Real Numbers

In: Business and Management

Submitted By cbindustrial
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Números reales y aritmética de ordenador



El conjunto de los números reales

El conjunto de los números reales

El conjunto de los números reales
1.1 El conjunto de los números reales 5 1.2 Naturales, enteros, racionales e irracionales 8 1.3 Valor absoluto 9 1.4 El principio de inducción 10 1.5 Intervalos y conjuntos destacados 13 1.6 Ejercicios 14

Existen diferentes formas de formalizar el conjunto de los números reales aunque se pueden agrupar en dos variantes: constructivos y axiomáticos. Los primeros son demasiado laboriosos para un curso de Cálculo y, por este motivo, hemos preferido dejarlos de lado. En su lugar, hemos asumido que el conjunto de los números reales es conocido por el lector y elegimos la definición axiomática de este conjunto.

1.1 El conjunto de los números reales
Vamos a definir el conjunto de los números reales, R, en términos de qué sabemos hacer con sus elementos, qué propiedades tienen. Estas propiedades que vamos a presentar aquí se llaman axiomas y, por supuesto, no son todas las propiedades de los números reales sino las mínimas, y es que a partir de ellas se obtienen el resto de propiedades.
Es difícil que, si alguien nos pregunta, seamos capaces de dar una respuesta clara de qué es un número pero sí somos capaces de decir qué cosas podemos hacer con ellos.
En el conjunto de los números reales tenemos definidas varias operaciones. La primera que todos aprendemos es la suma.

Suma de números reales
Las suma verifica, entre otras, las siguientes propiedades. Sean a, b y c números reales cualesquiera.
a) Propiedad asociativa: a + (b + c) = (a + b) + c.
b) Propiedad conmutativa: a + b = b + a.
c) Existencia de elemento neutro: a + 0 = a.
d) Existencia de elemento opuesto: a + (−a) = 0.
Estas cuatro propiedades se resumen diciendo que (R, +) es un grupo abeliano o conmutativo.

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