Lecture

Absolute Value: Definition and Properties

Graph Chat

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.

Description

This lecture covers the definition and basic properties of the absolute value function for real numbers. It explains how to calculate the absolute value of a number, the basic properties such as the absolute value of a product and the triangle inequality.

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

Official source

Quiz

Related concepts (28)

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.

Continuous function

In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is .

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.

Definable real number

Informally, a definable real number is a real number that can be uniquely specified by its description. The description may be expressed as a construction or as a formula of a formal language. For example, the positive square root of 2, , can be defined as the unique positive solution to the equation , and it can be constructed with a compass and straightedge. Different choices of a formal language or its interpretation give rise to different notions of definability.

Extended real number line

In mathematics, the affinely extended real number system is obtained from the real number system by adding two infinity elements: and where the infinities are treated as actual numbers. It is useful in describing the algebra on infinities and the various limiting behaviors in calculus and mathematical analysis, especially in the theory of measure and integration. The affinely extended real number system is denoted or or It is the Dedekind–MacNeille completion of the real numbers.

Related lectures (64)

Real Numbers Sequences

Explores real number sequences, convergence, limits, boundedness, and monotonicity.

Continuity Definition: Epsilon and Delta MOOC: Analysis I (part 4) : Limit of a function, continuous functions MOOC: Analysis I

Covers the definition of continuity at a point using epsilon and delta.

Real Numbers: Sets and Operations

Covers the basics of real numbers and set theory, including subsets, intersections, unions, and set operations.

Analyse I: Summary

Covers real numbers, complex numbers, numerical sequences, series, real functions, function limits, derivatives, Taylor series, integrals, and growth rates of functions.

Limits and Exponential Functions