GRADE 7 MATH LEARNING GUIDE LESSON 12: SUBSETS OF
... 3. What do you call the subset of real numbers that includes negative numbers (that came from the concept of “opposites” and specifically used in describing debt or below zero temperature) and is united with the whole numbers? Give examples. Expected Answer: Integers A third subset is the integers. ...
... 3. What do you call the subset of real numbers that includes negative numbers (that came from the concept of “opposites” and specifically used in describing debt or below zero temperature) and is united with the whole numbers? Give examples. Expected Answer: Integers A third subset is the integers. ...
real analysis - Atlantic International University
... A function is monotone if it only increases or only decreases, f increases monotonically (is monotonic increasing) if x y , implies f x that = f y A function f decreases monotonically (is monotonic decreasing) if x y A function f is strictly monotonically increasing is x y , implies tha ...
... A function is monotone if it only increases or only decreases, f increases monotonically (is monotonic increasing) if x y , implies f x that = f y A function f decreases monotonically (is monotonic decreasing) if x y A function f is strictly monotonically increasing is x y , implies tha ...
Transcendental values of the digamma function
... Let us remark that the qth cyclotomic polynomial Φq (x) is irreducible over an algebraic number field K if and only if K ∩ Q(ζq ) = Q, where ζq denotes a primitive qth root of unity. Indeed, the extension K(ζq )/K is Galois with group isomorphic to the subgroup of Gal(Q(ζq )/Q) having K ∩ Q(ζq ) as ...
... Let us remark that the qth cyclotomic polynomial Φq (x) is irreducible over an algebraic number field K if and only if K ∩ Q(ζq ) = Q, where ζq denotes a primitive qth root of unity. Indeed, the extension K(ζq )/K is Galois with group isomorphic to the subgroup of Gal(Q(ζq )/Q) having K ∩ Q(ζq ) as ...
4.2 Euclid`s Classification of Pythagorean Triples
... A triple of positive integers (x, y, z) is called a Pythagorean triple if the integers satisfy the equation x2 + y 2 = z 2 . Such a triple is called primitive if x, y, z have no common factor. For instance, (3, 4, 5) and (5, 12, 13) are primitive triples, whereas (6, 8, 10) is not primitive, but is ...
... A triple of positive integers (x, y, z) is called a Pythagorean triple if the integers satisfy the equation x2 + y 2 = z 2 . Such a triple is called primitive if x, y, z have no common factor. For instance, (3, 4, 5) and (5, 12, 13) are primitive triples, whereas (6, 8, 10) is not primitive, but is ...
![arXiv:math/0511682v1 [math.NT] 28 Nov 2005](http://s1.studyres.com/store/data/014696627_1-8def914a5ac3ed74bde3727e1309931c-300x300.png)
Arithmetic and Geometric Sequences
... difference, and let b1 , b2 , . . . , bn , . . . be a geometric sequence with a positive common ratio. Prove that there exists numbers x and y such that bn = xy an for all n ≥ 1. (So, the arithmetic and geometric progressions are related after all!). 24. Two trains are 100 miles apart on the same tr ...
... difference, and let b1 , b2 , . . . , bn , . . . be a geometric sequence with a positive common ratio. Prove that there exists numbers x and y such that bn = xy an for all n ≥ 1. (So, the arithmetic and geometric progressions are related after all!). 24. Two trains are 100 miles apart on the same tr ...
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"".