. [1]) that a sufficient condition for an infinite Cantor ex ...
... because not all primes would cancel with the prime factors of q. Thus, formula (2) would give 1 = A -pB whence, in view of A + (a/s) = pn (B + (b/s)) it would follow that pn= (a +s)/b <2s, in contradiction to (i). It is well known (see, e.g. [1]) that a sufficient condition for an infinite Cantor ex ...
LAWS OF LARGE NUMBERS FOR PRODUCT OF RANDOM
... of random variables. A statistical application to a problem in geometric probability is also provided. ...
... of random variables. A statistical application to a problem in geometric probability is also provided. ...
The Arithmetic-Geometric Mean
... Suppose you want to compute the average of a set of numbers. There are a number of ways of doing this; for example, if you arrange the numbers in order in a list, the value in the middle of the list is called the median average. You may have a set of numbers in which the same number occurs more than ...
... Suppose you want to compute the average of a set of numbers. There are a number of ways of doing this; for example, if you arrange the numbers in order in a list, the value in the middle of the list is called the median average. You may have a set of numbers in which the same number occurs more than ...
Lesson 3.9 – Intro to Sequences ppt
... find values of other terms. • Explicitly defined sequences provide the function that will generate each term. • An = 2n + 3 or Bn = 5(3)n • For arithmetic sequences, we define explicit formulas similarly to linear functions. • Why do you think we can use linear functions to help us define arithmetic ...
... find values of other terms. • Explicitly defined sequences provide the function that will generate each term. • An = 2n + 3 or Bn = 5(3)n • For arithmetic sequences, we define explicit formulas similarly to linear functions. • Why do you think we can use linear functions to help us define arithmetic ...
SRWColAlg6_0P_02
... If S is a set, the notation a S means that a is an element of S. b S means that b is not an element of S. • For example, if Z represents the set of integers, then –3 Z but π Z. ...
... If S is a set, the notation a S means that a is an element of S. b S means that b is not an element of S. • For example, if Z represents the set of integers, then –3 Z but π Z. ...
06. Naive Set Theory
... (ii) Construct a real between 0 and 1 that is not listed in the table: (a) Go down the “diagonal” of the table starting at the first digit in the decimal expansion of the first real. (b) Write “3” if the digit in the diagonal is a 4; write “4” if the digit in the diagonal is anything else. our examp ...
... (ii) Construct a real between 0 and 1 that is not listed in the table: (a) Go down the “diagonal” of the table starting at the first digit in the decimal expansion of the first real. (b) Write “3” if the digit in the diagonal is a 4; write “4” if the digit in the diagonal is anything else. our examp ...
Full text
... Since then, several authors proved general theorems on fractions that can be represented as series Involving Fibonacci numbers and general n-Bonacci numbers [1, 2, 3, 4 ] . In the present paper we will prove a theorem which includes as special cases all the earlier results. We introduce some notatio ...
... Since then, several authors proved general theorems on fractions that can be represented as series Involving Fibonacci numbers and general n-Bonacci numbers [1, 2, 3, 4 ] . In the present paper we will prove a theorem which includes as special cases all the earlier results. We introduce some notatio ...
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"".