SISTEMA DE NÚMEROS REALES (INTRODUCCIÓN)

Introduction to the Real Number System

Understanding Sets and Systems

• The speaker introduces the topic, clarifying that a set is a collection of elements, while a system includes both a set and rules for interaction among its elements.

• The necessity of studying the real number system is questioned, with three main reasons proposed for its importance.

Ideal Representation of Quantities

• The real number system is described as ideal for representing quantities derived from experiments and measurements, overcoming limitations found in natural numbers, integers, and rational numbers.

• Natural numbers (1, 2, 3...) cannot represent negative quantities; integers include negatives but lack rational comparisons. Rational numbers allow comparisons between magnitudes with non-zero denominators.

Mathematical Laboratory Concept

• The speaker likens the real number system to a mathematical laboratory where one can manipulate and observe behaviors of different objects within this unidimensional space.

• Examples are provided on how to locate points on a real line and perform operations like union, intersection, difference using geometric representations.

Building New Coordinate Systems

• The third reason emphasizes that the real number system enables the construction of new coordinate systems such as two-dimensional and three-dimensional spaces useful in various mathematical studies.

• These systems adhere to properties studied within the realm of real numbers and are essential in fields like vector spaces and linear algebra.

Defining Real Numbers

• A clear distinction is made between sets and systems; laws governing operations (addition & multiplication) define the structure of the real number system.

• The set of real numbers (denoted by R) results from combining rational and irrational numbers associated with addition and multiplication operations.

Axioms Governing Real Numbers

Properties of Operations

• Formal definition: The real number system consists of R along with addition (+), multiplication (·), and an order relation (<).

• Key axioms include commutative properties for addition (a + b = b + a) and multiplication (a · b = b · a).

Associative Properties

• Associative properties state that grouping does not affect outcomes:

• For addition: a + (b + c) = (a + b) + c

• For multiplication: a cdot (b cdot c)= (a cdot b)cdot c

Distributive Property

• Distributive property combines both operations: a cdot(b + c)= a cdot b + a cdot c.

Existence of Identity Elements

• There exist additive identity (0): a + 0 = a, multiplicative identity (1): a · 1 = a.

Inverses in Real Numbers

• Each element has an additive inverse (-a); however, only non-zero reals have multiplicative inverses (1/a) ensuring product equals identity.

Understanding Real Numbers and Their Axioms

Positive and Negative Real Numbers

• The discussion begins with the classification of real numbers into positive and negative, defining positive reals as those greater than 0 and negative reals as those less than 0.

• Axiom 7 states that the sum of two positive real numbers will also be a positive real number, reinforcing the closure property under addition for this subset.

• Similarly, multiplying two positive real numbers results in another positive real number, highlighting the multiplicative closure property.

Properties of Real Numbers

• Axiom 8 emphasizes that any non-zero real number is either a positive or a negative real number, ensuring that each element can only take one value within this classification.

• Axiom 9 addresses order properties by stating that zero does not belong to the set of positive reals, which is fundamental for establishing further properties of the real number system.

Completeness and Supremum

• The completeness axiom guarantees that every non-empty set of upper-bounded real numbers has a least upper bound (supremum), ensuring no elements are missing from the set.

• This axiom implies that when subsets are formed from real numbers, they will always contain their supremum, thus maintaining integrity within the structure of real numbers.

Summary of Real Number System

• In summary, the system of real numbers is defined by four components: addition (+), multiplication (·), order (≤), and it adheres to axioms related to field properties, order properties, and completeness.