Lecture

Important Property of Archimedean Bodies

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Description

This lecture discusses an important property of Archimedean bodies, specifically focusing on the Archimedean property. It covers the proposition that for all x in a set, there exists a positive real number n such that n*x is greater than y. The demonstration involves constructing a logical argument to prove the proposition true.

In MOOCs (2)

Analysis I

This course covers : Part 1: Prelude, basic concepts, real numbers. Part 2: Introduction to complex numbers. Part 3: Real number sequences I and II. Part 4: Limit of a function, continuous functions. Part 5: Continuous functions and derivable functions, the derivative function. Part 6: Studies of functions, limited developments. Part 7: Indefinite and definite integrals, integration (selected chapters).

Analysis I (part 1): Prelude, basic concepts, real numbers

Can you calculate the value of sin(π/4)? Does the identity ln(xy)=ln(x)+ln(y) sound familiar? In the prelude we recall and expand on a few such notions, in particular the trigonometric functions sin, cos and tan and their properties, the concept of reciprocal functions such as exp and ln, some calculation rules concerning powers, logarithms and roots, sets and functions. Next we discuss number systems: based on the "intuitive" notion of the natural integers N={0,1,2,3,...}, we rigorously define the rational

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Related concepts (22)

Real number

In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature. Here, continuous means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives.

Construction of the real numbers

In mathematics, there are several equivalent ways of defining the real numbers. One of them is that they form a complete ordered field that does not contain any smaller complete ordered field. Such a definition does not prove that such a complete ordered field exists, and the existence proof consists of constructing a mathematical structure that satisfies the definition. The article presents several such constructions. They are equivalent in the sense that, given the result of any two such constructions, there is a unique isomorphism of ordered field between them.

Argument

An argument is a series of sentences, statements or propositions some of which are called premises and one is the conclusion. The purpose of an argument is to give reasons for one's conclusion via justification, explanation, and/or persuasion. Arguments are intended to determine or show the degree of truth or acceptability of another statement called a conclusion. Arguments can be studied from three main perspectives: the logical, the dialectical and the rhetorical perspective.

Logical form

In logic, logical form of a statement is a precisely-specified semantic version of that statement in a formal system. Informally, the logical form attempts to formalize a possibly ambiguous statement into a statement with a precise, unambiguous logical interpretation with respect to a formal system. In an ideal formal language, the meaning of a logical form can be determined unambiguously from syntax alone. Logical forms are semantic, not syntactic constructs; therefore, there may be more than one string that represents the same logical form in a given language.

Logical consequence

Logical consequence (also entailment) is a fundamental concept in logic which describes the relationship between statements that hold true when one statement logically follows from one or more statements. A valid logical argument is one in which the conclusion is entailed by the premises, because the conclusion is the consequence of the premises.

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