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Full text

Section 3.2 Complex Numbers
Section 3.2 Complex Numbers

Standard (Index) Form
Standard (Index) Form

here
here

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Complex numbers
Complex numbers

Inequalities (65.1 KB)
Inequalities (65.1 KB)

... Inequalities (i.e., expressions involving the signs > (”is greater than”) or < (”is less than”) are readily visualised geometrically on the real number line (e.g., 4 > 3, 1 12 < 2). The aspects featured in the following two examples are important. ...
6:00 PM June 26, 2011 1. Find all real-valued functions
6:00 PM June 26, 2011 1. Find all real-valued functions

25.4 Sum-product sets
25.4 Sum-product sets

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5th Grade Math Power Standards at a Glance

... to divide unit fractions by whole numbers and whole numbers by unit fractions ...
The Fundamental Theorem of Algebra
The Fundamental Theorem of Algebra

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lecture5
lecture5

... • Primitive math operations (i.e. +,-,*, / on ints, doubles, etc) • Accessing simple variables (and data members) • Accessing an array, A[i] ...
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Lecture 3

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PDF

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More Revision for tests

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Non-standard Simplex Problems

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Leadership_Lesson_Pl..

(˜P ∨ ˜Q) are tautologically equivalent by constructing a truth
(˜P ∨ ˜Q) are tautologically equivalent by constructing a truth

Number Theory * Introduction (1/22)
Number Theory * Introduction (1/22)

Chapter 11: The Non-Denumerability of the Continuum
Chapter 11: The Non-Denumerability of the Continuum

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Scientific Notation PowerPoint

Exercises for Unit I V (The basic number systems of mathematics)
Exercises for Unit I V (The basic number systems of mathematics)

Full text - The Fibonacci Quarterly
Full text - The Fibonacci Quarterly

... E x a m p l e . Consider 9{x) = ( 3 * i ) . . . (3Yn)[(X + 1 .= Y?) A • • • A (X + n = Yn2)] so that 9(X) asserts that X - f l , X + 2 , . . . , X + n are perfect squares within the field. In this case one can take k = 2 with /xi = l / 2 n and / j 2 = 1. The first value, /ii, stands for the fields o ...
< 1 ... 58 59 60 61 62 63 64 65 >

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"".
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